COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONSVol 29 Nos 5 amp 6 pp 903ndash923 2004

Local Existence for the Dumbbell Modelof Polymeric Fluids

Tiejun Li Hui Zhang and Pingwen Zhang

LMAM and School of Mathematical SciencesPeking University Beijing PR China

ABSTRACT

A local existence and uniqueness theorem is proved for a micro-macro modelfor polymeric fluid as well as the property of the solution The polymer stresstensor is given by an integral which involves the solution of a diffusion equationthe coefficient of this diffusion equation depends on the gradient of the solutionof the NavierndashStokes equation

Key Words Dumbbell model NavierndashStokes equation FokkerndashPlank equation

AMS Subject Classification 76A05 76D03 35Q35

1 INTRODUCTION

In this paper we prove the well-posedness of coupled kineticndashhydrodynamicmodels for polymeric fluids These models differ from traditional hydrodynamicmodels by taking explicitly into account the micromechanical structure of thepolymers The simplest class of micromechanical models which is capable ofreproducing some aspects of polymeric flow behavior is the class of dumbbell

lowastCorrespondence Hui Zhang LMAM and School of Mathematical Sciences PekingUniversity Yiheyun Road 5 Haidian District Beijing 100871 PR China Fax 86-10-62767146 E-mail huizhangmathpkueducn

903

DOI 101081PDE-120037336 0360-5302 (Print) 1532-4133 (Online)Copyright copy 2004 by Marcel Dekker Inc wwwdekkercom

ORDER REPRINTS

904 Li Zhang and Zhang

models In these models each polymer molecule is represented as two beads joinedtogether by elastic spring or rigid rod For dilute polymer solutions elastic dumbbellmodels have been exclusively used for complex flow simulations The configurationof the spring then specifies the conformation of the polymer

An incompressible fluid is subject to the following system of equations

ut + u middot u + p = u + middot for x isin (1)

middot u = 0 for x isin (2)

u = 0 for x isin = (3)

where is a bounded domain in 3 with smooth boundary Here p and u denotethe pressure and the velocity field respectively All of the coefficients are normalizedto one in the equations In contrast to traditional models of complex fluids whichexpress polymer stress using empirical constitutive relations expresses thepolymer stress in terms of the microscopic conformations of the polymers

= gQotimesQ =int3

gQotimesQdQ (4)

where obeys the FokkerndashPlanck equation

t + u middot = minusQ middot middotQminus gQ+Q13 (5)

Here we use subscript Q to denote the derivatives with respect to Q without thesubscript the differentiation is understood to be in x Q is the independent variablein the configuration space of the dumbbell and (depending on xQ and t) isconfiguration distribution function = uT gQ is a vector function whichdenotes the spring force between two beads We refer the reader to Bird et al (1987)Doi and Edwards (1986) and Risken (1984) for the details

An elastic dumbbell model has six configuration degrees of freedom Two typesof elastic dumbbells are important in flow modeling The first is the Hookeandumbbell In this model the connector is a Hookean spring and the connector forcegQ is given by gQ = Q In this special case we get from (1)ndash(5) a reduced systemof equations for u and

ut + u middot u + p = u + middot middot u = 0 (6)

t + u middot minus uT minus u + minus I = 013 (7)

In this way one eliminates Q as an independent variable This is the well-knownOldroyd-B model Its well-posedness has been studied by Guillopeacute and Saut (1990)Lions and Masmoudi (2000) and its numerical simulation has been completed byFeigl et al (1995) and Laso and Oumlttinger (1993) However their methods do notseem to extend to more general cases when closed systems of equations such as(6)ndash(7) are not available The second type of model is done by using the force law

gQ = Q1minus QQ0

2

where Q the length of the connector is not allowed to exceed some fixed Q0 Thisis known as the FENE model for Finitely Extensible Non-linear Elastic dumbbell

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Dumbbell Model of Polymeric Fluids 905

In general the spring force law should be nonlinear vector function Assumethe nonlinear force

gQ = fQ2Q (8)

where f satisfies the following condition

(G) The function f is C-smooth from 0 to 0 and there existnumber ge 0 and k gt 0 such that limQrarr fQ2Q = k Moreover

lim supQrarr

f primeQ2Qminus2 le C

and higher derivatives of f have at most polynomial growth as Q rarr

The recent resurgence of interest on (1)ndash(5) comes from chemical engineerswho are interested in designing stochastic modeling techniques for polymeric fluidsOne of the most popular approach uses the so-called Brownian configurationalfields (BCF) which is a stochastic field variable Qx t that describes the localconformation of the polymers Hulsen et al (1997) For the simplest dumbbellmodel Q satisfies

Qt + u middot Q = uTQminus gQ+ vt (9)

where vt is the temporal Gaussian white noise Equation (6) is the FokkerndashPlanckequation for (9) For smooth solutions the systems (1)ndash(5) and (1) (4) (9) areequivalent For one-dimensional shear flows the convergence numerical analysiswas carried out in Li and Zhang (2002) and Jourdain et al (2002) for linear springforce and well-posedness in Jourdain et al (2004) for nonlinear spring force usingthe specific structure of the shear flow system To extend such analysis to highdimension a crucial step (Li and Zhang 2003) has been done to analyze the localwell-posedness of the coupled system (1) (4) (9) with periodic boundary condition

A local existence and uniqueness theorem for solutions of Euler equationcoupled with kinetic theory of polymeric fluid was proved by Renardy (1990 1991)Energy methods were used to show the well-posedness of the Dirichlet initialboundary value problem for incompressible hypoelastic materials Since is aprobability density L1 spaces are natural for the Q dependence The frameworkis nice but many important details for the estimates are missed out Thesedetails are important for further analysis in particular numerical analysis of therecently proposed multiscale numerical methods In addition the commonly usedformulation of the dumbbell model (1)ndash(5) (Bird et al 1987 Doi and Edwards1986 Guillopeacute and Saut 1990 Jourdain et al 2002 Lions and Masmoudi 2000)is different from the one analyzed in Renardy (1991) where the model does notinclude a solvent viscosity Therefore it is of interest to supplement the work ofRenardy (1991) with a detailed well-posedness analysis for the (1)ndash(5) This is themain objective of the present paper

Suppose the system is supplied with the initial value

ux 0 = u0 xQ 0 = 0xQ13 (10)

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906 Li Zhang and Zhang

and define the space for the distribution function

n0 = 3 rarr

int1+ QnQdQ lt

(11)

moreover we let nk be the space of all whose derivatives with respect to Q upto order k lie in n0 Finally set

k =⋂n=0

nk (12)

with the natural topology of a Frechet space The space k consists of functionswhich together with their derivatives vanish at infinite order as Q rarr Next westate the main results

The main assumptions of the theorem are

(A1) The domain isin 3 is bounded and is of class C4(A2) u0 isin H4(A3) 0 isin H42 where H

kl stands for⋂

n=0 Hknl Moreover

0 ge 0 andint0xQdQ = 1 for every x isin

In addition we need compatibility conditions between the initial data andthe incompressibility and boundary conditions So we assume the followingcompatibility conditions

(C1) div u0 = 0 and u0 = 0 on (C2) u1 vanishes on

where u1 = utt=0

Theorem 1 Assume that (A1)ndash(A3) (C1)ndash(C2) and (G) hold then there exists T prime gt 0such that the problem (1)ndash(5) and (10) has a unique solution with the regularity

u isin2⋂

k=0

Hk0 T prime H4minus2k (13)

isin1⋂

k=0

Hk0 T prime H3minus2k (14)

isin1⋂

k=0

Hk0 T prime H3minus2k013 (15)

The paper is organized as follows In the Sec 2 we define iterative scheme ofthe system to obtain the existence of the solution The scheme alternates betweensolving an equation of the same type as encountered in incompressible elasticity andsolving a linear diffusion equation The Sec 3 is devoted to giving the detailed proofthe main lemmas where the estimates of with respect to Lagrangian variable arecompleted first then the corresponding estimates with respect to Eulerian variableare obtained We draw the conclusion in Sec 4

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Dumbbell Model of Polymeric Fluids 907

2 ITERATIVE CONSTRUCTION OF SOLUTION

In this section we construct an iterative scheme of the system (1)ndash(3) and (5)to obtain the existence of the solution Given an iterate um we determine um+1 bysolving the equations

um+1t + um middot um+1 + pm+1 = um+1 + middot m (16)

middot um+1 = 0 (17)

with the initial condition um+1x 0 = u0x and boundary condition um+1 = 0where

m =int3

gQotimesQm dQ13 (18)

Meanwhile for given um we determine m+1 from the initial value problem

m+1t + um middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (19)

m+1xQ 0 = 0xQ (20)

where m = umT Our eventual task is to show that the mapping um 13rarr um+1

is a contraction in an appropriate complete space of functions The fixed point ofthe mapping is the solution we seek

