Cauchy convergence schemes for some nonlinear partial ...negh/Preprints/[5] Cauchy...that the Galerkin solutions of certain nonlinear equations are Cauchy. Of course we do not avoid

Applicable AnalysisVol. 90, No. 1, January 2011, 85–102

Cauchy convergence schemes for some nonlinear partialdifferential equationsy

Nathan Glatt-Holtz* and Roger Temam

Department of Mathematics, The Institute for Scientific Computing and AppliedMathematics Indiana University, Bloomington 47405, USA

Communicated by Y. Xu

(Received 4 February 2010; final version received 21 February 2010)

Motivated by ongoing work in the theory of stochastic partial differentialequations we develop direct methods to infer that the Galerkin approx-imations of certain nonlinear partial differential equations are Cauchy (andtherefore convergent). We develop such a result for the Navier–Stokesequations in space dimensions two and three, and for the primitiveequations in space dimension two. The analysis requires novel estimates forthe nonlinear portion of these equations and delicate interpolation resultsconcerning subspaces.

Keywords: nonlinear PDE; primitive equations; Navier–Stokes; Galerkinapproximations; compactness methods; fractional spaces; interpolation;comparison estimates; stochastic PDE

AMS Subject Classifications: 35Q86; 35Q30; 46B70

1. Introduction

Galerkin approximations are commonly used to prove the existence of solutions of(linear and) nonlinear evolution equations. After constructing such approximations,one derives a priori estimates for these solutions. These a priori estimates produceweak convergence results which are sufficient to pass to the limit in the linear case. Inthe nonlinear case, one may pass to the limit by applying a classical compactnesscriterion [1,2]. In any case by proving that, in certain spaces, the Galerkinapproximations are (sub-sequentially) convergent we implicitly deduce that they areCauchy in the corresponding spaces.

In this article, we tackle the following unusual question, namely to prove directlythat the Galerkin solutions of certain nonlinear equations are Cauchy. Of course wedo not avoid the (classical) part of convergence concerning the a priori estimates.Besides its intrinsic interest, the question that we address is motivated by ongoingwork, [3,4], concerning nonlinear stochastic partial differential equations (SPDEs).

*Corresponding author. Email: [email protected] to Professor Vsevolod Alekseevich Solonnikov on the occasion of his 75thbirthday.

ISSN 0003–6811 print/ISSN 1563–504X online

! 2011 Taylor & FrancisDOI: 10.1080/00036811003735956http://www.informaworld.com

Downloaded By: [Glatt-Holtz, Nathan] At: 22:15 21 February 2011

For such systems, particularly those that are forced by nonlinear white noise driventerms, the a priori estimates may not be carried out ‘pathwise’, that is ! by !, in theunderlying probability space. The classical compactness methods are thereforecomplicated by the addition of stochastic terms and novel approaches are required inorder to pass to the limit. In the aforementioned articles [3,4], Cauchy convergence isestablished to overcome this difficulty. On this point the approach is very similar tothe deterministic case. As such we thought it was worthwhile to separate this analysiswhich has some intricate elements.

In what follows, and in parallel to the aforementioned articles [3,4], we prove theCauchy convergence of the Galerkin approximations for the 2D primitive equations(PEs) and for the Navier–Stokes equations in space dimensions 2 and 3. The resultthat we obtain is local in time for the PEs and 3D Navier–Stokes equations andglobal in time for the 2D Navier–Stokes Equations.

The analysis below is organized as follows: in Section 2, we recall the 2D PEs andtheir mathematical setting. In Section 3, we establish the Cauchy convergence. Due tothe discretization of the basic equations, stray elements depending on only one of theorders appear for these estimates. We address the magnitude of these terms by makinguse of the generalized Poincare inequality. The analysis for this point requires bothnovel estimates on the nonlinear portion of the PEsH1 (presented in Lemma 2.1), andthe usage of fractional order spaces to avoid difficulties with the boundary. Section 4addresses this point and makes extensive application of delicate interpolation resultsfrom [5]. Section 5 establishes similar results for the Navier–Stokes equations.

2. Review of the 2D PEs and their mathematical setting

The PEs are widely regarded as a fundamental description of geophysical scale fluidflow. They provide the analytical core of large general circulation models (GCMs)that are at the forefront of numerical simulations of the earth’s oceans andatmosphere (see e.g. [6]). Beyond their significance in geophysics, these systems havean intricate analytical structure that has led to a large body of work in themathematics community. See [7–15] and the recent surveys [16,17].

The two-dimensional version of these equations takes the form:

@tu! u@xu! w"u#@zu$ !Du$ f v! @xps $ "Tg#0

Z 0

z@xTd !z

% Fu ! $u"v,T # _W1, "2:1a#

@tv! u@xv! w"u#@zv$ !Dv! f u % Fv ! $v"v,T # _W2, "2:1b#

w"u# %Z 0

z@xu d !z,

Z 0

$hu dz % 0, "2:1c#

@tT! u@xT! w"u#@zT$ %DT % FT ! $T "v,T # _W3: "2:1d#

Here (v,w)% (u, v,w), T denote, respectively, the flow field and the temperature of thefluid being modelled, v is the horizontal velocity. The coefficients !, % account for themolecular viscosity and the rate of heat diffusion, g is the gravitational constant and #0

86 N. Glatt-Holtz and R. Temam


represents the mean density of the fluid; f, which is a function of the earth’s rotationand the local latitude is taken constant here for simplicity. The terms Fu, Fv and FT

correspond to external sources of horizontal momentum and heat. They vanish in theocean (and Fu, Fv in the atmosphere); they are added for mathematical generality or toconsider nonhomogeneous boundary conditions (which we do not do here).

