Boundar Layer Estimates for Channel Flow

8/11/2019 Boundar Layer Estimates for Channel Flow

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BOUNDARY LAYER ASSOCIATED WITH A CLASS OF 3D

NONLINEAR PLANE PARALLEL CHANNEL FLOWS

ANNA L MAZZUCATO1

Department of Mathematics, Penn State University

McAllister Building University Park, PA 16802, U.S.A.

Email:[email protected]

DONGJUAN N IU 2 3

School of Mathematical Sciences, Capital Normal University

Beijing 100048, P. R. ChinaEmail: [email protected]

XIAOMINGWAN G 4

208 James J. Love Building, Department of Mathematics,

Florida State University, 1017 Academic Way, Tallahassee, FL 32306-4510, USA

Email: [email protected]

1. INTRODUCTION

The dynamics of the viscous incompressible flow is governed by the clas-

sical incompressible Navier-Stokes equations (NSE) for Newtonian fluids

[11, 19]:v

t + (v )v v+ p= F, v= 0, (1.1)

where v is the Eulerian fluid velocity, p is the kinematic pressure, is thekinematic viscosity and F is a (given) applied external body force. The sys-

tem is equipped with an initial condition v0 and the no-slip no-penetration

boundary condition

vD

= 0. (1.2)

If the kinematic viscosity is small (or the Reynolds number is large) such

as in air and water, we may formally set the viscosity to zero in the Navier-

Stokes system and we arrive at the Euler system for incompressible inviscid

Date: March 7, 2011.1Supported in part by National Science Foundation grant DMS 0708902.2Corresponding author.3Supported in part by National Tian Yuan grant, China (No. 10926069) and National

Youth grant, China (No. 11001184).4Supported in part by the National Science Foundation, a COFRA award from FSU, and

a 111 project from the Chinese Ministry of Education at Fudan University.

1


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2 BOUNDARY LAYER FOR CHANNEL FLOWS

flows:v0

t + (v0

)v0

+ p0

=F

, v0

= 0. (1.3)The Euler system is equipped with the same initial condition but a different,

no-penetration, boundary condition

v0 n

D

= 0, (1.4)

since we have dropped the highest order spatial derivative. In (1.4), n is the

normal vector field ofD.This heuristic limit can not be valid uniformly over space due to the dis-

parity of boundary conditions. This disparity usually leads to the emergence

of a thin layer near the boundary of the domain, the so-called boundary

layer, where the viscous flow makes a sharp transition from the almost in-

viscid flow in the interior of the domain to the zero value at the boundary

[18].

Boundary layers are of great importance since it is the vorticity generated

in the boundary layer and later advected into the main stream that drives the

flow in many physical applications. On the other hand, the existence of

a boundary layer renders the inviscid limit problem a particularly singular

one and hence its analysis is a challenge.

Standard classical approach to the boundary layer problem is by approx-

imating the viscous (NSE) solution within the boundary layer via the so-

called Prandtl equation [16]. Here we take a slightly different approach and

derive Prandtl-type effective equation for thecorrectorthat approximatesv v0 [23, 6]. This alternative approach has the advantage that the match-ing procedure is conceptually simple: the sum of the inviscid solution and

the corrector is a natural candidate for an approximation to the viscous solu-

tion. The analysis of the boundary layer problem then consists of the study

of the Prandtl-type equation, and proof of convergence of the approximate

solution to the exact solution of the Navier-Stokes system.

There exists an abundant literature on boundary layer theory [15, 18] and

the vanishing viscosity limit (we refer in particular to [1, 2, 3, 4, 7, 8, 9,

10, 12, 13] and references therein). However, few examples of nonlinear

boundary layers exist in the literature (see [17] for the case of half space

in the analytical setting, [21, 22] for the case of channel flow with uniforminjection and suction at the boundary among others). The main purpose of

this manuscript is to investigate the boundary layer associated with a class

of nonlinear plane parallel flows. This family of nonlinear solutions to the

Navier-Stokes system was introduced in [24] where the vanishing viscos-

ity limit was established. More recently, the boundary layer for this class

of flows was studied in [14] using semiclassical expansions for heat equa-

tions with drifts without referring to the Prandtl theory, and certain L(L)


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BOUNDARY LAYER FOR CHANNEL FLOWS 3

convergence results derived. In both cases, Couette-type flows are consid-

ered, with nonhomogeneous characteristic data. The current manuscript

follows the Prandtl type approach, and presents a more detailed analysis ofthe boundary associated with this class of exact solutions and hence will go

beyond the results derived in those two previous works. Our main result

(Theorem 5.2) provides error bounds for the approximation of the NSE so-

lution given by the Euler solution plus the corrector, and hence convergence

rates in the vanishing viscosity limit (Corollary 5.3 and 5.4). We consider

bothL and Sobolev H1 bounds, which give information on the possiblegrowth of normal derivatives in the boundary layer. We establish these re-

sults under some compatibility conditions on the initial and boundary data

and body force.

The manuscript is organized as follows. In Section 2 we recall the family

of nonlinear 3D plane parallel channel flow. Section 3 is devoted to formal

asymptotic expansion of this class of flows at small viscosity utilizing the

Prandtl type (corrector) approach. We construct an approximate solution to

the Navier-Stokes system utilizing the solution to the Prandtl type system

(corrector) and the solution to the Euler system in Section 4. The main

convergence result is provided in Section 5. Improved estimates with higher

order approximations are provided in Section 6. Decay estimates of the

correctors are furnished in Appendix A.