We will consider the mapping in the function space SM T with metricdmiddot middot showed next SM T is the set of all functions u times 0 T rarr 3 with thefollowing properties

u isin2⋂

k=0

Hk0 TH4minus2k (21)

u04 + u12 + u20 le M (22)

middot u = 0 (23)

u = 0 on (24)

ux 0 = u0 utx 0 = u1x13 (25)

Here middot kl denotes the norm in Hk0 THl The function u0 u1 lie in H4and H2 respectively On SM T we define the metric

du1 u2 = u1 minus u204 + u1 minus u212 + u1 minus u22013 (26)

Lemma 1 If M is chosen large enough SM T is not empty Moreover the metric dis complete on SM T

The proof will be given along the line of the argument of Lemma 1 by Renardy(1990) with slight modifications

ORDER REPRINTS

908 Li Zhang and Zhang

The contraction of the mapping is established by proving the next fivelemmas

Lemma 2 Assume that the bounds of the following kind hold

um04 + um12 + um20 le M (27)

m03 + m11 le K (28)

Then (16)ndash(18) has a solution

um+1 isin2⋂

k=0

Hk0 TH4minus2k (29)

and we have

um+104 + um+112 + um+120 le 1M TK (30)

where 1M TK may depend on the initial value u0 and u1 and it is bounded if theparameters M T and K are bounded

Lemma 3 Consider (16) and a second equation

vm+1t + vm middot vm+1 + qm+1 = vm+1 + middot m middot vm+1 = 0 (31)

and the initial and boundary conditions are

vm+1x 0 = u0x vm+1 = 0 on 13 (32)

Here we assume vm isin SM T vm = um m = m for t = 0 and the assumptions ofLemma 2 also hold for (31) (with the same constants MK) Then we have an estimateof the form

um+1 minus vm+104 + um+1 minus vm+112 + um+1 minus vm+120le 2M TK middot um minus vm04 + um minus vm12 + um minus vm20+m minus m03 + m minus m11

where 2M TK is similar as 1M TK in Lemma 2 moreoverlimTrarr0 2M TK = 0

Lemma 4 Given um isin SM T there exists a unique solution of (19)ndash(20) which hasthe regularity

m+1 isin1⋂

k=0

Hk0 TH3minus2k013 (33)

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Dumbbell Model of Polymeric Fluids 909

Let middot n be the norm in

1⋂k=0

Hk0 TH3minus2kn013 (34)

We have an estimate of the form

m+1n le K1nM T (35)

where K1 is similar to 1 in Lemma 2

In fact we obtain the regularity of m+1 is better than (33) in the proof ofLemma 4 But we only need the estimate of m+1 in the norm middot n so we onlyemphasize the estimate of (33)

Let us consider a second equation of the same form

m+1t + vm middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (36)

m+1xQ 0 = 0xQ13 (37)

Here m = vmT and we assume vmt=0 = u0 In addition to (19) the followingresult holds

Lemma 5 Let um vm isin SM T be given Then for every n we have an estimate ofthe form

m+1 minus m+1n le K2nM Tdum vm (38)

where K2 is similar to 2 in Lemma 3

Lemma 6

m isin1⋂

k=0

Hk0 TH3minus2k (39)

and

m minus m03 + m minus m11 le K2 + 2M Tdumminus1 vmminus113

provided that m m isin ⋂1k=0 H

k0 TH3minus2k0 and m = int3 gQotimesQm dQ

Here is defined in the condition (G)

By combining Lemmas 2ndash6 it follows easily that is a contraction in SM Tif M is chosen sufficiently large and T is chosen sufficiently small then Theorem 1follows immediately Next we will be concerned with the proofs of lemmas

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910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

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Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

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912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

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Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

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914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

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916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

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Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

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918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

904 Li Zhang and Zhang

models In these models each polymer molecule is represented as two beads joinedtogether by elastic spring or rigid rod For dilute polymer solutions elastic dumbbellmodels have been exclusively used for complex flow simulations The configurationof the spring then specifies the conformation of the polymer

An incompressible fluid is subject to the following system of equations

ut + u middot u + p = u + middot for x isin (1)

middot u = 0 for x isin (2)

u = 0 for x isin = (3)

where is a bounded domain in 3 with smooth boundary Here p and u denotethe pressure and the velocity field respectively All of the coefficients are normalizedto one in the equations In contrast to traditional models of complex fluids whichexpress polymer stress using empirical constitutive relations expresses thepolymer stress in terms of the microscopic conformations of the polymers

= gQotimesQ =int3

gQotimesQdQ (4)

where obeys the FokkerndashPlanck equation

t + u middot = minusQ middot middotQminus gQ+Q13 (5)

Here we use subscript Q to denote the derivatives with respect to Q without thesubscript the differentiation is understood to be in x Q is the independent variablein the configuration space of the dumbbell and (depending on xQ and t) isconfiguration distribution function = uT gQ is a vector function whichdenotes the spring force between two beads We refer the reader to Bird et al (1987)Doi and Edwards (1986) and Risken (1984) for the details

An elastic dumbbell model has six configuration degrees of freedom Two typesof elastic dumbbells are important in flow modeling The first is the Hookeandumbbell In this model the connector is a Hookean spring and the connector forcegQ is given by gQ = Q In this special case we get from (1)ndash(5) a reduced systemof equations for u and

ut + u middot u + p = u + middot middot u = 0 (6)

t + u middot minus uT minus u + minus I = 013 (7)

In this way one eliminates Q as an independent variable This is the well-knownOldroyd-B model Its well-posedness has been studied by Guillopeacute and Saut (1990)Lions and Masmoudi (2000) and its numerical simulation has been completed byFeigl et al (1995) and Laso and Oumlttinger (1993) However their methods do notseem to extend to more general cases when closed systems of equations such as(6)ndash(7) are not available The second type of model is done by using the force law

gQ = Q1minus QQ0

2

where Q the length of the connector is not allowed to exceed some fixed Q0 Thisis known as the FENE model for Finitely Extensible Non-linear Elastic dumbbell

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 905

In general the spring force law should be nonlinear vector function Assumethe nonlinear force

gQ = fQ2Q (8)

where f satisfies the following condition

(G) The function f is C-smooth from 0 to 0 and there existnumber ge 0 and k gt 0 such that limQrarr fQ2Q = k Moreover

lim supQrarr

f primeQ2Qminus2 le C

and higher derivatives of f have at most polynomial growth as Q rarr

The recent resurgence of interest on (1)ndash(5) comes from chemical engineerswho are interested in designing stochastic modeling techniques for polymeric fluidsOne of the most popular approach uses the so-called Brownian configurationalfields (BCF) which is a stochastic field variable Qx t that describes the localconformation of the polymers Hulsen et al (1997) For the simplest dumbbellmodel Q satisfies

Qt + u middot Q = uTQminus gQ+ vt (9)

where vt is the temporal Gaussian white noise Equation (6) is the FokkerndashPlanckequation for (9) For smooth solutions the systems (1)ndash(5) and (1) (4) (9) areequivalent For one-dimensional shear flows the convergence numerical analysiswas carried out in Li and Zhang (2002) and Jourdain et al (2002) for linear springforce and well-posedness in Jourdain et al (2004) for nonlinear spring force usingthe specific structure of the shear flow system To extend such analysis to highdimension a crucial step (Li and Zhang 2003) has been done to analyze the localwell-posedness of the coupled system (1) (4) (9) with periodic boundary condition

A local existence and uniqueness theorem for solutions of Euler equationcoupled with kinetic theory of polymeric fluid was proved by Renardy (1990 1991)Energy methods were used to show the well-posedness of the Dirichlet initialboundary value problem for incompressible hypoelastic materials Since is aprobability density L1 spaces are natural for the Q dependence The frameworkis nice but many important details for the estimates are missed out Thesedetails are important for further analysis in particular numerical analysis of therecently proposed multiscale numerical methods In addition the commonly usedformulation of the dumbbell model (1)ndash(5) (Bird et al 1987 Doi and Edwards1986 Guillopeacute and Saut 1990 Jourdain et al 2002 Lions and Masmoudi 2000)is different from the one analyzed in Renardy (1991) where the model does notinclude a solvent viscosity Therefore it is of interest to supplement the work ofRenardy (1991) with a detailed well-posedness analysis for the (1)ndash(5) This is themain objective of the present paper

Suppose the system is supplied with the initial value

ux 0 = u0 xQ 0 = 0xQ13 (10)

ORDER REPRINTS

906 Li Zhang and Zhang

and define the space for the distribution function

n0 = 3 rarr

int1+ QnQdQ lt

(11)

moreover we let nk be the space of all whose derivatives with respect to Q upto order k lie in n0 Finally set

k =⋂n=0

nk (12)

with the natural topology of a Frechet space The space k consists of functionswhich together with their derivatives vanish at infinite order as Q rarr Next westate the main results