We consider the evolution of (2.1) over a rectangular domain M! (0,L)"(#h, 0) and label the parts of the boundary !i! (0,L)" {0}, !b! (0,L)" {#h} and!l! {0,L}" (#h, 0). We posit the physically realistic boundary conditions

@zv$ !vv ! 0, w ! 0, @zT$ !TT ! 0, on !i, %2:2a&

v ! 0, @xT ! 0, on !l, %2:2b&

v ! 0, w ! 0, @zT ! 0, on !b: %2:2c&

The equations and boundary conditions (2.1), (2.2) are supplemented by initialconditions for u, v and T, that is

u ! u0, v ! v0, T ! T0, at t ! 0: %2:3&

For a detailed treatment of the physical derivation and significance of (2.1), (2.2)see [18,19].

We next review the mathematical setting for (2.1)–(2.3). For the most part ourpresentation and notations follow the recent survey [16], to which we refer the readerfor a more detailed treatment.

The main function spaces used are defined as follows. Take:

H :! U ! %u, v,T &2L2%M&3 :Z 0

#hu dz ! 0

! ":

We equip H with the inner product1

%U,U ]& :!Z

Mv ' v]dM$

Z

MTT ]dM, U ! %v,T &, U ] ! %v],T ]&:

Here and below we shall make use of the vertical averaging operator P" ! 1h

R 0#h "% "z&d "z

and its orthogonal complement Q"!"#P". Note that the projection operator # :L2(M)3!H may be explicitly written according to U ! (Qu, v,T ). We also define

V :! U ! %u, v,T &2H1%M&3 :Z 0

#hu dz ! 0, v ! 0 on !l [ !b

! ":

Here we take the inner product ((', '))! #((', '))1$$((', '))2 where for given U! (v,T ),U ]! (v ],T ])2V,

%%U,U ]&&1 :!Z

M@xv ' @xv] $ @zv ' @zv] dM$ !v

Z

!i

v ' v] dx,

%%U,U ]&&2 :!Z

M@xT @xT

] $ @zT @zT] dM$ !T

Z

!i

TT ] dx:

Applicable Analysis 87


Note that under these definitions a Poincare type inequality jUj!CkUk holds for allU2H1(M)3"V. Moreover the norms k # kH1, k # k may be seen to be equivalent overall of H1(M)3.

Even if U is very regular, several terms in the abstract formulation of (2.1) do notbelong to V (see (2.11), (2.14), (2.15) below). As such, we shall also make use of theadditional auxiliary spaces:

~V :$ U $ %u, v,T &2H1%M&3 :Z 0

'hu dz $ 0, v $ 0 on !l

! ",

Z :$ U $ %u, v,T &2H1%M&3 : v $ 0 on !l

# $:

As for V we endow both spaces with the norm k # k. One may verify that " also mapsZ onto ~V and is continuous on H1(M)3.

Some intermediate order space are employed in the analysis below. Following thenotation of [5, Chapter 1, Section 2.1] we define, for s2 [0, 1],

~Vs $ ( ~V,H )1's, Vs $ (V,H )1's: %2:4&Proposition 2.1 establishes the result important for our analysis that Vs $ ~Vs fors2 (0, 1/2).

Finally we take V(2)$H 2(M)3\V and equip this space with the classical H 2(M)norm which we denote by j # j(2).

The linear second-order terms in the equation are captured in the Stokes-typeoperator A which is understood as a bounded operator from V to V 0 viahAU,U ]i$ ((U,U ])). The additional terms in the variational formulation of thisportion of the equation term capture the Robin boundary condition (2.2a). Theymay be formally derived by multiplying '!Du, '!Dv, '"DT in (2.1a), (2.1b), (2.1d)by test functions u ], v ], T ], integrating over M and integrating by parts. We shallmake use of the subspace D(A)*V(2) given by

D%A& $ fU $ %v,T &2V%2& : @zv+ #vv $ 0, @zT+ #TT $ 0 on !i,

@xT $ 0 on !l, @zT $ 0 on !b, g:On this space we may extend A to an unbounded operator by defining

AU $'!QDu'!Dv'"DT

0

B@

1

CA, U2D%A&: %2:5&

Since A is self adjoint, with a compact inverse A'1 :H!D(A) we may apply thestandard theory of compact, symmetric operators to guarantee the existence of anorthonormal basis {#k}k,0 for H of eigenfunctions of A with the associatedeigenvalues {$k}k,0 forming an unbounded, increasing sequence. Note that by theregularity results in [20] or [21] we have #k2D(A)*V(2). Define

Hn $ spanf#1, . . . ,#ng:Take Pn and Qn$ I'Pn to be the projections from H onto Hn and its orthogonalcomplement, respectively. For all m4n we let P n

m $ Pm ' Pn.We shall also make use of the fractional powers of A. Given #40, take