Throughout the paper, we use C to denote generic constants that mayvary line by line, but is independent of the kinematic viscosity.

2. NONLINEAR PLANE-PARALLEL CHANNEL FLOWS

We start by recalling the ansatz for a plane-parallel channel flow intro-

duced in [24]. That is, we look for solutions of the fluid equations of the

form:

v(t,x,y ,z ) = (u1(t, z), u2(t,x,z), 0) (2.1)

in an infinitely long horizontal channel, but we impose periodicity in the

horizontal coordinatesx, y. We hence reduce to work in the spatial domainQ:= [0, L] [0, L] [0, 1], whereLis the horizontal period. This assump-tion ensures uniqueness of the solution to the fluid equations. Flows of theform (2.1) are automatically divergence free.

The symmetry of the solution is preserved by both the Navier-Stokes and

Euler evolution if the initial condition v0and body force F satisfy the same

ansatz, that is:

v|t=0= v0(x,y ,z ) = (a(z), b(x, z), 0), (2.2)

F= (f1(t, x), f2(t,x,z), 0). (2.3)


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We will denote by v the solution of the Navier-Stokes system (1.1) with

viscosity and by v0 the solution of the Euler system (1.3). Periodicity

in the horizontal directions is complemented by boundary conditions in thevertical variablez. For NSE, we prescribed the fluid velocity at the channelwalls together with a non-penetration condition:

v|z=0 = 0(t,x,y), v|z=1 = 1(t,x,y),

where i(t,x,y) := (i1(t), i2(t, x), 0), i= 0, 1.

It is not difficult to show that imposing the symmetry (2.1) on the solution

system reduces NSE to the following weakly non-linear system:

tu1 zzu1 = f1,

tu2+u1xu2 xxu2 zzu2=f2, (2.4)on := [0, L] [0, 1]. We remark that although the above system is a 2 by2system, the plane-parallel flows we study are not two-dimensional.

In addition, we will always assume that the initial data, boundary data

and forcing satisfy certain compatibility conditions. We recall the zero-

order compatibility condition which takes the form

i(0, x , y) = v0(x,y ,i), i= 0, 1, (2.5)

and the first-order compatibility condition

ti1(0) =zza(i) +f1(0, i)

ti2(0, x) +a(i)xb(x, i) =xxb(x, i) +zzb(x, i) +f2(0, x , i), (2.6)

where i = 0, 1. These compatibility conditions prevent the formation ofan initial layer in the NSE evolution due to the difference in boundary val-

ues between the initial data and the fluid velocity at any positive time. In

[10, 14], the boundary layer is analyzed without assuming any compatibil-

ity condition. In this case, extra vorticity is produced at the boundary in the

limit of vanishing viscosity, and in general only L bounds can be read-ily obtained for the correctors. While the zero-order condition is uniform

in , the first-order conditions in general imply that the boundary data i,i = 0, 1, is dependent on for t > 0 (we assume the forcing is given andindependent of viscosity). We notice however that this undesirable de-pendence can be eliminated if the second derivatives of the initial data (aandb) vanish at the channel walls, that is, the initial condition has a linearprofile near the walls. In addition, We would like to point out that only the

zero order compatibility condition is needed if we are only interested in the

leading order expansion of the channel flow.


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Since we are working in a domain that is periodic in the horizontal direc-

tions, we will employ the Sobolev spaces, for m

Z+,

Hm(Q) =Hmper(Q) := {f :Q R| f L2(Q),|| m,fperiodic in the horizontal directions}.

We denote with Hm(Q)the subspace of functions in Hm(Q)that are con-stant in y. With abuse of notation, we write (Hm(Q))2 Hm(Q) forspaces of vector fields. As customary, H10 is the space of functions in H

1

that vanish at the boundary in trace sense.

Due to the weak coupling in the system (2.4), well-posedness is easily

established. For instance, v L(H1())3 under the assumption that v0and i belong to H1() and f := (f1, f2) L(0,; H1())3. We donot address this point in detail here, and refer for example to [14, 24] forfurther discussion.

By formally taking the limit 0, NSE become the Euler system (1.3).We continue to assume periodicity in the horizontal directions x, y, but im-pose the no-penetration condition (1.4) at the channel walls. We observe

that solutions of the form (2.1) automatically satisfy the no-penetration con-

dition. Under the plane-parallel symmetry, the Euler system reduces to the

following weakly non-linear system

tu01=f1,

tu02+u

01xu

02=f2, (2.7)

in. We take the same initial condition (2.2) for both Euler and Navier-Stokes:

v0|t=0= v0. (2.8)

The solution of (2.7) is obtained by solving an ordinary differential equa-

tion and a transport equation. Therefore, the solution is regular provided

the initial data is regular enough. For example, if v0 Hm(), andf L(0, T; Hm()), m > 5 as we will assume throughout, then v0 C(0, T; Hm()).

In the rest of the paper, we will focus on the analysis of the reduced

systems (2.4) and (2.7) on. For this purpose, we set

u(t,x,z) := (u1(t, z), u

2(t,x,z)),

u0(t,x,z) := (u01(t, z), u

02(t,x,z)),

f(t,x,z) := (f1(t, z), f2(t,x,z)),

i(t, x) := (i1(t), i2(t, x)), i= 0, 1,

u0(t,x,z) = (a(x), b(x, z)).