The main assumptions of the theorem are

(A1) The domain isin 3 is bounded and is of class C4(A2) u0 isin H4(A3) 0 isin H42 where H

kl stands for⋂

n=0 Hknl Moreover

0 ge 0 andint0xQdQ = 1 for every x isin

In addition we need compatibility conditions between the initial data andthe incompressibility and boundary conditions So we assume the followingcompatibility conditions

(C1) div u0 = 0 and u0 = 0 on (C2) u1 vanishes on

where u1 = utt=0

Theorem 1 Assume that (A1)ndash(A3) (C1)ndash(C2) and (G) hold then there exists T prime gt 0such that the problem (1)ndash(5) and (10) has a unique solution with the regularity

u isin2⋂

k=0

Hk0 T prime H4minus2k (13)

isin1⋂

k=0

Hk0 T prime H3minus2k (14)

isin1⋂

k=0

Hk0 T prime H3minus2k013 (15)

The paper is organized as follows In the Sec 2 we define iterative scheme ofthe system to obtain the existence of the solution The scheme alternates betweensolving an equation of the same type as encountered in incompressible elasticity andsolving a linear diffusion equation The Sec 3 is devoted to giving the detailed proofthe main lemmas where the estimates of with respect to Lagrangian variable arecompleted first then the corresponding estimates with respect to Eulerian variableare obtained We draw the conclusion in Sec 4

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 907

2 ITERATIVE CONSTRUCTION OF SOLUTION

In this section we construct an iterative scheme of the system (1)ndash(3) and (5)to obtain the existence of the solution Given an iterate um we determine um+1 bysolving the equations

um+1t + um middot um+1 + pm+1 = um+1 + middot m (16)

middot um+1 = 0 (17)

with the initial condition um+1x 0 = u0x and boundary condition um+1 = 0where

m =int3

gQotimesQm dQ13 (18)

Meanwhile for given um we determine m+1 from the initial value problem

m+1t + um middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (19)

m+1xQ 0 = 0xQ (20)

where m = umT Our eventual task is to show that the mapping um 13rarr um+1

is a contraction in an appropriate complete space of functions The fixed point ofthe mapping is the solution we seek

We will consider the mapping in the function space SM T with metricdmiddot middot showed next SM T is the set of all functions u times 0 T rarr 3 with thefollowing properties

u isin2⋂

k=0

Hk0 TH4minus2k (21)

u04 + u12 + u20 le M (22)

middot u = 0 (23)

u = 0 on (24)

ux 0 = u0 utx 0 = u1x13 (25)

Here middot kl denotes the norm in Hk0 THl The function u0 u1 lie in H4and H2 respectively On SM T we define the metric

du1 u2 = u1 minus u204 + u1 minus u212 + u1 minus u22013 (26)

Lemma 1 If M is chosen large enough SM T is not empty Moreover the metric dis complete on SM T

The proof will be given along the line of the argument of Lemma 1 by Renardy(1990) with slight modifications

ORDER REPRINTS

908 Li Zhang and Zhang

The contraction of the mapping is established by proving the next fivelemmas

Lemma 2 Assume that the bounds of the following kind hold

um04 + um12 + um20 le M (27)

m03 + m11 le K (28)

Then (16)ndash(18) has a solution

um+1 isin2⋂

k=0

Hk0 TH4minus2k (29)

and we have

um+104 + um+112 + um+120 le 1M TK (30)

where 1M TK may depend on the initial value u0 and u1 and it is bounded if theparameters M T and K are bounded

Lemma 3 Consider (16) and a second equation

vm+1t + vm middot vm+1 + qm+1 = vm+1 + middot m middot vm+1 = 0 (31)

and the initial and boundary conditions are

vm+1x 0 = u0x vm+1 = 0 on 13 (32)

Here we assume vm isin SM T vm = um m = m for t = 0 and the assumptions ofLemma 2 also hold for (31) (with the same constants MK) Then we have an estimateof the form

um+1 minus vm+104 + um+1 minus vm+112 + um+1 minus vm+120le 2M TK middot um minus vm04 + um minus vm12 + um minus vm20+m minus m03 + m minus m11

where 2M TK is similar as 1M TK in Lemma 2 moreoverlimTrarr0 2M TK = 0

Lemma 4 Given um isin SM T there exists a unique solution of (19)ndash(20) which hasthe regularity

m+1 isin1⋂

k=0

Hk0 TH3minus2k013 (33)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 909

Let middot n be the norm in

1⋂k=0

Hk0 TH3minus2kn013 (34)

We have an estimate of the form

m+1n le K1nM T (35)

where K1 is similar to 1 in Lemma 2

In fact we obtain the regularity of m+1 is better than (33) in the proof ofLemma 4 But we only need the estimate of m+1 in the norm middot n so we onlyemphasize the estimate of (33)

Let us consider a second equation of the same form

m+1t + vm middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (36)

m+1xQ 0 = 0xQ13 (37)

Here m = vmT and we assume vmt=0 = u0 In addition to (19) the followingresult holds

Lemma 5 Let um vm isin SM T be given Then for every n we have an estimate ofthe form

m+1 minus m+1n le K2nM Tdum vm (38)

where K2 is similar to 2 in Lemma 3

Lemma 6

m isin1⋂

k=0

Hk0 TH3minus2k (39)

and

m minus m03 + m minus m11 le K2 + 2M Tdumminus1 vmminus113

provided that m m isin ⋂1k=0 H

k0 TH3minus2k0 and m = int3 gQotimesQm dQ

Here is defined in the condition (G)

By combining Lemmas 2ndash6 it follows easily that is a contraction in SM Tif M is chosen sufficiently large and T is chosen sufficiently small then Theorem 1follows immediately Next we will be concerned with the proofs of lemmas

ORDER REPRINTS

910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 905

In general the spring force law should be nonlinear vector function Assumethe nonlinear force

gQ = fQ2Q (8)

where f satisfies the following condition

(G) The function f is C-smooth from 0 to 0 and there existnumber ge 0 and k gt 0 such that limQrarr fQ2Q = k Moreover

lim supQrarr

f primeQ2Qminus2 le C

and higher derivatives of f have at most polynomial growth as Q rarr

The recent resurgence of interest on (1)ndash(5) comes from chemical engineerswho are interested in designing stochastic modeling techniques for polymeric fluidsOne of the most popular approach uses the so-called Brownian configurationalfields (BCF) which is a stochastic field variable Qx t that describes the localconformation of the polymers Hulsen et al (1997) For the simplest dumbbellmodel Q satisfies

Qt + u middot Q = uTQminus gQ+ vt (9)

where vt is the temporal Gaussian white noise Equation (6) is the FokkerndashPlanckequation for (9) For smooth solutions the systems (1)ndash(5) and (1) (4) (9) areequivalent For one-dimensional shear flows the convergence numerical analysiswas carried out in Li and Zhang (2002) and Jourdain et al (2002) for linear springforce and well-posedness in Jourdain et al (2004) for nonlinear spring force usingthe specific structure of the shear flow system To extend such analysis to highdimension a crucial step (Li and Zhang 2003) has been done to analyze the localwell-posedness of the coupled system (1) (4) (9) with periodic boundary condition

A local existence and uniqueness theorem for solutions of Euler equationcoupled with kinetic theory of polymeric fluid was proved by Renardy (1990 1991)Energy methods were used to show the well-posedness of the Dirichlet initialboundary value problem for incompressible hypoelastic materials Since is aprobability density L1 spaces are natural for the Q dependence The frameworkis nice but many important details for the estimates are missed out Thesedetails are important for further analysis in particular numerical analysis of therecently proposed multiscale numerical methods In addition the commonly usedformulation of the dumbbell model (1)ndash(5) (Bird et al 1987 Doi and Edwards1986 Guillopeacute and Saut 1990 Jourdain et al 2002 Lions and Masmoudi 2000)is different from the one analyzed in Renardy (1991) where the model does notinclude a solvent viscosity Therefore it is of interest to supplement the work ofRenardy (1991) with a detailed well-posedness analysis for the (1)ndash(5) This is themain objective of the present paper

Suppose the system is supplied with the initial value

ux 0 = u0 xQ 0 = 0xQ13 (10)

ORDER REPRINTS

906 Li Zhang and Zhang

and define the space for the distribution function

n0 = 3 rarr

int1+ QnQdQ lt

(11)

moreover we let nk be the space of all whose derivatives with respect to Q upto order k lie in n0 Finally set

k =⋂n=0

nk (12)

with the natural topology of a Frechet space The space k consists of functionswhich together with their derivatives vanish at infinite order as Q rarr Next westate the main results

The main assumptions of the theorem are

(A1) The domain isin 3 is bounded and is of class C4(A2) u0 isin H4(A3) 0 isin H42 where H

kl stands for⋂

n=0 Hknl Moreover

0 ge 0 andint0xQdQ = 1 for every x isin

In addition we need compatibility conditions between the initial data andthe incompressibility and boundary conditions So we assume the followingcompatibility conditions