D%A#& $ U2H :X

k

$2#k jUkj2 51( )

, %2:6&



where Uk! (U,!k). On this set we may define A! according to

A!U !X

k

"!kUk!k, for U !X

k

Uk!k: "2:7#

Accordingly we equip V!!D(A!/2) with the norm

jUj! ! jA!=2Uj !X

k

"!kjUkj2 !1=2

: "2:8#

It is direct to verify that D(A1/2)!V and that the associated norms k $ k and j $ j1 areidentical. Moreover, by exploiting the regularity theory developed in [20], we mayalso infer the equivalence of j $ j(2) and j $ j2 on D(A). Classically, we have thegeneralized (and reverse) Poincare estimates

jPnUj2!2 % "!2&!1n jPnUj2!1 , jQnUj2!1 %

1

"!2&!1n

jQnUj2!2 , "2:9#

for any !15!2.The following result, whose proof we postpone to Section 4, is used in Theorem 3.1.

PROPOSITION 2.1 Let !2 (0, 1/2). Then V!! ~V! !D"A!=2# and therefore the estimate

jUj! % jUj1&!kUk! "2:10#

holds for all U2 ~V

Note that in some previous works, the second component of the pressure(cf [16, Section 2]) is included in the definition of the principal linear operator A.Since this breaks the symmetry of A we relegate such terms to a separate, lower orderoperator Ap :V!H defined by

ApU !&Q

R 0z @xT d "z00

0

@

1

A: "2:11#

If U2D(A), ApU2 ~V and we have the estimates

jApUj % ckUk, kApUk % cjUj"2#: "2:12#

We next capture the nonlinear portion of (2.1). In accordance with (2.1c) and(2.2) (w! 0 on #i) we define the diagnostic function:

w"U# ! w"u# !Z 0

z@xu d "z, U ! "u, v,T #2V: "2:13#

For U! (v,T )2V, U ]! (v ],T ])2V(2) we take B(U,U ])!B1(U,U ])'B2(U,U ])where

B1"U,U ]# :!Q"u@xu ]#u@xv ]

u@xT ]

0

B@

1

CA !B11"u, u ]#

B21"u, v ]#

B31"u,T ]#

0

B@

1

CA "2:14#



and

B2!U,U ]" :#Q!w!u"@zu ]"w!u"@zv ]

w!u"@zT ]

0

B@

1

CA #B12!u, u ]"

B22!u, v ]"

B32!u,T ]"

0

B@

1

CA: !2:15"

The following lemma summarizes some properties of B needed in the sequel.2

LEMMA 2.1 B is well-defined as a bilinear continuous map from V$V(2) into H and:

(i) for any U2V, U ]2V(2) and U[2H

jhB!U,U ]",U [ij % ckUkkU ]k1=2jU ]j1=2!2" jU[j: !2:16"

In particular, for U2V(2),

jB!U"j2 % kUk3jUj!2"; !2:17"

(ii) for U2V(2), B!U"2 ~V & ~V1=4 # V1=4 # D!A1=8" and satisfies the estimate

kB!U"k2 %ckUkjUj3!2": !2:18"

Proof The estimate (2.16) may be established with the now standard anisotropictype techniques. See [21] or [14]. To the best of our knowledge, the H1 estimate for Bgiven in (ii) is new. To prove it, we observe that, for U# (u, v,T )2V(2),B(U )#!B(U ) where

B!U" #u@xu' w@zu

u@xv' w@zv

u@xT' w@zT

0

B@

1

CA, !2:19"

and ! is the orthogonal projector from L2(M)3 onto H. Note that ! is alsocontinuous from H1((M))3 into itself (see e.g. [22] for the similar but more involvedresult for the Navier-Stokes equations). Hence it suffices, to prove (2.17), to showthat kB(U )k is bounded by the right-hand side of (2.17) for a suitable constant c.

To estimate the L2-norm of the gradient of B(U ) we ought to estimate theL2-norm of terms like

u@2u, !@u"2, !@w"!@u", w@2u, !2:20"

where @# @x or @z (i.e. @# @/@x or @/@z). The most delicate terms are terms like(@w)(@u) and w @2u that we estimate using the anisotropic inequalities as in [16,21].Hence for e.g. (@w)(@u):

j@w@ujL2!M" #Z

M!@w"2!@u"2 dM

%Z L

0j@wj2L1

zj@uj2L2

zdx

%Z L

0j@wj2L1

zdx

! "j@uj2L2

zL1x:



Using the expression (2.13) of w!w(u), for respectively @! @z, @x, we bound the firstterm above by j@xujL2(M)j@xzujL2(M) or by hj@xxuj2L2"M#, which in both cases aredominated by jUj2"2#:

The term j@uj2L2zL

12is estimated as follows [16,21]:

j@ujL2zL

1x! j!jL1

x$ cj!j1=2L2

xj@x!j1=2L2

x, where ! !

Z 0

%hj@uj2 d !z

! "1=2

:

Hence, pointwise,

@x! !1

!