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We assume tacitly throughout the rest of the paper that all functions are

periodic in thex variable, so that boundary conditions will be given only at

the channel walls, that is, forz= 0andz= 1.

3. PRANDTL-TYPE EQUATIONS FOR CORRECTORS

The approach to a rigorous boundary layer analysis that we take is to

derive effective Prandtl-type equations for acorrectorthat approximates the

difference between the NSE solution(u, 0)and the Euler solution(u0, 0).We assume that the NSE solution is well approximated by

uapp(t,x,z) := uou(t,x,z) + 0(t,x,

z

) + u,0(t,x,1 z

), (3.1)

whereu

ou

is the so-called outer solution, that is, the vector field expectedto represent the fluid velocity outside of the boundary layers, while 0 and

u,0 are the correctors respectively near the lower (z= 0) and upper (z=1) wall of the channel. We make the ansatz that the correctors take thefollowing form

0(t,x, z

) = (01(t,

z

), 02(t,x, z

)),

u,0(t,x,1 z

) = (u,01 (t,

1 z

), u,02 (t,x,1 z

)). (3.2)

This form corresponds to the zero order in a formal asymptotic expansion in

powers of of the difference between the NSE and Euler solutions in eachboundary layer. Introducing the stretched variables Z= z

andZu = 1z

we see that the correctors must satisfy the following matching conditions

0j 0asZ ; u,0j 0as Zu , j = 1, 2, (3.3)in order for the vanishing viscosity limit to hold.

In addition, we define

:= [0, L] [0,)as the spatial domain occupied by the correctors0(t,x,Z) andu,0(t,x,Zu).

Inserting (3.1) into (2.4) and (2.4) and dropping lower-order terms in ,

we obtain the systems of equations that uou and the correctors must satisfyrespectively.

(1) The outer solution uou satisfies the reduced Euler equations (2.7)

with the initial data

u0|t=0 = u0. (3.4)

Consequently, by uniqueness of the solution to the reduced system,

we can identify uou u0.


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(2) The lower corrector 0 = (01, 02)satisfies

t01 ZZ

01 = 0, (3.5a)

t02 ZZ02+ 01x02+ u01(t, 0)x02+ 01xu02(t,x, 0) = 0, (3.5b)

(01, 02)|Z=0= (01(t) u01(t, 0), 02(t, x) u02(t,x, 0)), (3.5c)

(01, 02)|Z== 0, (3.5d)

(01, 02)|t=0= (0, 0). (3.5e)

(3) The upper corrector u,0 = (u,01 , u,02 )satisfies

tu,01 ZuZuu,01 = 0,

tu,02 ZuZu

u,02 +

u,01 x

u,02 +u

01(t, 1)x

u,02 +

u,01 xu

02(t,x, 1) = 0,

(u,01 , u,02 )|Zu=0= (11(t) u01(t, 1), 12(t, x) u02(t,x, 1)),

(u,01 , u,02 )|Zu= = 0,

(u,01 , u,02 )|t=0 = 0. (3.6)

The well-posedness of the above systems is readily established, so that (3.1)

gives a well-defined vector field. Furthermore, the correctors exhibit certain

rates of decay in the stretched variables, which in turn will be used to estab-

lish error bounds for the approximate solution uapp. The solvability of the

systems (3.5) and (3.6) along with the decay properties of the correctors are

discussed in Appendix A.

4. APPROXIMATE SOLUTIONS

In order to derive error bounds for the approximate solution to NSE intro-

duced in Section 3 above, it is more convenient to modify (3.1) so that the

boundary condition (2.4) is met exactly. Such a modification is well-known

in the literature (see [5], [20], [21], [22], [24] for instance).

Let(z)be a smooth function defined on[0, 1]such that(z) = 1whenz

[0, 1

3] and (z) = 0 when z

[1

2, 1]. We have (z)(1

z)

0

whenz [0, 1].Next, we define a truncated approximation uapp

(t,x,z) =(uapp1 (t, z),u

app2 (t,x,z))to the NSE solution, where

uapp1 (t, z) :=u

01(t, z) +(z)

01(t,

z

) +(1 z)u,01 (t,1 z

), (4.1a)

uapp2 (t,x,z) :=u

02(t,x,z) +(z)

02(t,x,

z

) +(1 z)u,02 (t,x,1 z

).

(4.1b)


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Then uapp satisfies the following system

tuapp

1

zz uapp

1 =f

1+A

+B,

tuapp2 + u

app1 xu

app2 xxuapp2 zz uapp2 =f2+D+E+F, (4.2)

where

A= 2[(z)Z01+ (1 z)Zuu,01 ], (4.3a)

B= [zzu01+

(z)01+

(1 z)u,01 ], (4.3b)D= (z)((z) 1)01x02+ (1 z)((1 z) 1)u,01 xu,02 , (4.3c)E=

[(z)(Z01xzu

02(t,x, 0) +zu

01(t, 0)Zx

02)

(1 z)(zu01(t, 1)Zuxu,02 +01xzu02(t,x, 1)Zu) 2(z)Z02 2(1 z)Zuu,02 ], (4.3d)

F =((z)xx02 (1 z)xxu,02 xxu02 zzu02 02

(1 z)u,02 ). (4.3e)

The corresponding initial and boundary conditions are respectively

uapp|t=0 = u0,

uapp|z=0 = 0(t, x), uapp|z=1=1(t, x). (4.4)

Both uapp and the truncated uapp depend on viscosity , but for sake ofnotation we do not explicitly show it.