(C1) div u0 = 0 and u0 = 0 on (C2) u1 vanishes on

where u1 = utt=0

Theorem 1 Assume that (A1)ndash(A3) (C1)ndash(C2) and (G) hold then there exists T prime gt 0such that the problem (1)ndash(5) and (10) has a unique solution with the regularity

u isin2⋂

k=0

Hk0 T prime H4minus2k (13)

isin1⋂

k=0

Hk0 T prime H3minus2k (14)

isin1⋂

k=0

Hk0 T prime H3minus2k013 (15)

The paper is organized as follows In the Sec 2 we define iterative scheme ofthe system to obtain the existence of the solution The scheme alternates betweensolving an equation of the same type as encountered in incompressible elasticity andsolving a linear diffusion equation The Sec 3 is devoted to giving the detailed proofthe main lemmas where the estimates of with respect to Lagrangian variable arecompleted first then the corresponding estimates with respect to Eulerian variableare obtained We draw the conclusion in Sec 4

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 907

2 ITERATIVE CONSTRUCTION OF SOLUTION

In this section we construct an iterative scheme of the system (1)ndash(3) and (5)to obtain the existence of the solution Given an iterate um we determine um+1 bysolving the equations

um+1t + um middot um+1 + pm+1 = um+1 + middot m (16)

middot um+1 = 0 (17)

with the initial condition um+1x 0 = u0x and boundary condition um+1 = 0where

m =int3

gQotimesQm dQ13 (18)

Meanwhile for given um we determine m+1 from the initial value problem

m+1t + um middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (19)

m+1xQ 0 = 0xQ (20)

where m = umT Our eventual task is to show that the mapping um 13rarr um+1

is a contraction in an appropriate complete space of functions The fixed point ofthe mapping is the solution we seek

We will consider the mapping in the function space SM T with metricdmiddot middot showed next SM T is the set of all functions u times 0 T rarr 3 with thefollowing properties

u isin2⋂

k=0

Hk0 TH4minus2k (21)

u04 + u12 + u20 le M (22)

middot u = 0 (23)

u = 0 on (24)

ux 0 = u0 utx 0 = u1x13 (25)

Here middot kl denotes the norm in Hk0 THl The function u0 u1 lie in H4and H2 respectively On SM T we define the metric

du1 u2 = u1 minus u204 + u1 minus u212 + u1 minus u22013 (26)

Lemma 1 If M is chosen large enough SM T is not empty Moreover the metric dis complete on SM T

The proof will be given along the line of the argument of Lemma 1 by Renardy(1990) with slight modifications

ORDER REPRINTS

908 Li Zhang and Zhang

The contraction of the mapping is established by proving the next fivelemmas

Lemma 2 Assume that the bounds of the following kind hold

um04 + um12 + um20 le M (27)

m03 + m11 le K (28)

Then (16)ndash(18) has a solution

um+1 isin2⋂

k=0

Hk0 TH4minus2k (29)

and we have

um+104 + um+112 + um+120 le 1M TK (30)

where 1M TK may depend on the initial value u0 and u1 and it is bounded if theparameters M T and K are bounded

Lemma 3 Consider (16) and a second equation

vm+1t + vm middot vm+1 + qm+1 = vm+1 + middot m middot vm+1 = 0 (31)

and the initial and boundary conditions are

vm+1x 0 = u0x vm+1 = 0 on 13 (32)

Here we assume vm isin SM T vm = um m = m for t = 0 and the assumptions ofLemma 2 also hold for (31) (with the same constants MK) Then we have an estimateof the form

um+1 minus vm+104 + um+1 minus vm+112 + um+1 minus vm+120le 2M TK middot um minus vm04 + um minus vm12 + um minus vm20+m minus m03 + m minus m11

where 2M TK is similar as 1M TK in Lemma 2 moreoverlimTrarr0 2M TK = 0

Lemma 4 Given um isin SM T there exists a unique solution of (19)ndash(20) which hasthe regularity

m+1 isin1⋂

k=0

Hk0 TH3minus2k013 (33)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 909

Let middot n be the norm in

1⋂k=0

Hk0 TH3minus2kn013 (34)

We have an estimate of the form

m+1n le K1nM T (35)

where K1 is similar to 1 in Lemma 2

In fact we obtain the regularity of m+1 is better than (33) in the proof ofLemma 4 But we only need the estimate of m+1 in the norm middot n so we onlyemphasize the estimate of (33)

Let us consider a second equation of the same form

m+1t + vm middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (36)

m+1xQ 0 = 0xQ13 (37)

Here m = vmT and we assume vmt=0 = u0 In addition to (19) the followingresult holds

Lemma 5 Let um vm isin SM T be given Then for every n we have an estimate ofthe form

m+1 minus m+1n le K2nM Tdum vm (38)

where K2 is similar to 2 in Lemma 3

Lemma 6

m isin1⋂

k=0

Hk0 TH3minus2k (39)

and

m minus m03 + m minus m11 le K2 + 2M Tdumminus1 vmminus113

provided that m m isin ⋂1k=0 H

k0 TH3minus2k0 and m = int3 gQotimesQm dQ

Here is defined in the condition (G)

By combining Lemmas 2ndash6 it follows easily that is a contraction in SM Tif M is chosen sufficiently large and T is chosen sufficiently small then Theorem 1follows immediately Next we will be concerned with the proofs of lemmas

ORDER REPRINTS

910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

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ORDER REPRINTS

906 Li Zhang and Zhang

and define the space for the distribution function

n0 = 3 rarr

int1+ QnQdQ lt

(11)

moreover we let nk be the space of all whose derivatives with respect to Q upto order k lie in n0 Finally set

k =⋂n=0

nk (12)

with the natural topology of a Frechet space The space k consists of functionswhich together with their derivatives vanish at infinite order as Q rarr Next westate the main results

The main assumptions of the theorem are

(A1) The domain isin 3 is bounded and is of class C4(A2) u0 isin H4(A3) 0 isin H42 where H

kl stands for⋂

n=0 Hknl Moreover

0 ge 0 andint0xQdQ = 1 for every x isin

In addition we need compatibility conditions between the initial data andthe incompressibility and boundary conditions So we assume the followingcompatibility conditions

(C1) div u0 = 0 and u0 = 0 on (C2) u1 vanishes on

where u1 = utt=0

Theorem 1 Assume that (A1)ndash(A3) (C1)ndash(C2) and (G) hold then there exists T prime gt 0such that the problem (1)ndash(5) and (10) has a unique solution with the regularity

u isin2⋂

k=0

Hk0 T prime H4minus2k (13)

isin1⋂

k=0

Hk0 T prime H3minus2k (14)

isin1⋂

k=0

Hk0 T prime H3minus2k013 (15)

The paper is organized as follows In the Sec 2 we define iterative scheme ofthe system to obtain the existence of the solution The scheme alternates betweensolving an equation of the same type as encountered in incompressible elasticity andsolving a linear diffusion equation The Sec 3 is devoted to giving the detailed proofthe main lemmas where the estimates of with respect to Lagrangian variable arecompleted first then the corresponding estimates with respect to Eulerian variableare obtained We draw the conclusion in Sec 4

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 907

2 ITERATIVE CONSTRUCTION OF SOLUTION

In this section we construct an iterative scheme of the system (1)ndash(3) and (5)to obtain the existence of the solution Given an iterate um we determine um+1 bysolving the equations

um+1t + um middot um+1 + pm+1 = um+1 + middot m (16)

middot um+1 = 0 (17)

with the initial condition um+1x 0 = u0x and boundary condition um+1 = 0where

m =int3

gQotimesQm dQ13 (18)

Meanwhile for given um we determine m+1 from the initial value problem

m+1t + um middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (19)

m+1xQ 0 = 0xQ (20)

where m = umT Our eventual task is to show that the mapping um 13rarr um+1

is a contraction in an appropriate complete space of functions The fixed point ofthe mapping is the solution we seek

We will consider the mapping in the function space SM T with metricdmiddot middot showed next SM T is the set of all functions u times 0 T rarr 3 with thefollowing properties

u isin2⋂

k=0

Hk0 TH4minus2k (21)

u04 + u12 + u20 le M (22)

middot u = 0 (23)

u = 0 on (24)

ux 0 = u0 utx 0 = u1x13 (25)

Here middot kl denotes the norm in Hk0 THl The function u0 u1 lie in H4and H2 respectively On SM T we define the metric

du1 u2 = u1 minus u204 + u1 minus u212 + u1 minus u22013 (26)

Lemma 1 If M is chosen large enough SM T is not empty Moreover the metric dis complete on SM T

The proof will be given along the line of the argument of Lemma 1 by Renardy(1990) with slight modifications

ORDER REPRINTS

908 Li Zhang and Zhang

The contraction of the mapping is established by proving the next fivelemmas

Lemma 2 Assume that the bounds of the following kind hold

um04 + um12 + um20 le M (27)

m03 + m11 le K (28)