Z 0

%h@u@x@u d !z ) j@x!j $

Z 0

hj@x@uj2 d !z

! "1=2

,

and

j@x!jL2x$ j@2ujL2"M#:

In the end

j@ujL2zL

11$ cj@uj1=2L2"M#j@

2uj1=2L2"M# $ ckUk1=2jUj1=2"2# ,

and j@w@ujL2(M) is bounded by a term like the right-hand side of (2.17).The proof is similar for a term like w@2u, and, as we said, the other terms are

easier to handle. g

We capture the Coriolis forcing with the bounded operator E :H!H given by

EU :!%Q f v

f u

0

0

B@

1

CA: "2:21#

We observe that E is also continuous from V to ~V and that

jEUj $ f jUj, kEUk $ f kUk: "2:22#

For the external forcing terms Fu, Fv, FT we let:

F !QFu

Fv

FT

0

B@

1

CA:

For Theorem 3.1 below we assume F 2L2loc"&0,1#,H# and that U0! (u0, v0,T )2V.

With the definitions (2.5), (2.11), (2.14), (2.15) and (2.21) we may reformulate (2.1) asan abstract equation:

dU

dt' AU' ApU' B"U# ' EU ! F, U"0# ! U0: "2:23#

We finally recall the definition of the Galerkin approximations associated with (2.23).We say that, U (n)2C([0,1); Hn) is a solution of the Galerkin system of order n if:

dU"n#

dt' AU "n# ' Pn"ApU

"n# ' B"U "n## ' EU "n## ! PnF

U "n#"0# ! PnU0: "2:24#g



3. Comparison estimates for the Galerkin system

In this section, we establish that the solutions of (2.24) are ‘locally Cauchy’ in strongtype spaces. The result (and proof) may be seen as an alternative to the classicalAubin compactness theorem [1] although this is not our primary motivation. Note inthis connection that we infer the convergences of the Galerkin approximationswithout any estimates on dU (n)/dt.

THEOREM 3.1 Suppose that F2L2loc!"0,1#,H#, U02V. Then there exists t0$ 4 0

independent of n and depending only on the data, such that, {U (n)}n%1, the solutions of(2.24), are Cauchy (and hence convergent) in C!"0, t0$&;V # \ L2!"0, t0$&;D!A##

Proof By taking the inner product of (2.24) with AU (n) and applying (2.12), (2.16),(2.22) we classically infer that

d

dtkU !n#k2 ' jAU !n#j2 ( c1!1' jPnFj2 ' kU !n#k6#: !3:1#

With kU (n)(0)k) kPnU0k( kU0k and jPnF j( jF j, we infer from (3.1) that1' kU (n)(t)k2( 2(1' kU0k2) for an initial period during which

d

dtkU !n#k2 ( c1!jFj2 ' !1' kU !n#k2#3#,

so that

1' kU !n#!t#k2 ( 1' kU0k2 ' c1

Z t

0jF!s#j2 ds' 23c1t!1' kU0k2#3,

and thus 1' kU (n)(t)k2( 2(1' kU0k2) holds until the time t0$ where

c1

Z t0$

0jF!s#j2 ds' 23c1t

0$!1' kU0k2#3 ) 1' kU0k2: !3:2#

Returning to (3.1) we obtain the classical a priori estimates which we write inthe form:

supn

supt2 "0,t0$&

kU !n#k2 'Z t0$

0jAU !n#j2 dt

!

51: !3:3#

Fix m4n and denote R(m,n))U (m)*U (n). Subtracting the equations for U (m) andU (n) and taking an inner product with AR(m,n) gives

1

2

d

dtkR!m, n#k2 ' jAR !m, n#j2

) *hPmB!U !m## * PnB!U !n##,AR !m, n#i* hPmApU

!m# * PnApU!n#,AR !m, n#i

* hPmEU!m# * PnEU

!n#,AR !m, n#i' hP n

mF,AR!m, n#i: !3:4#

We consider first the nonlinear term:

hPmB!U !m## * PnB!U !n##,AR !m, n#i) hB!R!m, n#,U !m## ' B!U !n#,R!m, n## ' P n

mB!U!n##,AR !m, n#i

:) J1 ' J2 ' J3: !3:5#



For J1, (2.16) yields the estimate

jJ1j ! ckR"m, n#kkU "m#k1=2jU "m#j1=2"2# jAR"m, n#j

! 1

10jAR"m, n#j2 $ cjAU "m#j2kR"m, n#k2: "3:6#

On the other hand, (2.16) also implies

jJ2j ! CkU "n#kkR"m, n#k1=2jR"m, n#j1=2"2# jAR"m, n#j

! 1

10jAR"m, n#j2 $ ckU "n#k4kR"m, n#k2: "3:7#

Finally, by applying (2.9), (2.10) with !% 1/4 and then making use of (2.17), (2.18)we estimate the third term

jJ3j !1

10jAR"m, n#j2 $ cjP n

mB"U"n##j2

! 1

10jAR"m, n#j2 $ c

"1=4n

jB"U "n##j21=4

! 1

10jAR"m, n#j2 $ c

"1=4n

jB"U "n##j3=2kB"U "n##k1=2

! 1

10jAR"m, n#j2 $ c

"1=4n

jB"U "n##jkB"U "n##k

! 1

10jAR"m, n#j2 $ c

"1=4n

"kU "n#k3=2jAU "n#j1=2#"kU "n#k1=2jAU "n#j3=2#

! 1

10jAR"m, n#j2 $ c

"1=4n

kU "n#k2jAU "n#j2: "3:8#

At this juncture we underline that the second inequality is justified sinceB"U "n##2 ~V1=4%V1=4%D"A1=8# (see Proposition 2.1).