5. ERROR ESTIMATES AND CONVERGENCE RATES

We are now ready to prove our main result, that is, error bounds for the

approximationuapp of the true NSE solution, which then yield convergence

rates as viscosity vanishes. Later, we will improve upon these results by

including more terms from an asymptotic expansion in power of

in boththe outer solution and the correctors.

The approximation error is given by uerr(t,x,z) = u(t, z) uapp, andit satisfies the following system of equations

tuerr1

zzu

err1 =

(A+B), (5.1)

tuerr2 +u

err1 xu

app2 +u

1xu

err2 xxuerr2 zzuerr2 = (D+E+ F),

(5.2)

whereA throughFare given in (4.3), with boundary conditions and initialdata

uerr|z=0= 0, uerr|z=1= 0,

uerr|t=0= 0. (5.3)


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A key technical result is an anisotropic Sobolev embedding contained

in the following lemma, which is proved in [5] (Corollary 7.3), and [20]

(Remark 4.2) for instance.

Lemma 5.1(Temam&Wang). For allu H10 ()uL() (5.4) C(u

1

2

L2zu

1

2

L2+ zu

1

2

L2xu

1

2

L2+ u

1

2

L2xzu

1

2

L2),

where the left-hand or both sides of the inequality could be infinite.

The main results of this paper are contained in the next theorem.

Theorem 5.2. Let u0 Hm(), i H2(0, T; Hm()),i = 0, 1, andfL(0, T; H

m

()), m > 5, satisfy the zero-order compatibility condition(2.5). Then, there exists positive constantsCi, i = 1, 2, 3, independent of such that for any solution u of the system(2.4)with initial condition u0and boundary data i,

u uappL(0,T;L2()) C1 34 , (5.5)u uappL(0,T;H1()) C2 14 , (5.6)u uappL((0,T)) C3

, (5.7)

where uapp is given in equation(4.1).

We do not concern ourselves with optimizing the regularity imposed onthe data in Theorem 5.2, since our aim is to investigate the boundary layer

which is present even for smooth data.

Before proceeding with the proof of the theorem, we state some immedi-

ate consequences.

Corollary 5.3. Under the hypotheses of Theorem 5.2, the following optimal

convergence rate holds:

C11

4 u u0L(0,T;L2()) C214 , (5.8)whereC1, C2 are constants depending on u0, f and

i, i = 1, 2, but inde-

pendent of.

Corollary (5.8) is a consequence of the estimate 0L(0,T;L2()) 14 .We can similarly establish convergence rates of the Navier-Stokes to the

Euler solution. These rates recover and improve upon some of the results

of [14, 24].

Corollary 5.4. Under the hypotheses of Theorem 5.2, there exist positive

constantsCi, i = 1, 2, independent of such that for any (0, 1) such


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that as 0,

u

u0

L(0,T;H1()) C1 1

4 ,u u0L((0,T))) C2

,

where = [0, L] [, 1 ].Proof of Theorem 5.2. We first employ the results of the Appendix and stan-

dard energy estimates to derive error bounds in L([0, T], L2()) andL([0, T], H1()). We then apply the anisotropic Sobolev inequality (5.4)to obtain bounds inL([0, T] ).

Multiplying (5.1) byuerr1 and integrating by parts over, we obtain that

1

2

d

dtuerr

1 2L2(0,1)+zu

err

1 2L2(0,1) =

1

0 (A+B)uerr

1 dz

2 2

3

1

3

|(Zuu,01 +Z01) +(01+ u,01 )||uerr1 | dz

+u01H2(0,1)uerr1 L2() (5.9) uerr1 L2(0,1)

18

7

4 (Zu2Zuu,01 L2(0,)+ Z2Z01L2(0,))+ (u,01 L2(0,)+ 01L2(0,)+ u01H2(0,1))

,

where some of the terms on the right hand side of the last inequality are

estimates as exemplified below (the limits of integration are determined by

the support properties of the cut-off function): 23

1

3

|Z01(t, z

)uerr1 (t, z)| dz

94uerr1 L2(0,1)(

23

1

3

Z4|Z01(t, Z)|2

dZ)1

2

95

4

4uerr1 L2(0,1)Z2Z01L2(0,).

Applying Cauchy and then Gronwalls inequalities to (5.9) gives:

uerr1 L(0,T;L2(0,1))+ zuerr1 L2(0,T;L2(0,1)) 18 (Zu2Zuu,01 L2(0,T;L2(0,))+ Z2Z01L2(0,T;L2(0,))

+ 01L2(0,T;L2(0,))+ u,01 L2(0,T;L2(0,))+ u01L2(0,T;H2(0,1))) (5.10) C1 ,

where the constantC1 depends on u01L(0,T;H2(0,1)), andT explicitly, butis independent of by Lemma A.1 and A.2.


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Next, we multiply (5.1) by zzuerr1 and integrate over:

12

ddtzuerr1 2L2(0,1)+zzuerr1 2L2(0,1)

18(Zu2Zuu,01 L2(0,T;L2(0,))+ Z2Z01L2(0,T;L2(0,))+ 01L2(0,T;L2(0,))+ u,01 L2(0,T;L2(0,))

+ u01L2(0,T;H2(0,1)))zzuerr1 L2(0,T;L2(0,1))

8zzuerr1 2L2(0,1)+ 2C21, (5.11)

by Cauchys inequality and Lemma A.1 and A.2 again. Integrating (5.11)

in time gives

zuerr1 L(0,T;L2(0,1))+

4zzuerr1 L2(0,T;L2(0,1)) 2C1

. (5.12)

Therefore, we conclude that

uerr1 L(0,T;L2(0,1)) C1,uerr1 L(0,T;H1(0,1)) 2C1

,

uerr1 L((0,T)(0,1)) (5.13)

uerr

1 1

2

L(0,T;L2

(0,1))uerr

1 1

2

L(0,T;H1

(0,1))2C1

3

4 .