Then (16)ndash(18) has a solution

um+1 isin2⋂

k=0

Hk0 TH4minus2k (29)

and we have

um+104 + um+112 + um+120 le 1M TK (30)

where 1M TK may depend on the initial value u0 and u1 and it is bounded if theparameters M T and K are bounded

Lemma 3 Consider (16) and a second equation

vm+1t + vm middot vm+1 + qm+1 = vm+1 + middot m middot vm+1 = 0 (31)

and the initial and boundary conditions are

vm+1x 0 = u0x vm+1 = 0 on 13 (32)

Here we assume vm isin SM T vm = um m = m for t = 0 and the assumptions ofLemma 2 also hold for (31) (with the same constants MK) Then we have an estimateof the form

um+1 minus vm+104 + um+1 minus vm+112 + um+1 minus vm+120le 2M TK middot um minus vm04 + um minus vm12 + um minus vm20+m minus m03 + m minus m11

where 2M TK is similar as 1M TK in Lemma 2 moreoverlimTrarr0 2M TK = 0

Lemma 4 Given um isin SM T there exists a unique solution of (19)ndash(20) which hasthe regularity

m+1 isin1⋂

k=0

Hk0 TH3minus2k013 (33)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 909

Let middot n be the norm in

1⋂k=0

Hk0 TH3minus2kn013 (34)

We have an estimate of the form

m+1n le K1nM T (35)

where K1 is similar to 1 in Lemma 2

In fact we obtain the regularity of m+1 is better than (33) in the proof ofLemma 4 But we only need the estimate of m+1 in the norm middot n so we onlyemphasize the estimate of (33)

Let us consider a second equation of the same form

m+1t + vm middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (36)

m+1xQ 0 = 0xQ13 (37)

Here m = vmT and we assume vmt=0 = u0 In addition to (19) the followingresult holds

Lemma 5 Let um vm isin SM T be given Then for every n we have an estimate ofthe form

m+1 minus m+1n le K2nM Tdum vm (38)

where K2 is similar to 2 in Lemma 3

Lemma 6

m isin1⋂

k=0

Hk0 TH3minus2k (39)

and

m minus m03 + m minus m11 le K2 + 2M Tdumminus1 vmminus113

provided that m m isin ⋂1k=0 H

k0 TH3minus2k0 and m = int3 gQotimesQm dQ

Here is defined in the condition (G)

By combining Lemmas 2ndash6 it follows easily that is a contraction in SM Tif M is chosen sufficiently large and T is chosen sufficiently small then Theorem 1follows immediately Next we will be concerned with the proofs of lemmas

ORDER REPRINTS

910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 907

2 ITERATIVE CONSTRUCTION OF SOLUTION

In this section we construct an iterative scheme of the system (1)ndash(3) and (5)to obtain the existence of the solution Given an iterate um we determine um+1 bysolving the equations

um+1t + um middot um+1 + pm+1 = um+1 + middot m (16)

middot um+1 = 0 (17)

with the initial condition um+1x 0 = u0x and boundary condition um+1 = 0where

m =int3

gQotimesQm dQ13 (18)

Meanwhile for given um we determine m+1 from the initial value problem

m+1t + um middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (19)

m+1xQ 0 = 0xQ (20)

where m = umT Our eventual task is to show that the mapping um 13rarr um+1

is a contraction in an appropriate complete space of functions The fixed point ofthe mapping is the solution we seek

We will consider the mapping in the function space SM T with metricdmiddot middot showed next SM T is the set of all functions u times 0 T rarr 3 with thefollowing properties

u isin2⋂

k=0

Hk0 TH4minus2k (21)

u04 + u12 + u20 le M (22)

middot u = 0 (23)

u = 0 on (24)

ux 0 = u0 utx 0 = u1x13 (25)

Here middot kl denotes the norm in Hk0 THl The function u0 u1 lie in H4and H2 respectively On SM T we define the metric

du1 u2 = u1 minus u204 + u1 minus u212 + u1 minus u22013 (26)

Lemma 1 If M is chosen large enough SM T is not empty Moreover the metric dis complete on SM T

The proof will be given along the line of the argument of Lemma 1 by Renardy(1990) with slight modifications

ORDER REPRINTS

908 Li Zhang and Zhang

The contraction of the mapping is established by proving the next fivelemmas

Lemma 2 Assume that the bounds of the following kind hold

um04 + um12 + um20 le M (27)

m03 + m11 le K (28)

Then (16)ndash(18) has a solution

um+1 isin2⋂

k=0

Hk0 TH4minus2k (29)

and we have

um+104 + um+112 + um+120 le 1M TK (30)

where 1M TK may depend on the initial value u0 and u1 and it is bounded if theparameters M T and K are bounded

Lemma 3 Consider (16) and a second equation

vm+1t + vm middot vm+1 + qm+1 = vm+1 + middot m middot vm+1 = 0 (31)

and the initial and boundary conditions are

vm+1x 0 = u0x vm+1 = 0 on 13 (32)

Here we assume vm isin SM T vm = um m = m for t = 0 and the assumptions ofLemma 2 also hold for (31) (with the same constants MK) Then we have an estimateof the form

um+1 minus vm+104 + um+1 minus vm+112 + um+1 minus vm+120le 2M TK middot um minus vm04 + um minus vm12 + um minus vm20+m minus m03 + m minus m11

where 2M TK is similar as 1M TK in Lemma 2 moreoverlimTrarr0 2M TK = 0

Lemma 4 Given um isin SM T there exists a unique solution of (19)ndash(20) which hasthe regularity

m+1 isin1⋂

k=0

Hk0 TH3minus2k013 (33)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 909

Let middot n be the norm in

1⋂k=0

Hk0 TH3minus2kn013 (34)

We have an estimate of the form

m+1n le K1nM T (35)

where K1 is similar to 1 in Lemma 2

In fact we obtain the regularity of m+1 is better than (33) in the proof ofLemma 4 But we only need the estimate of m+1 in the norm middot n so we onlyemphasize the estimate of (33)

Let us consider a second equation of the same form

m+1t + vm middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (36)

m+1xQ 0 = 0xQ13 (37)

Here m = vmT and we assume vmt=0 = u0 In addition to (19) the followingresult holds

Lemma 5 Let um vm isin SM T be given Then for every n we have an estimate ofthe form

m+1 minus m+1n le K2nM Tdum vm (38)

where K2 is similar to 2 in Lemma 3

Lemma 6

m isin1⋂

k=0

Hk0 TH3minus2k (39)

and

m minus m03 + m minus m11 le K2 + 2M Tdumminus1 vmminus113

provided that m m isin ⋂1k=0 H

k0 TH3minus2k0 and m = int3 gQotimesQm dQ

Here is defined in the condition (G)

By combining Lemmas 2ndash6 it follows easily that is a contraction in SM Tif M is chosen sufficiently large and T is chosen sufficiently small then Theorem 1follows immediately Next we will be concerned with the proofs of lemmas

ORDER REPRINTS

910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

908 Li Zhang and Zhang

The contraction of the mapping is established by proving the next fivelemmas

Lemma 2 Assume that the bounds of the following kind hold

um04 + um12 + um20 le M (27)

m03 + m11 le K (28)

Then (16)ndash(18) has a solution

um+1 isin2⋂

k=0

Hk0 TH4minus2k (29)

and we have

um+104 + um+112 + um+120 le 1M TK (30)

where 1M TK may depend on the initial value u0 and u1 and it is bounded if theparameters M T and K are bounded

Lemma 3 Consider (16) and a second equation

vm+1t + vm middot vm+1 + qm+1 = vm+1 + middot m middot vm+1 = 0 (31)

and the initial and boundary conditions are

vm+1x 0 = u0x vm+1 = 0 on 13 (32)

Here we assume vm isin SM T vm = um m = m for t = 0 and the assumptions ofLemma 2 also hold for (31) (with the same constants MK) Then we have an estimateof the form

um+1 minus vm+104 + um+1 minus vm+112 + um+1 minus vm+120le 2M TK middot um minus vm04 + um minus vm12 + um minus vm20+m minus m03 + m minus m11

where 2M TK is similar as 1M TK in Lemma 2 moreoverlimTrarr0 2M TK = 0

Lemma 4 Given um isin SM T there exists a unique solution of (19)ndash(20) which hasthe regularity

m+1 isin1⋂

k=0

Hk0 TH3minus2k013 (33)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 909

Let middot n be the norm in

1⋂k=0

Hk0 TH3minus2kn013 (34)

We have an estimate of the form

m+1n le K1nM T (35)

where K1 is similar to 1 in Lemma 2

In fact we obtain the regularity of m+1 is better than (33) in the proof ofLemma 4 But we only need the estimate of m+1 in the norm middot n so we onlyemphasize the estimate of (33)

Let us consider a second equation of the same form

m+1t + vm middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (36)

m+1xQ 0 = 0xQ13 (37)

Here m = vmT and we assume vmt=0 = u0 In addition to (19) the followingresult holds