For the terms involving the Coriolis operator E, we make a second application of(2.10) with !% 1/4 and then apply (2.22) to observe:

hPmEU"m# & PnEU

"n#,AR"m, n#ij

! jhER"m, n#,AR"m, n#ij$ jhPnmEU

"n#,AR"m, n#ij

! c"jR"m, n#jjAR"m, n#j$ jQnEU"n#jjARm, nj#

! 1

10jAR"m, n#j2 $ c"kR"m, n#jj2 $ jQnEU

"n#j2#

! 1

10jAR"m, n#j2 $ c"kR"m, n#k2 $ 1

"1=4n

jEU"n#j21=4#

! 1

10jAR"m, n#j2 $ c"kR"m, n#k2 $ 1

"1=4n

jEUj3=2kEU"n#k1=2#

! 1

10jAR"m, n#j2 $ c"kR"m, n#k2 $ 1

""1=4#n

kU"n#k2# "3:9#



Along the same lines, but more involved, the term involving Ap is estimated with(2.10), (2.12)

jhPmApU!m" # PnApU

!n", AR!m, n"ij

$ jhApR!m, n",AR!m, n"ij% jh!Pm # Pn"ApU

!n",AR!m, n"ij

$ 1

10jAR!m, n"j2 % ckR!m, n"k2 % jhA1=8ApU

!n", !Pm # Pn"A7=8R!m, n"ij

$ 1

10jAR!m, n"j2 % ckR!m, n"k2 % jApU

!n"j1=4jAR!m, n"j

$ 1

10jAR!m, n"j2 % ckR!m, n"k2 % kApU

!n"k 1

!1=8n

jAR!m, n"j"

$ 1

10jAR!m, n"j2 % ckR!m, n"k2 % 1

!1=8n

jU!n"j!2"jAR!m, n"j

$ 2

10jAR!m, n"jAR!m, n"j2 % c kR!m, n"k2 % 1

!1=4n

jAU!n"j2! "

: !3:10"

Once again the second inequalities in both (3.9), (3.10) are justified since EU (n),ApU

(n)2D(A1/4). Summing up the estimates above we conclude:

d

dtkR!m, n"k2 % jAR !m, n"j2

$ c!1% jAU !m"j2 % kU !n"k4"kR!m, n"k2

% c

!1=4n

!1% kU !n"k2"!1% jAU !n"j2" % cjP nmFj

2

:& "m,nkR!m, n"k2 % Gm,n: !3:11"

By applying (3.3) we observe that

supm,n

Z t0'

0"m,n ds51, !3:12"

and

limm,n!1

Z t0'

0Gm,n ds & 0: !3:13"

We therefore drop the jAR(m,n)j2 term in (3.11) and with the Gronwall lemma,we conclude that for every t $ t0',

kR!m, n"!t"k2 $expZ t

0"m,n ds

! "kP n

mU0k2 %Z t

0Gm,n ds

! "

$c kP nmU0k2 %

Z t0'

0Gm,n ds

! ": !3:14"



This proves that U (n) is Cauchy in C!"0, t0#$;V %. Returning to (3.11) and integratingfrom 0 to t0# yields the estimate

Z t0#

0jAR!m, n%j2 & kP n

mU0k2 ' supt2 "0,t0#$

kRm,nk2 !Z t0#

0!m,n ds'

Z t0#

0Gm,n ds: !3:15%

We conclude that U (n) is Cauchy in L2!"0, t0#$;D!A%%. The proof is complete. g

4. Interpolation results

In this section we prove Proposition 2.1. The work consists in showing theequivalence of the spaces V2", ~V2", D(A") for "2 [0, 1/4]. This follows as a directconsequence of two technical results, Propositions 4.1 and 4.2 which we establishbelow. The inequality (2.10) is merely a restatement of the classical interpolationinequality [5, Proposition 2.3].

The first result is the following:

PROPOSITION 4.1 Let

Z :( U ( !u, v,T %2H1!M%3 : v ( 0 on !l

! ",

Zb :( U ( !u, v,T %2H1!M%3 : v ( 0 on !l [ !b

! ", !4:1%

and define, for s2 [0, 1],

Hs0!M%3 :( "H1

0!M%3,L2!M%3$1)s,

Zsb :( "Zb,L

2!M%3$1)s,

Zs :( "Z,L2!M%3$1)s,

Hs!M%3 :( "H1!M%3,L2!M%3$1)s:

Then,

Hs0!M%3 * Zs

b * Zs * Hs!M%3, !4:2%

with the spaces being equipped with equivalent norms. Moreover, for all s2 (0, 1/2),

Hs0!M%3 ( Zs

b ( Zs ( Hs!M%3: !4:3%

Proof The proof draws on classical results in [5]. If @M were smooth, the main step,to establish (4.4), would follow precisely from [5, Theorem 11.1]. Here, due to its fourcorners, M is not smooth and further analysis is required.