Multiplying (5.2) by uerr2 , integrating over, and noticing thatu1 is in-

dependent ofx, we obtain that

1

2

d

dtuerr2 2L2()+zuerr2 2L2()+xuerr2 2L2()

=

uerr1 xuapp2 u

err2 dx

Duerr2 dx

Euerr2 dx

F uerr2 dx

=:J1+J2+J3+J4. (5.14)

We bound each term on the right-hand side separately:

J1 C1 (xu02L()+ x02L()+ xu,02 L())uerr2 L2() C1C2 uerr2 L2(), (5.15)

withC2 a constant that depends on u02L(0,T;H2+s()),s >0, but is inde-pendent of. Above, we have used (5.13) in the first inequality and LemmaA.2 in the second one.


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The support properties of and similar techniques as those used in (5.9)imply the following bound for J2

J2 L0

(

23

1

3

(|01x02| + |u,01 xu,02 |)|uerr2 |dzdx

9 uerr2 L2()(01L(0,)Z2x02L2() (5.16)+ 9u,01 L(0,)Zu2xu,02 L2()).

Similarly, we deal withJ3,J4as follows

J3 34uerr2 L2()(u02H3+s)Z01L2(0,)+ u01H2(0,1)Zx02L2()+ u02H3+s()Zuu,01 L2(0,)+ u01H2(0,1)Zxu,02 L2()

+ Z0

2L2()+ Zuu,0

2 L2()), (5.17)wheres >0 is arbitrary, and

J4 uerr2 L2()(xx02L2()+ xxu,02 L2()+ u02H2()+ 02L2()+ u,02 L2()). (5.18)

By substituting (5.15)-(5.18) into (5.14), applying first Cauchy and then

Gronwalls inequalities, we have

uerr2 L(0,T;L2())+

x,zuerr2 L2(0,T;L2()) C53

4 , (5.19)

with C5a constant that depends on T, u01L(0,T;H2+s(0,1)), and u02L(0,T;H3+s()),but is independent of, where in the last inequality we have applied the re-sults in Appendix A once again.

Similarly, multiplying both sides of (5.2) by xxuerr2 and integrating byparts over gives

1

2

d

dtxuerr2 2L2()+x,zxuerr2 2L2()

uerr1 L2()xuerr2 L2()(xxu02L()+ xx02L()+ xxu,02 L())+(Z201L(0,)xx02L2()+ Zu2u,01 L(0,)xxu,02 L2())+

3

4xuerr2 L2()(Z01L2(0,)u02H4+s()+ u01H2(0,1)Zxx02L2()

+ xZ0

2L2()+ Zu

u,0

1 L2(0,)u0

2H4+s()+ Zxxu,02 L2()u01H2(0,1)+ xZu,02 L2()) (5.20)

+xuerr2 L2()(xxx02L2()+ xxxu,02 L2()+ u02H3()+ x02L2()+ xu,02 L2()).

from which it follows that

xuerr2 L(0,T;L2())+

x,z(xuerr2 )L2(0,T;L2()) C63

4 , (5.21)


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whereC6depends onT, u01L(0,T;H2(0,1))and u02L(0,T;H4+s()),s >0,but is independent of.

Analogous calculations give

1

2

d

dtzuerr2 2L2()+zuerr2 2L2()

uerr1 L2()(xu02L()+ x02L()+ xu,02 L())+ u1L()xuerr2 L()zzuerr2 L2()+zz02L2()(01L(0,)Z2x02L2()+ u,01 L(0,)Zu2xu,02 L2()) (5.22)

+3

4

zzu

err2

L2()(

Z

01L2(0,

)

u02H3+s()

+ u01H2(0,1)Zx02L2()+ Zuu,01 L2(0,)u02H3+s()+ u01H2(0,1)Zuxu,02 L2()+ Z02L2()+ Zuu,02 L2())+zzuerr2 L2()(xx02L2()+ xxu,02 L2()+ u02H2()+ 02L2()+ u,02 L2()),

which imply, utilizing (5.13) and (5.21),

zuerr2 L(0,T;L2)+

x,zzuerr2 L2(0,T;L2()) C71

4 , (5.23)

where C7depends on u1L((0,T)), u01L(0,T;H2+s(0,1))and u02L(0,T;H3+s()),independent of. A uniform bound onu1L((0,T))in terms off1L((0,T)),a(z)L(0,1), and 01(t)L(0,T),follows from the maximum principle forthe heat equation.

In the above calculations, we cannot improve the bound of order 3

4 , since

we cannot perform any integration by parts in the right-hand side involving

second or mixed derivatives inz, as zuerr2 may not vanish at the boundary.

Proceeding in a similar fashion we also have

xxu

err2

L(0,T;L2())

C8

3

4 ,

zxuerr2 L(0,T;L2()) C814 , (5.24)

with C8depending on T, u02L(0,T;H5+s(0,1)), u01L(0,T;H2()), and u1L((0,T)),independent of.