Lemma 5 Let um vm isin SM T be given Then for every n we have an estimate ofthe form

m+1 minus m+1n le K2nM Tdum vm (38)

where K2 is similar to 2 in Lemma 3

Lemma 6

m isin1⋂

k=0

Hk0 TH3minus2k (39)

and

m minus m03 + m minus m11 le K2 + 2M Tdumminus1 vmminus113

provided that m m isin ⋂1k=0 H

k0 TH3minus2k0 and m = int3 gQotimesQm dQ

Here is defined in the condition (G)

By combining Lemmas 2ndash6 it follows easily that is a contraction in SM Tif M is chosen sufficiently large and T is chosen sufficiently small then Theorem 1follows immediately Next we will be concerned with the proofs of lemmas

ORDER REPRINTS

910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 909

Let middot n be the norm in

1⋂k=0

Hk0 TH3minus2kn013 (34)

We have an estimate of the form

m+1n le K1nM T (35)

where K1 is similar to 1 in Lemma 2

In fact we obtain the regularity of m+1 is better than (33) in the proof ofLemma 4 But we only need the estimate of m+1 in the norm middot n so we onlyemphasize the estimate of (33)

Let us consider a second equation of the same form

m+1t + vm middot m+1 = minusQ middot m middotQminus gQm+1+Q

m+1 (36)

m+1xQ 0 = 0xQ13 (37)

Here m = vmT and we assume vmt=0 = u0 In addition to (19) the followingresult holds

Lemma 5 Let um vm isin SM T be given Then for every n we have an estimate ofthe form

m+1 minus m+1n le K2nM Tdum vm (38)

where K2 is similar to 2 in Lemma 3

Lemma 6

m isin1⋂

k=0

Hk0 TH3minus2k (39)

and

m minus m03 + m minus m11 le K2 + 2M Tdumminus1 vmminus113

provided that m m isin ⋂1k=0 H

k0 TH3minus2k0 and m = int3 gQotimesQm dQ

Here is defined in the condition (G)

By combining Lemmas 2ndash6 it follows easily that is a contraction in SM Tif M is chosen sufficiently large and T is chosen sufficiently small then Theorem 1follows immediately Next we will be concerned with the proofs of lemmas

ORDER REPRINTS

910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

910 Li Zhang and Zhang

3 PROOF OF LEMMAS

31 Estimates of u

In this subsection we will give the proof of Lemmas 2 and 3

Let um = v um+1 = w and q = pm+1 (16) can be rewritten as

wt + v middot w + q = w + f middot w = 0 (40)

w = 0 on wx 0 = u013 (41)

Here f = middot m Next setting z = wt then (40) will be transformed to

z = w minus v middot w minus q + f middot w = 013 (42)

After utilizing the operator t to (42) we obtain

zt = zminus v middot zminus + hw w ft (43)

middot z = 0 z = 0 on (44)

zx 0 = z0x = u113 (45)

Here hw w ft = minusvt middot w + ft and = qtIf we know z we can solve the Stokesrsquo problem (42) to obtain w Meanwhile

we will obtain the solution z by solving the parabolic problem (43)ndash(45) when w isgiven Next we will solve these two problems by using the Galerkin approximation

Let V be the space of all divergence-free vector in H10 and let ii isin be

a basis for V We seek an approximation to z of the form

zN x t =Nsum

n=1

ntnx13 (46)

For given zN let wN be the corresponding approximate solution of (42) Now wesolve the following approximate version of (43) For n = 1 2 13 13 13 N we require that

intn

zNt + v middot zN minuszN minus PNhwnwn ft

dx = 0 (47)

and we impose the initial conditions

zN x 0 = PNz0x13 (48)

Here PN is the orthogonal projection in V onto the span of 1 2 13 13 13 N Equations (47) and (48) is an initial value problem for a linear system of first-orderODErsquos and existence and uniqueness of the solutions are trivial

To obtain an estimate uniform in N we replace n by zN It is a standardprocedure to obtain the following estimate (Temam 1995)

zN t2L2 +int T

0zN s2L2 ds le PNz02L2 + C

int T

0h2L2 ds13 (49)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 911

This implies that

zN isin L0 T L2 cap L20 T V (50)

if h isin L20 T L2 which will be showed later Now we replace N by zNt toobtainint T

0zNt 2L2 ds + zN2L2 le C

int T

0h2L2 + zN0 2 (51)

which implies

zN isin L0 T V zNt isin L20 T L2 (52)

if h isin L20 T L2 From (50) and (52) it is obtained that the regularitythrough zN of Eq (43) (Evans 1998)

zN2H2 le C(zNt 2L2 + h2L2 + zN2L2

)13 (53)

Therefore

zN isin L0 T V cap L20 TH2 (54)

provided that h isin L20 T L2In order to show that h isin L20 T L2 the same Galerkin technique is

applied to wrsquos Eq (40) wN satisfies

wNt = wN minus v middot wN + PNf minus q middot wN = 0 (55)

wN = 0 on wNx 0 = u013 (56)

Similar as the estimate (53) we have

wN2H2 le C(wN

t 2L2 + f2L2 + wN2L2

)13 (57)

provided that f isin L20 T L2 and u0 isin H2 which implies that h isinL20 T L2 when f isin H10 T L2

Higher regularity of wN can be obtained similarly Recalling the definition off and the condition on we have f isin L20 TH2 capH10 T L2 andcorrespondingly we obtain

wN isin2⋂

k=0

Hk0 TH4minus2k capW 10 TH113 (58)

Passing N rarr we complete the proof of Lemma 2Proof of Lemma 3 is based on the same type of estimates Let Um+1 =

um+1 minus vm+1 sm+1 = pm+1 minus qm+1 m = m minus m Then Um+1 satisfies

Um+1t + vm middot Um+1 + sm+1 = Um+1 + middot m +Um middot vm+113

Then along the argument of Lemma 2 with slight modification we can obtainLemma 3 by studying the equation of Um+1 since vm+1 satisfies the result ofLemma 2

ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

912 Li Zhang and Zhang

32 Estimates of

Proof of Lemma 6 From the definition of it is straightforward to obtain theestimate of it from the assumption of It is easy to check that

2L20TL2 le Cint T

0

int

( int3

Q2fQdQ)2

dx dt13

From the condition (G) and isin L20 T L20 we obtain the estimate of inL20 T L2 By Lemma 5 and 6 we can obtain

m minus mL20T le Cm minus m+2

le K2 + 2M Tdumminus1 vmminus113

Higher derivatives can be obtained similarly

33 Estimates of with Respect to Lagrangian Variables

We will obtain the estimate of with respect to Lagrangian variables first thentranslate them to the Eulerian variables Consider the flow map

tx t = umx t t x 0 = (59)

where denotes the Lagrangian coordinates and define Q t = m+1

x tQ t then (5) can be rewritten in the form

tQ t = minusQ middot middotQminus gQ+Q (60)

Q 0 = 0Q13 (61)

In this subsection we still denote ux t tT by with the loss of confusion Itfollows from the maximum principle that positivity is preserved and by integratingboth sides of (60) we find thatint

3Q tdQ =

int3

Q 0dQ = 1 for all t13

Because the coefficients of Eq (60) are unbounded standard existence results forparabolic equations cannot be used Now we are strongly motivated by the workof Renardy (1991) to use a sequence of approximating problems with boundedcoefficients for which we derive uniform estimates

Let Q be a C-function such that 0 = 1 is a monotone decreasingfunction of Q and Q = Qminus for large Q where is a sufficiently largenumber For N isin let N Q = QN We now consider (60) the approximateproblem

tNQ t = minusQ middot N Q middotQminus gQN +QN (62)

NQ 0 = 0Q13 (63)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 913

The existence of solutions of the Cauchy problem (62) and (63) is standard resultssince the coefficients of (62) are bounded Next we will consider the estimates N

and its derivatives with respect to and Q In order for the convenience of thenotation we will denote

i1i2131313imj1j2131313jnN = mQi1

Qi2131313Qim

nj1 j2 131313jnN (64)

and specially we define

i1i2131313im0N = mQi1

Qi2131313Qim

N 0j1j2131313jnN = nj1 j2 131313jn

N 13 (65)

The summation convention is assumed in the following content

331 The Estimate of N

Multiplying (62) with Q2n and integrating by parts yields

t

int3

Q2nN dQ = 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nminus2N Q

times Q middot middotQminusQ middot gQN dQ13 (66)

Pay attention that Q middot gQ is nonnegative thus we obtain

t

int3

Q2nN dQ le 4n2 + 2nint3

Q2nminus2N dQ+ 2nint3

Q2nN dQ13

(67)

Application of the Gronwallrsquos inequality and u isin SM T yields

int3

Q2nN dQ le KnM Tint31+ Q2n0 dQ (68)

where KnM T is bounded if nM T is bounded so we have

N isin L0 T L0 if 0 isin H2013 (69)