The inclusions (4.2) are a direct consequence of the definitions. Let D(M) be thecollection of all C1 functions with compact support in M. Since,

D!M%+H10!M%+Hs

0!M%+Hs!M%,

the second point (4.3) follows if we can show that

D!M% is dense in Hs!M%: !4:4%



To this end fix u2Hs(M). We consider a particular covering of M by O0!M,and by balls Oj, j" 1, centred on the boundary @M. We first choose four balls O1,O2, O3 and O4 centred at the corners, each with radius small enough so as to avoidtouching the opposite sides. Then, if necessary, we choose further balls all of whichavoid the corners to complete the covering. Let 1!

Pj !j be a partition of unity

subordinated to {Oj}j. We need to show that the elements !1u, !2u, !3u, !4u, thosecorresponding to the corners, may be approximated by functions in D(M). Anyremaining elements !ju are treated exactly as in [5].

Take g!1u to be the extension of !1u by zero outside of M. As a consequence ofLemma 4.1 established below, we see that g!1u is in Hs(R2). Consider now a smoothmollifier function " with support included in the ball B :!B((#1, #1), 1/4) centred at(#1, #1) with radius 1/4 (see e.g. [23] for basic properties). As #! 0 we have the "# $v! v in H1(R2) (resp. L2(R2)), for every v2H1(R2), (resp. L2(R2)). Hence, byinterpolation, "# $ v converges to v in Hs(R2), for every v2Hs(R2). Moreover wehave that supp("# $ v) is included in supp("#)% supp(v) [24]. Note that "# is supportedby B#!B((##,##), #/4). In particular, we have "# $ g!1u converges to g!1u in Hs(Rd)

and that (for all sufficiently small values of #) "# $ g!1u is a family of smooth

functions, compactly supported in M. We conclude that "# $ g!1ujM 2D&M' and

converges to g!1ujM!!1u in Hs(M).With similar arguments we may treat the remaining corner elements !2u, !3u, !4u.

The proof is therefore complete. g

The proof of Proposition 4.1 required the following Lemma. An analogous resultfor the case of smooth domains was established in [5, Theorem 11.4].

LEMMA 4.1 Suppose 0( s51/2. The extension ~v of any element v2Hs(M) to be zerooutside M defines a linear continuous operator mapping from Hs(M) into Hs(R2).

Proof We proceed by making some judicious applications of certain extension andrestriction operators. Given v2H1(M) we define Ev on (#L, 0)) (#h, 0) by setting

Eu&x, z' ! u&#x, z', # L5 x5 0, # h5 z5 0:

By similar reflections we are able to define Ev on the strip Rx) (#h, 0). Let N 1 beany open bounded set with smooth boundary so that M*N 1*Rx) (#h, 0). It isclear that E is linear continuous from H1(M) to H1(N 1) and from L2(M) to L2(N 1).Hence, by interpolation, E is continuous from Hs(M) to Hs(N 1). Next forv2Hs(N 1) define S1v to be the extension of v by zero outside of N 1. By [5, Theorem11.4], S1 continuously maps Hs(N 1) into Hs(Rx)Rz). Now take N 2 to be openbounded with smooth boundary so that M*N 2* [0,L])Rz. We define R to be therestriction operator to N 2. Observe, again by interpolation, that R is a continuousmap from Hs(Rx)Rz) into Hs(N 2). Finally, for v2Hs(N 2) we define S2v to beextension of v by zero outside of N 2. Again by [5, Theorem 11.4], S2 is linearcontinuous from Hs(N 2) into Hs(Rx)Rz). Summarizing we see that S2RS1E islinear and continuous from Hs(M) into Hs(R2). With the observation thatS2RS1Ev ! ~v we conclude the desired result. g

We come to the second proposition which is an application of [5, Theorem 14.3].We note that this result relies on holomorphic interpolation which, for our context,gives the same spaces as Hilbert interpolation [5, Theorem 14.1, Remark 14.2].



PROPOSITION 4.2 For all s2 [0, 1],

Vs ! Zsb \H, ~Vs ! Zs \H: "4:5#

Remark 4.1 We may endow both Z and Zb with the k $ k norm defined in Section 2.As noted above ! is continuous on H1"M#3 % Z,Zb,H1

0"M#. This observation isused in the proof of Lemma 4.2.

Proof In this proof we follow closely the notations used in [5]. We begin byaddressing Vs. We define

X ! Zb, Y ! " ! L2"M#3, "4:6#

and observe that these spaces satisfy (14.18) of [5]. We take @ :! I&!, recalling that! is the projection operator from L2(M) onto H (see Section 2). Let

X ! f0g ! Y, # ! L2"M#d:

Trivially, Y,X ' #, are each Banach and @2L(",#) which are the requirements of(14.19), (14.20) in [5]. Following the notations set in (14.21), (14.22) of [5] weobserve that

"X#@,X ! fU2Zb : @U ! 0g ! fU2Zb : !U ! Ug ! Zb \H ! V, "4:7#

and

"Y#@,Y ! fU2L2"M#3 : !U ! Ug ! H: "4:8#

It remains to address (14.23), the remaining condition necessary to apply[5, Theorem 14.3]. We take

~X :! @Zb~Y :! @L2"M#d ! H? "4:9#

and observe that each space is Banach. This is clear for ~Y and for the space ~X wemake use of the fact that @ is a projection operator, i.e. that @2U! @U over allU2L2(M)3. With this in mind, suppose that {Un}n(0 is a convergent sequence in @Zb

and denote the limit by U. Clearly there must exist a sequence fU ]ngn(0 in Zb such

that @U ]n ! U (with the convergence understood in the topology defined by k $ k).