Collecting (5.19), (5.21) and (5.22), we have that

uerr2 L(0,T;H1()) C1

4 . (5.25)


14/22


Finally, the anisotropic Sobolev embedding (5.4), together with inequal-

ities (5.19), (5.21) and (5.22), gives

uerr2 L((0,T)) C(uerr2 1

2

L(0,T;L2())zuerr2 1

2

L2(0,T;L2())

+ zuerr2 1

2

L(0,T;L2())xuerr2 1

2

L(0,T;L2()) (5.26)

+ uerr2 1

2

L(0,T;L2())xzuerr2 1

2

L(0,T;L2()))

C.With this estimate, we conclude the proof of Theorem 5.2.

6. IMPROVED CONVERGENCE RATE

It is possible to derive convergence rates of higher order in than those ofSection 5 by including more terms in the asymptotic expansion for the outer

solution and the correctors. (We recall that we showed the convergence rate

is of order 3

4 , 1

4 , 1

4 forL2, H1, L norm respectively.) The higher-orderexpansions can be also used to show the optimality of the convergence rate.

In this section, we illustrate this idea by presenting the corresponding

results up to the first-order expansion.

Accordingly, we replace (3.1) with the following ansatz

uapp,1(t,x,z) := uou(t,x,z) + ulc(t,x, z ) + uuc(t,x,

1

z ), (6.1)

where

uou(t,x,z) = u0(t,x,z) +u1(t,x,z)is the outer solution, validin = [0, L] [0, 1];

ulc(t,x, z) =0(t,x, z

) +

1(t,x, z

)is the lower corrector,

defined in= [0, L] [0,); uub(t,x, 1z

) = u,0(t,x, 1z

) +

u,1(t,x, 1z

) is the upper

corrector, also defined in.

As before, the correctors must satisfy the matching conditions:

i 0asZ ; u,i 0as Zu , (6.2)wherei = 0, 1andZ= z

andZu = 1z

are the stretched variables.

Next we derive the systems satisfied by the outer solution and the correc-

tors. By consistency, the terms at leading order in both the outer solution

and the correctors are given by the Euler solution u0, and 0, u,0 con-

structed in Section 3 respectively. The first-order terms are given below:


15/22


(1) The first-order term of the outer solution u1(t,x,z) = (u11(t, z),u12(t,x,z))satisfies the following system of transport equations

tu11 = 0,

tu12+u

11xu

02+u

01xu

12= 0,

(u11, u12)|t=0= (0, 0). (6.3)

Given the regularity of the Euler solution u0, it is immediate to seethat(u11, u

12) 0.

(2) The first-order term in the lower corrector 1(t,x,Z) = (11, 12)

satisfies the system

t11

ZZ

11 = 0,

t12+ u01(t, 0)x

12+

01x

12+

11(x

02+ xu

02(t,x, 0))

ZZ12 = Z(012xzu02(t,x, 0) +zu01(t, 0)x02),(11,

12)|Z=0 = (u11(t, 0),u12(t,x, 0)) = 0,

(11, 12)|Z= = (0, 0), (11, 12)|t=0= (0, 0),

from which it follows that 11 0. Hence, the equations of1 reduceto

11 = 0,

t1

2+ u0

1(t, 0)x

1

2+ 0

1x

1

2 ZZ

1

2= Z(012xzu02(t,x, 0) +zu01(t, 0)x02), (6.4)

12|Z=0 = 0, 12|Z== 0,12|t=0= 0.

(3) By symmetry, the first-order term in the upper correctoru,1(t,x,Zu) =(u,11 ,

u,12 )satisfies the following system

u,11 = 0,

tu,12 +u

01(t, 1)x

u,12 +

01x

u,12 ZuZuu,12

= Zu(u,01 2xzu02(t,x, 1) +zu01(t, 1)xu,02 ), (6.5)u,12 |Zu=0,= 0, u,12 |t=0= 0.

Solvability and regularity of linear parabolic systems such as (6.4) and

(6.5) is well known and can be established here following arguments similar

to those in the Appendix for the zero-order correction. The first order com-

patibility condition (2.6) between initial and boundary data and the force is

used to improve the regularity of the correctors; for instance, we can show


16/22


thatt01 L(0, T; L2(0,))andxxx12 L2(0, T; H2()). We omit

the proof and refer to [25] for more details.

As in Section 3, it is convenient for the analysis to modify the definitionof the approximate solution uapp,1 so that the boundary conditions are ex-

actly met. We therefore define a truncated approximation uapp,1(t,x,z) =(uapp,11 (t, z),u

app,12 (t,x,z))by

uapp,11 (t, z) :=u

01(t, z) +(z)

01(t,

z

) +(1 z)u,01 (t,1 z

), (6.6a)

uapp,12 (t,x,z) :=u

02(t,x,z) +(z)(

02(t,x,

z

) +

12(t,x,z

))

+(1

z)(u,0

2 (t,x,

1 z

) +

u,1

2 (t,x,

1 z

)), (6.6b)

where is the cut-off function used in Section 4. We note that, since11 =u11= 0,u

app,11 uapp1 in equation (4.1a).