332 The Estimate of mQN m = 1 2 3

Differentiating (62) with respect to Qi we obtain the following equationfor i0

N

t

i0N = minusQ middot [N Q middotQminus gQ

i0N

]+Qi0N +i (70)

i0N Q 0 = Qi

0Q (71)

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

914 Li Zhang and Zhang

where

i = minusQiN jk middotQk minus gjQ+ N Qji minus Qi

gjQj0N

minusQ middot QiN Q middotQminus gQN

= +i +minus

i 13 (72)

Here +i = i or 0 and minus

i = minusi and 0Now we decompose

i0N =

i0N+ minus

i0Nminus in which

i0N+ and

i0Nminus are solutions

of the following problems

t

i0N+ = minusQ middot [N Q middotQminus gQ

i0N+

]+Qi0N+ ++

i (73)

i0N+ Q 0 = maxQi

0Q 0 (74)

and respectively

t

i0Nminus = minusQ middot [N Q middotQminus gQ

i0Nminus

]+Qi0Nminus +minus

i (75)

i0Nminus Q 0 = maxminusQi

0Q 013 (76)

Since i0N+ and

i0Nminus are both positive we can now proceed as the estimate of N

We obtain

t

int3

Q2ni0N+ dQ = 4n2 + 2n

int3

Q2nminus2i0N+ dQ

+ 2nint3

Q2nminus2N QQ middot middotQminusQ middot gQi0N+ dQ

+int3

Q2n+i dQ13 (77)

By using (72) we have

int3

Q2n+i dQ le

int3

Q2nN Q middot QiN Q middotQminus gQdQ

+int3

Q2n(j0N+ +

j0Nminus

)Qi

N jkQk + gjQ+ N Qji + Qi

gjQdQ (78)

The first integral on the right hand side of (78) involves only N and no derivativesof N it can be controlled using (68) The integrand in the second integral can

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 915

be estimated for large Q by a constant times N QMQ2n + Q2nminus1gQsumj

j0N+ +

j0Nminus Thus we obtain

t

int3

Q2ni0N+ dQ

le C1nMint31+ Q2ni0

N+ dQ+ C2nM

minus 2nint

Q2nminus2NQ middot gQi0N+ dQ+ K3

int31+ Q2nminus1N gQ

+MN Q2nsumj

j0N+ +

j0Nminus dQ13 (79)

Similar estimate can be done for i0Nminus Summing up all the index i and the +minus

equations we obtain an inequality of quantityint3 Q2n sumi i0

N dQ Noticing whenn is sufficiently large the third integral may be controlled by the second integralthus we obtain

int31+ Q2nsum

i

i0N dQ le KnM T

( int31+ Q2nQ0 + 0dQ

)13

(80)

It implies

N isin L(0 T L(1

))if 0 isin H2

(1

)13 (81)

By the same way we may obtain for k = 2

N isin L(0 T L(2

))if 0 isin H2213 (82)

333 The Estimate of N t

We take the absolute value to the two sides of the symbol ldquo=rdquo to (62) andmultiply it by 1+ Q2n and integrate it in Q space It will be obtained that

N t isin L(0 T L(0

))if 0 isin H2

(2

)13 (83)

334 The Estimate of N

In the following we will show thatint3 N dQ is bounded Differentiating (62)

with respect to i yields the equation

t

0iN = minusQ middot [N Q middotQminus gQ

0iN

]+Q0iN +i

0iN Q 0 = i0Q

(84)

ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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ORDER REPRINTS

916 Li Zhang and Zhang

where

i = minusQ middot N Qi middotQN

= minusQ middot N Qi middotQN minus N Qi middotQ middot QN

= +i +minus

i 13 (85)

Here +i and minus

i have the similar definition as beforeLet 0i

N = 0iN+ minus

0iNminus where 0i

N+ and 0iNminus are the solutions of the following

problems

t

0iNplusmn = minusQ middot [N Q middotQminus gQ

0iNplusmn

]+Q0iNplusmn +plusmn

i (86)

0iNplusmn Q 0 = i0

plusmn13 (87)

Because 0iN+ and

0iNminus are positive we can proceed as above once more Observing

that there are only N and j0N terms involved in i and using isin L20 T

L (see formula (119)) we will obtainint31+ Q2nsum

i

0iN dQ le KnM T

( int31+ Q2n0 + 0dQ

)13

(88)

That shows

N isin L0 T L0 if 0 isin H3213 (89)

335 The Estimate of QN and 2QN

Next we will estimate the mixed derivatives of N Differentiating (84) withrespect to Qj yields the equation of ji

N

t

jiN = minusQ middot [N Q middotQminus gQ

jiN

]+QjiN +ji (90)

jiN Q 0 = Qj

i0Q (91)

where

ji = minusQjN lk middotQk minus glQ+ N Qlj minus Qj

glQliN

minusQ middot QjN Q middotQminus gQ

0iN

minusQ middot QjNi middotQNminus Ql

N QiljN

minusQ middot [N Qi middotQj0N

]= +

ji +minusji 13 (92)

Here +ji and minus

ji have the similar definition as before

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 917

Let jiN =

jiN+ minus

jiNminus where

jiN+ and

jiNminus are the solutions of the following

problems

t

jiNplusmn = minusQ middot [N Q middotQminus gQ

jiNplusmn

]+QjiNplusmn +plusmn

ji (93)

jiNplusmn Q 0 = Qj

i0plusmn13 (94)

Because jiN+ and

jiNminus are positive we can proceed as before The terms involving

liN in ji may be controlled by the terms from (90) by integrating by parts as in

(79) and the other terms has been estimated in former subsections Then by using and belong to L20 T L we find

int31+ Q2nsum

ij

∣∣jiN

∣∣dQ

le KnM T

( int3

(1+ Q2n)

[ 1sumlm=0

lQ

m 0Q t

]dQ

)13 (95)

That shows

N isin L0 T L1 if 0 isin H3213 (96)

By the same way we can obtain that

N isin L0 T L2 if 0 isin H3213 (97)

336 The Estimate of Nt

We take absolute value to both sides of the symbol ldquo=rdquo to (84) and multiply itby Q2n and integrate it to Q Moreover we have to integrate it to since the factori of i in (84) is only belong to L20 T L After the utilization of (96)and (97) it will be obtained that

Nt isin L20 T L0 if 0 isin H3213 (98)

This yields

N isin H10 TH10 (99)

if 0 isin H32

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

918 Li Zhang and Zhang

337 The Estimate of 2N

Next we estimate the high order derivative of N with respect to Differentiating (84) with respect to j yields the equation

t

0ijN = minusQ middot [N Q middotQminus gQ

0ijN

]+Q0ijN +ij (100)

0ijN Q 0 = 2ij0Q13 (101)

where

ij = minusQ middot N Q2ij middotQNminus Q middot [N Qi middotQ0jN

]minusQ middot [N Qj middotQ

0iN

]= +

ij +minusij 13 (102)

Here +ij and minus

ij have the similar definition as beforeAs before we decompose 0ij

N = 0ijN+ minus

0ijNminus where 0ij

N+ and 0ijNminus satisfy

t

0ijNplusmn = minusQ middot [N Q middotQminus gQ

0ijNplusmn

]+Q0ijNplusmn +plusmn

ij (103)

0ijNplusmn Q 0 = 2ij0

plusmn13 (104)

From (102) we have

int31+ Q2n+

ij dQ

le 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)13 (105)

Multiplying both sides of (103) by 1+ Q2n integrating to Q and summing themaltogether we obtain

t

int31+ Q2n0ij

N dQ

le KM nint31+ Q2n0ij

N dQ

+ 2

( int31+ Q2nN dQ+

int31+ Q2nsum

k

k0N dQ

)

+ ( int

31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

kl

klN dQ

)(106)

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 919

Now we multiplyint31+ Q2n0ij

N dQ to (106) and integrate it in space It canbe obtained

t

int

( int31+ Q2n0ij

N dQ)2

d

le KM nint

( int31+ Q2n0ij

N dQ)2

d+ C1

int2

2 d+ C22L 13

(107)

Here we used thatint31+ Q2nN dQ

int31+ Q2nsum

k

k0N dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

kl

klN dQ

belong to L0 Ttimes and the boundedness of By using isin L20 TL and Gronwallrsquos inequality we obtain

N isin L0 TH20 if 0 isin H3213 (108)

338 The Estimate of 3N

Differentiating (100) with respect to yields the equation

t

0ijkN = minusQ middot [N Q middotQminus gQ

0ijkN

]+Q0ijkN + ijk (109)

0ijkN Q 0 = 3ijk0Q (110)

where

ijk = minusQ middot N Q3ijk middotQNminus Q middot [N Q2ij middotQ0kN

]minusQ middot [N Q2ik middotQ

0jN

]minus Q middot [N Q2jk middotQ0iN

]minusQ middot [N Qi middotQ

0jkN

]minus Q middot [N Qj middotQ0ikN

]minusQ middot [N Qk middotQ

0ijN

]= +

ijk + minusijk13 (111)