Since @! I&! is continuous onH1(M)3, we infer that @2U ]n ! @U. Thus, since @ is a

projection this in turn implies @U ]n ! @U so that @U!U. We thus infer that @Zb is

equal to its closure and hence is Banach. Since @ is continuous both in L2(M)3 andH1(M)3 it follows directly that @2L"X, ~X# \ L"Y, ~Y#. This is the requirement of(14.23), (ii). For (14.23), (iii) we take G :! @ and r :! 0. With these definitions one maysee by inspection that the identity @G!!!) r!, holds for every !2 ~X) ~Y.

Gathering the above observations we now infer from Theorem 14.3 that

Vs ! *V,H +1&s ! *"X#@,X, "Y#@,X+1&s

! "*X,Y+1&s#@,*X,Y+1&s! "*Zb,L

2"M#3+1&s#@,X ! "Zsb#@,X

! fU2Zsb : !U!Ug!Zs

b \H: "4:10#

To address ~Vs the proof follows exactly as above by replacing Z by Zb in (4.6),(4.7), (4.9), (4.10) and by replacing ~V by V in (4.7), (4.10). g



5. Cauchy convergence for the Navier-Stokes equations

We finally consider the Navier-Stokes equations in space dimension d! 2, 3, on abounded domain M:

@tU" #U $ r%U& !DU" rp ! F, #5:1a%

r $U ! 0, #5:1b%

U#0% ! U0, #5:1c%

UjM ! 0: #5:1d%

The system (5.1) describes the flow of a viscous incompressible fluid. HereU! (u1, . . . , ud), p and ! represent the velocity field, the pressure and the coefficientof kinematic viscosity, respectively. We assume that M has a smooth boundary @M.

We begin by recalling the abstract setting for (5.1). See e.g. [22] for a detailedtreatment. Let H :! {U2L2(M)d : r $U! 0, U $ n! 0}, where n is the outer pointingnormal to @M; H is endowed as a Hilbert space with the L2 inner product and norm.The Leray–Hopf projector, PH, is defined as the orthogonal projection of L2(M)d

onto H. Also define V :! fU2H10#M%d : r $U ! 0g, with the inner product

((U,U ]))!RM rU $rU ] dM. Due to the Dirichlet boundary condition, (5.1d), the

Poincare inequality jUj' ckUk holds for U2V justifying this defintion.The linear portion of (5.1) is captured in the Stokes operator A!&PH D, which is

an unbounded operator from H to H with the domain D(A)!H 2(M)\V. As above,one can prove the existence of an orthonormal basis {!k}k(0 for H of eigenfunctionsof A with the associated eigenvalues {"k}k(0 forming an unbounded increasingsequence. Define Hn! span{!1, . . . ,!n} and take Pn to be the projection from Honto this space. We let Qn :! I&Pn and Pm

n :! Pm & Pn for every mn.As above (cf (2.6), (2.7), (2.8)) we may define the fractional powers A#. We

denote the domains of these operators by D(A#) and associate norms j $ j2#! jA# $ j.With these definitions and notations the generalized direct and inverse Poincareestimates hold as in (2.9). Also in parallel to the presentation above we let~V ! H \H1#M%d and define the intermediate spaces ~Vs ! ) ~V,H *1&s, Vs! [V,H]1&s.As previously in Lemma 2.1 we see that

V2# ! ~V2# ! D#A#% for all #2 #0, 1=4% #5:2%

and that

jUj# ' jUj1&#kUk#H1#M% whenever U2 ~V: #5:3%

Indeed, due to [5, Theorem 11.1] we have that

Hs0#M%d ! Hs#M%d

for all s2 [0, 1/2]. As such (5.2) and (5.3) follow directly once we establish that, fors2 (0, 1)

~Vs ! Hs#M% \H, Vs ! Hs0#M% \H: #5:4%



Perusing the proof of Lemma 4.2 we see that (5.4) is established in exactly the samemanner modulo with some minor notational changes.

The nonlinear portion of (5.1) is given by B!U,U ]" :# PH!!U $ r"U ]" #PH!

Pdj#1 uj@jU

]", which is defined for U# (u1, . . . , ud)2V and U ]2D(A). We usethe following properties of B.

LEMMA 5.1 Suppose d# 2 or 3.