Then uapp,1 satisfies the following system

tuapp,11 zz uapp,11 =f1+A+B,

tuapp,12 + u

app,11 xu

app,12 xxuapp,12 zz uapp,12 (6.7)

=f2+D+ E+ F+ G,

whereA, B andD are as in (4.3a), (4.3b) and (4.3c), respectively, and E,F and Gare given by

E=

[(z)((z) 1)01x12+ (1 z)((1 z) 1)u,01 xu,12 2(z)Z02 2

(1 z)Zuu,02 ],

F =[(z)(Zx12zu

01(t, 0) +

1

2zzu

01(t, 0)Z

2x02+

1

201xzzu

02(t,x, 0)Z

2

xx

02) +(1

z)(

Zuzu

01(t, 1)x

u,12 +

1

2

zzu01(t, 1)(Z

u)2xu,02

+1

2u,01 x(zzu

02(t,x, 1))(Z

u)2 xxu,02 ) 2(z)Z

12

2(1 z)Zuu,12 (xxu02+zzu02)

02

(1 z)u,02 ],G=

3

2 [(z)12

(1 z)u,12 +(z)(1

2zzu

01(t, 0)Z

2x12 xx12)

+(1 z)( 12

zzu01(t, 1)(Z

u)2xu,12 xxu,12 ).


17/22


The system above is complemented by the following boundary and initial

conditions:

u

app,1

|t=0 = (a(z), b(x, z)),uapp,1|z=0=0, uapp,1|z=1=1.

(6.8)

Parallel to the analysis in Section 5, we define the approximation error

uerr(t,x,z) := u uapp,1.Then the error satisfies the following system

tuerr1 zz uerr1 = (A+B), (6.9)

tuerr2 + u

err1 xu

app,12 +u

1xu

err2 xxuerr2 zz uerr2

= (D+ E+ F+ G), (6.10)together with the following initial and boundary conditions

uerr

|z=0= 0, uerr

|z=1 = 0,uerr|t=0 = 0. (6.11)

Using the expansion (6.1), we can improve the convergence rate of The-

orem (5.2) under more regularity and compatibility conditions on the data.

Again, we do not optimize the regularity needed to establish the result.

Theorem 6.1. Letu0 Hm(), i H2(0, T; Hm()), i = 0, 1 andf L(0, T; Hm), m > 8, satisfy the compatibility conditions(2.5) and(2.6). Then, there exist positive constants C1, C2 independent of suchthat for any solution u of the system (2.4) with initial condition u0 and

boundary data i,

u uapp,1L(0,T;H1()) C1, (6.12)u uapp,1L((0,T)) C2 34 , (6.13)

whereuapp,1 is given in(6.6).

Parallel to Corollary 5.3, we also obtain optimal convergence rates.

Corollary 6.2. Under the hypotheses of Theorem 6.1, we have

C3

u u0 (z)0 (1 z)u,0L(0,T;H1) C4

,

whereC3andC4are constants depending on u0,i, i= 0, 1andf, but not

on.

In addition, we also improve the results in Corollary 5.4 as follows.

Corollary 6.3. Under the hypotheses of Theorem 5.2, there exists positive

constantsCi,i = 1, 2, independent of such that for any (0, 1)satisfy-ing

as 0,

u u0L(0,T;H1()) C1

,

u u0L((0,T))) C2 3

4 ,


18/22


where = [0, L] [, 1 ].We omit the proofs of Theorem 6.1 and Corollaries 6.2 and 6.3, which

are very similar to those in Section 4.

APPENDIX A. DECAY ESTIMATES OF THE CORRECTORS

In this appendix, we discuss the solvability of the systems of equations

satisfied by the correctors and establish their decay properties. For sim-

plicity, we will only state and prove the estimates needed in Section 5 and

6.

The correctors satisfy Prandtl-type effective equations and some of the

techniques we use are similar to those in [25], which deals with the reg-

ularity and the decay properties of solutions to the Prandtl-type equations

derived from the linearized compressible Navier Stokes equations.By symmetry in the problem between the lower and upper corrector, we

only deal with the correctors in the boundary layer at z = 0, 0 and 1.Below, we will be rather explicit in the dependence of constants on norms

of data in view of the applications to Section 5 and 6.

We begin by studying 01, which solves the following initial-boundaryvalue problem for the one-dimensional heat equation:

t01 ZZ01 = 0,

01|Z=0=01(t) u01(t, 0), 01|Z= = 0, (A.1)01|t=0

= 0.

Below, we denoteZ := Z2 + 1.Lemma A.1. Assume that01 L(0, T) andu01 L(0, T; H1(0, 1)).Then, for anyl Z+ there exists a constantCl > 0 depending on T suchthat

Zl01(L(0,T)) Cl(01L(0,T), u01(t, z)L(0,T;H1(0,1))), (A.2)Proof. We letG := 01() u01(0, )L(0,T).Then, by the maximum prin-ciple for the heat equation from (A.1), we have

01

L((0,T))

G.

Next, we define (t, Z) := GetZ. The function satisfies the followinginitial-boundary value problem for the heat equation on[0, +).

t ZZ= 0,|t=0=GeZ 0, |Z=0=Get > G, |Z= = 0. (A.3)

By the comparison principle for the heat equation it follows that

01L((0,T)) G eT. (A.4)


19/22


Then, for anyl 1,

Zl

0

1L((0,T)) GZl

eT

Z

L([0,+)) eT

0

1()u0

1(, 0)L(0,T),which gives the desired result. Above, we have used the property that

Zl eZ is uniformly bounded in Zfor anyl. We use Lemma A.1 to prove the decay properties of02. First, we recall

that02 satisfies

t02+

01x

02+ u

01(t, 0)x

02 ZZ02 = 01xu02(t,x, 0),

02|Z=0 = 02(t, x) u02(t,x, 0), 02|Z= = 0, (A.5)02|t=0= 0.