Here +ijk and minus

ijk have the similar definition as beforeAs before we decompose 0ijk

N = 0ijkN+ minus

0ijkNminus where 0ijk

N+ and 0ijkNminus

are satisfied

t

0ijkNplusmn = minusQ middot [N Q middotQminus gQ

0ijkNplusmn

]+Q0ijkNplusmn + plusmn

ijk (112)

0ijkNplusmn Q 0 = (

3ijk0

)plusmn13 (113)

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

920 Li Zhang and Zhang

By (111) we can find thatint31+ Q2n+

ijk dQ

le 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(114)

Multiplying both sides of (112) by 1+ Q2n and integrating to Q and summingthem altogether we obtain

t

int31+ Q2n0ijk

N dQ

le KM nint31+ Q2n0ijk

N dQ

times 3

( int31+ Q2nN dQ+

int31+ Q2nsum

l

l0N dQ

)

+ 2

( int31+ Q2nsum

l

0lN dQ+

int31+ Q2nsum

mn

mnN dQ

)

+ ( int

31+ Q2nsum

mn

0mnN dQ+

int31+ Q2n sum

lmn

lmnN dQ

)13

(115)

Now we multiplyint31+ Q2n0ijk

N dQ to (115) and integrate it to It can befound

t

int

( int31+ Q2n0ijk

N dQ)2

d

le KM nint

( int31+ Q2n0ijk

N dQ)2

d

+C1

int3

2 + 2 2d+ C22L (116)

Here we used that∣∣∣int31+ Q2nN dQ

int31+ Q2nsum

l

0lN dQ

int31+ Q2nsum

l

l0N dQ

int31+ Q2nsum

mn

mnN dQ

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 921

belong to L0 TH20 and the boundedness of Application ofGronwallrsquos inequality and isin L20 T L implies that

N isin L0 TH30 if 0 isin H3213 (117)

34 Estimate of with Respect to Eulerian Variable

341 The Estimate of Flow Map

Since we need the estimates of the derivatives of to x but now we have onlythose of to Eq (59) implies that we will have them if we can obtain the estimateof x t of (59)

Since um isin SM T we have um isin L0 Ttimes (see Evans 1998 pp 287)From Eq (59) we have

tx = um middot x13 (118)

By utilizing this equation we know x isin L0 Ttimes thus it is deduced that

= umx t tT middot x isin L20 T L13 (119)

Moreover t middot x = 0 implies

t = minus middot

tx middot x

minus113 (120)

Combining of (118) and (120) yields

t = middot um13 (121)

From um isin L0 Ttimes we obtain

isin L0 Ttimes13 (122)

Now by using (121)ndash(122) we have

t22L2 le C122L2 + C22um2L213 (123)

The Gronwallrsquos inequality implies

2 isin L0 T L2 (124)

when um isin SM T Similarly we can find

t32L2 le C132L2 + C222L2 + C33um2L213 (125)

It implies if um isin SM T

3 isin L0 T L213 (126)

ORDER REPRINTS

922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

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Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

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922 Li Zhang and Zhang

342 Estimate of N wrt x

By (122) (124) (126) and

NxQ t = N middot (127)

we have

NxQ t isin1⋂

k=0

Hk0 TH3minus2k0 (128)

when um isin SM T and

NQ t isin1⋂

k=0

Hk0 TH3minus2k013 (129)

343 The Limit of N When N rarr

All of the estimates above are uniform in N and the restriction to the initialdata is 0 isin

⋂2i=0 H

3i But when N passes to the limit of the sequence N

may not in⋂1

k=0 Hk0 TH3minus2k0 since k are based in L1-type norms and

the limit of a distributionally convergent bounded sequence in L1 may not be anL1-function but a singular measure In order to overcome the difficulty now we canutilize the technology in Renardy (1991) to improve the regularity of 0 This is thecause to suppose 0 isin H42 in the condition (A3) of Theorem 1 The detail issimilar to that in Renardy (1991)

The proof of Lemma 5 is based on the same type of estimates applied to thefunction minus We omit the details

4 CONCLUSION

A detailed well-posedness analysis for the dumbbell model of polymeric fluidsis finished in this paper The model considered is a coupled system of fluidsvelocity u and distribution density for polymeric fluids in kinetic theory ofpolymers The main theoretical framework is originally in the paper Renardy (1991)but a more detailed analysis is focused on the derivative estimate of withrespect to space variable In particular the L norm of

int31+ Q2nN dQint

31+ Q2nQN dQint31+ Q2nxN dQ and

int31+ Q2nxQN dQ

are essential for the proof which is the key point of the priori estimate This workis crucial for the numerical analysis of the recently proposed multiscale methods forsolving (1)ndash(5) (Jendrejak et al 2002)

ACKNOWLEDGMENTS

The authors are very grateful to Professor Weinan E for his helpful discussionsHui Zhang acknowledges the financial support of the Post-Doctoral ScienceFoundation of China Pingwen Zhang is partially supported by the special funds forMajor State Research Projects G1999032804 and National Science Foundation ofChina for Distinguished Young Scholars 10225103

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Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

ORDER REPRINTS

Dumbbell Model of Polymeric Fluids 923

REFERENCES

Bird R B Armstrong R C Hassager O (1987) Dynamics of Polymeric LiquidsVol 1 and 2 2nd ed New York John Wiley

Doi M Edwards S F (1986) The Theory of Polymer Dynamics Oxford OxfordUniversity Press

Evans L C (1998) Partial Differential Equations Providence Rhode Island AMSFeigl K Laso M Oumlttinger H C (1995) CONNFFESSIT approach for solving

a two-dimensional viscoelastic fluids problem Macromolecules 283261ndash3274Guillopeacute C Saut J C (1990) Existence results for the flow of viscoelastic fluids

with a differential constitutive law Nonlinear Anal TMA 15849ndash869Hulsen M A van Heel A P G van den Brule B H A A (1997) Simulation of

viscoelastic flows using Browmian configuration field J Non-Newtonian FluidMech 7079ndash101

Jendrejak R M de Pablo J J Graham M D (2002) A method for multiscalesimulation of flowing complex fluid J Non-Newtonian Fluid Mech 108123ndash142

Jourdain B Lelievre T Le Bris C (2002) Numerical analysis of micro-macrosimulations of polymeric fluid flows A simple case Math Models MethodsAppl Sci 12(9)1205ndash1243

Jourdain B Lelievre T Le Bris C (2004) Existence of solution for a micro-macro model of polymeric fluid The FENE model J Funct Anal 209162ndash193

Laso M Oumlttinger H C (1993) Calculation of viscoelastic flow using molecuarmodels The CONNFFESSIT approach J Non-Newtonian Fluid Mech471ndash20

Li T E W Zhang P (2002) Convergence of a stochastic method for themodeling of polymeric fluids Acta Mathematicae Applicatae Sinica EnglishSeries 18(4)529ndash536

Li T E W Zhang P (2003) Well-Posedness for the dumbbell model of polymericfluids Accepted by Comm Math Phys

Lions P L Masmoudi N (2000) Global solutios for some Oldroyd models ofnon-Newtonian flows Chin Ann Math 21B131ndash146

Renardy M (1990) Local existence of solutions of the Dirichlet initial boundaryproblem for incompressible hypoelastic materials SIAM J Math Anal211369ndash1385

Renardy M (1991) An existence theorem for model equations resulting fromkinetic theories SIAM J Math Anal 22(2)313ndash327

Risken H (1984) The Fokker-Planck Equation Methods of Solution and ApplicationsSpringer Verlag

Temam R (1995) Navier-Stokes Equations and Nonlinear Functional AnalysisPhiladelphia SIAM

Received June 2003Revised December 2003

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details

Request PermissionOrder Reprints

Reprints of this article can also be ordered at

httpwwwdekkercomservletproductDOI101081PDE120037336

Request Permission or Order Reprints Instantly

Interested in copying and sharing this article In most cases US Copyright Law requires that you get permission from the articlersquos rightsholder before using copyrighted content

All information and materials found in this article including but not limited to text trademarks patents logos graphics and images (the Materials) are the copyrighted works and other forms of intellectual property of Marcel Dekker Inc or its licensors All rights not expressly granted are reserved

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly Simply click on the Request Permission Order Reprints link below and follow the instructions Visit the US Copyright Office for information on Fair Use limitations of US copyright law Please refer to The Association of American Publishersrsquo (AAP) website for guidelines on Fair Use in the Classroom

The Materials are for your personal use only and cannot be reformatted reposted resold or distributed by electronic means or otherwise without permission from Marcel Dekker Inc Marcel Dekker Inc grants you the limited right to display the Materials only on your personal computer or personal wireless device and to copy and download single copies of such Materials provided that any copyright trademark or other notice appearing on such Materials is also retained by displayed copied or downloaded as part of the Materials and is not removed or obscured and provided you do not edit modify alter or enhance the Materials Please refer to our Website User Agreement for more details