(i) B is bilinear and continuous from V%D(A) to H. If U2V, U ]2D(A), andU[2H, then

!B!U,U ]",U ["!! !! & c jUj1=2kUk1=2kU ]k1=2jAU ]j1=2jU [j in d # 2,

kUkkU ]k1=2jAU ]j1=2jU [j in d # 3:

"!5:5"

(ii) If U2D(A), then B!U"2 ~V, and for d# 2, 3 we have,

kB!U"k2H1!M" & ckUkjAUj3 ' jUj1=2jAUj7=2, !5:6"

and, for every s2 (0, 1/2),

jB!U"j2~Vs& jB!U"j2!1(s"kB!U"k2s

H1!M"d

& c!kUk3(2sjAUj1'2s ' jUjs=2kUk3(3sjAUj1'!5=2"s": !5:7"

Remark 5.1 It is incorrectly stated in [4] Lemma 2.2 (iii) that B(U )2V whenu2D(A). Indeed B(U )# (U $r)U(rq where q# q(U ) is the unique element inH1(M) such that (U $r)U(rq2H. Hence rq and thus B(U ) do not necessarilyvanish on @M, so that B(U ) is not necessarily in V. This oversight inval-idates the estimate (3.17) upon which (3.18) and thus the conclusion of [4,Proposition 3.1(i)] rely.

Fortunately the estimate in [4, Lemma 2.2] can be replaced by (5.6) given here.Indeed the proof of [4, Lemma 2.2] gives valid estimates similar to (5.6) for (U $r)Uin H1(M)d. We know moreover that the Leray–Hopf projection of L2(M)d onto H iscontinuous in H1(M)d [22]. We may thus infer that B(U ) is also in H1(M)d andsatisfies the same estimates, proving (5.6). Using (5.2) and (5.3), (5.7) followsimmediately from (5.6)

In Theorem 5.1 we establish (5.12) which may be taken as a replacement of (3.17)in [4, Proposition 3.1(i)]. With minor changes in (3.18) this fixes the oversight in thisprevious work.

With the definitions for A and B and assuming that F 2L2loc!)0,1";H" and that

U02V we formulate (5.1) as the abstract evolution on H:

dU

dt' AU' B!U" # F U!0" # U0: !5:8"

As above we define the associated Galerkin approximations as the solutionsUn2C([0,1); Hn) of

dU!n"

dt' AU !n" ' PnB!U !n"" # Pn F U !n"!0" # PnU0: !5:9"

Similarly to Section 3 we have the Cauchy convergence of {U (n)}n*1.



THEOREM 5.1 Suppose d! 2, 3 and that F 2L2loc"#0,1$;H$, U02V. Then there

exists t0% 4 0 such that {U (n)}n&1 is Cauchy in C"#0, t0%';V $ \ L2"#0, t0%';D"A$$. In thecase d! 2 the result is global; more precisely {U (n)}n&1 is Cauchy inC"#0, t0%';V $ \ L2"#0, t0%';D"A$$ for every t0% 4 0.

Proof Multiplying (5.9) by AU (n), integrating and applying (5.5) leads to theestimates,

d

dtkU "n$k2 ( jAU "n$j2 ) c 1( jPnFj2 ( jU "n$j2kU "n$k4 d=2,

1( jPnFj2 ( kU "n$k6 d=3.

!"5:10$

From these inequalities we infer the existence of t0% 4 0 which satisfies uniformbounds as in (3.3). Note however that for d! 2 we can take t0% arbitrarily large.

Defining R(m,n)!U (m)*U (n) for all pairs of m4n the proof proceeds similarly toTheorem 3.1. The only significant difference comes when we estimate J3 coming fromthe analogue of (3.5). Consider any !2 [0, 1/5). Due to (5.2), D"A!$!V2!! ~V2!.Using (5.7) with s! 2!, we estimate

T3 )"

12jAR"m, n$j2 ( cjPm

n B"U"n$$j2

) "

12jAR"m, n$j2 ( c

#2!njQnB"U "n$$j22!

) "

12jAR"m, n$j2 ( c

#2!njB"U "n$$j2V2!

) "

12jAR"m, n$j2

( c

#2!n"kU "n$k3*4!jAU "n$j1(4! ( jU "n$j!kU "n$k3*6!jAU "n$j1(5!$

) "

12jAR"m, n$j2

( c

#2!n"kU "n$k

2"3*4!$1*4! ( jU "n$j 2!

1*5!kU "n$k2"3*6!$1*5! ( jAU "n$j2$: "5:11$

Note that these bounds are valid for both d! 2, 3. For definiteness we take !! 1/8and replace (3.8) with

T3 )"

12jAR"m, n$j2 ( c

#1=4n

"kU "n$k10 ( jU "n$j2=3kU "n$k12 ( jAU "n$j2$: "5:12$

With inconsequential changes to the definitions of $m,n, Gm,n in (3.11) we concludethe desired results. g

Acknowledgements

This work was partially supported by the National Science Foundation under the grantsNSF-DMS-0604235 andNSF-DMS-0906440, and by theResearch Fund of IndianaUniversity.

Notes

1. One sometimes also finds the more general definition (U,U ]) :!RM v + v] dM(

%RMTT ]dM with %40 fixed. This % is useful for the coherence of physical dimensions

and for (mathematical) coercivity. Since this is not needed here we take %! 1.



2. B is also continuous from V!V to V 0 and satisfies important cancellation properties andestimates. Since we are considering strong solutions in C([0, t],V)\L2([0, t],D(A)) theseproperties will not be needed here.

References

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