We begin with deriving bounds on Zlxxx02L(0,T;L2()),xxx02L((0,T)), and xxZ02L2(0,T;L2()), which are employed inSection 5.

Lemma A.2. Assume that02 H2(0, T; H5(0, L)), f2 L(0, T; H3+s(0, L)),s >0, u01 L(0, T; H1(0, L)) L([0, T)[0, L]), andu02 L(0, T; H5()).Then, for eachl Z+, there existsC1 > 0depending on f2L2(0,T;H3+s),u01L((0,T)(0,1)),02H1(0,T;H5(0,L)),u02L((0,T);H5()), T, and l, suchthat fori = 0, 1, 2, 3,

Z

lix

02

L(0,T;L2(

))+

Z

lixZ

02

L2(0,T;L2(

))

C1. (A.6)

Furthermore, for more regular data such thatu02L((0,T);H5()) andf2(t,x,z)L(0,T;H4+s()), s > 0, there exists a constant C2 > 0 suchthat fori = 0, 1, 2, 3,

Zlix02L((0,T)) C, (A.7)whereC2 depends on u01(t, z)L(0,T;H1(0,1)), 02H1+s(0,T;H5(0,L)),u02(t,x,z)L(0,T;H5+s()), f2(t,x,z)L(0,T;H4+s()),T, andl.Proof. Since ix

02, i= 1, 2, 3, satisfies the same equation as

02, with bound-

ary data given by derivatives ofi, i = 1, 2, we will establish only the

bounds for 02. Similar arguments apply for the derivatives provided thedata has sufficient regularity.

We will use again comparison estimates for parabolic equations. We

henceforth definew(t, Z) =02(t,x,Z) (02(t, x)u02(t,x, 0))eZ. Next,we observe that we have imposed enough regularity on the boundary data

and u0 such that u02 is a classical solution of (2.7) and we can extrapolate

the validity of this equation at z = 0 to obtain an equation satisfied byu02(t,x, 0)in t andx. Consequently, we derive the system satisfied by w as


20/22


follows:

tw+01xw+u

01(t, 0)xw ZZw=

01xu

02(t, , 0) +H,

w|Z=0 = 0, w|Z== 0, w|t=0= 0, (A.8)where

H(t,x,Z) := [t02(t, x) f2(t,x, 0) +01(t, Z)x02(t, x)01(t, Z)xu02(t,x, 0) +u01(t, 0)x02(t, x) +02(t, x) u02(t,x, 0)]eZ

acts as a forcing term.

Multiplying the first equation in (A.8) by Z2lwand integrating by partsover

gives

1

2

d

dtZlw2L2()+ ZlZw2L2()

Zlw2L2()+ u02(t,x, 0)H1(0,L)Zl01L2()ZlwL2()+ [t02L2(0,L)+ f2(t,x, 0)L2(0,L)+ |u01(t, 0)|02H1(0,L)+ (02H1(0,L)+ u02(t,x, 0)H1(0,L))

(Zl01L2(0,)+ ZleZL2(0,))]ZlwL2(),

where we have used the Sobolev inequality H1([0, L]) L([0, L]) andLemma A.1. By Gronwalls inequality, we then obtain that

ZlwL(0,T;L2())+ ZlZwL2(0,T;L2()) C(02H1(0,T;H1(0,L)), u02L2(0,T;H2(0,L)),

f2L2(0,T;L2()), u01L2(0,T;H1(0,L)), T , l),from which (A.6) easily follows.

Similarly, by multiplying both sides of the first equation in (A.8) by

p|w|p2w,p >2 and integrating by parts, we haved

dtwp

L

p

()+

4(p 1)p

Z|w|p

2

2L2

()

pu02(t, , 0)L(0,L)01Lp(0,)wp1Lp()+pHLp()wp1Lp()

.

We then conclude, by dividing for wp1Lp()

(which is finite by the regu-

larity of the system (A.8)), that

d

dtwLp() u02(t, , 0)L(0,L)01Lp(0,)+ HLp() (A.9)


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Taking the limitp and integrating in time yields similarly

wL((0,T)) C(01L(0,), u02L(0,T;H2+s()), 02H2((0,T)(0,L)),u01L((0,T;H1(0,L)), f2L((0,T;H1()), l).

Then the desired conclusion (A.7) follows.

Completely analogous results hold for u,01 and

u,02 , which we state below

for completeness.

Lemma A.3. Under the regularity assumptions on u0 andf2 of LemmasA.1 and A.2 and similar ones on12, it holds fori = 0, 1, 2, 3,

Zulu,01 (L(0,T)) C,Zulixu,02 L(0,T;L2())+ ZulixZu,02 L2(0,T;L2()) C,Zlixu,02 L((0,T)) C, (A.10)

where andCdepends on f2L(0,T;H4+s()), u01(t, z)L(0,T;H1(0,1)),12H2(0,T;H5(0,L)), u02(t,x,z)L((0,T);H5+s()),s >0,T, andl.

ACKNOWLEDGMENTS

The authors wish to thank the Institute for Mathematics and its Applica-

tions (IMA) at the University of Minnesota, where this work was partiallyconducted, for their hospitality and support. The IMA receives major fund-

ing from the National Science Foundation and the University of Minnesota.

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Boundar Layer Estimates for Channel Flow