navier-stokes equations in three-dimensional thin …navier-stokes equations in thin domains 5 (fp)...

Advances in Differential Equations. Vol. 1 (1996), no. 4, 499–546.

NAVIER-STOKES EQUATIONS IN THREE-DIMENSIONAL

THIN DOMAINS WITH VARIOUS BOUNDARY CONDITIONS

R. Temam 1,2 and M. Ziane 1

1 Department of Mathematics &The Institute for Scientific Computing and Applied Mathematics

Indiana University, Bloomington, Indiana 47405, USA2 Laboratoire d’Analyse Numérique. Université de Paris-Sud Orsay, France

e-mail: [email protected]

November 6, 1995

Abstract. In this work we develop new methods for studying the Navier-Stokes equations

in thin domains. We consider various boundary conditions and establish the global existenceof strong solutions when the initial data belong to “large sets”. Our work was inspired by the

recent interesting results of G. Raugel and G. Sell [RS 1,2,3] which, in the periodic case, give

global existence for smooth solutions of the 3D Navier-Stokes equations in thin domains forlarge sets of initial conditions. We extend their results in several ways: we consider numerous

boundary conditions and as it will appear hereafter, the passage from one boundary condition

to another one is not necessarily straightforward; also the set of initial data for which weobtain the global existence in time is larger in our case and its characterization is simpler.

The proof of our improved results is based on precise estimates of the dependence of someclassical constants on the thickness ǫ of the domain; e.g. Sobolev type constants and the

regularity constant for the corresponding Stokes problem.

As an application, we study the behavior of the average of the strong solution in the thindirection when the thickness of the domain goes to zero; we prove its convergence to the strong

solution of a 2D Navier-Stokes system of equations.

1991 Mathematics Subject Classification. 34C35, 35Q30, 76D05.

Key words and phrases. Navier-Stokes Equations, Thin domains, Global existence and regularity,Nonlinear partial differential equations, Sobolev inequalities in thin domains.

Typeset by AMS-TEX

2 R. TEMAM AND M. ZIANE

0. Introduction

We are concerned in this article with the Navier-Stokes equations of viscous incompress-ible fluids in three dimensional thin domains. Let Ωǫ be the thin domain Ωǫ = ω × (0, ǫ),where ω is a suitable domain in R2 and 0 < ǫ < 1.

Our aim is to study the global existence in time of the strong solutions of the 3D-Navier-Stokes equations (3DNSE) in the thin domains with various boundary conditions as well asthe study of the behavior of the average of the strong solutions in the thin direction whenthe thickness ǫ goes to zero.

The study of the global existence of strong solutions of the three-dimensional Navier-Stokes equations on thin domains was initiated by G. Raugel and G. Sell [RS1,2,3] . Theirwork is inspired by methods developed by J. Hale and G. Raugel [HR1,2] for reaction-diffusion equations and damped wave equations on thin domains. They used a dilationof the domain to obtain the dilated Navier-Stokes equations defined in the fixed domainΩ1 = ω × (0, 1). Then, assuming the space periodic boundary condition, they showedthat the dilated Navier-Stokes equations are a regular perturbation of the 2DNS whenthe thickness is small. They also obtained global existence of strong solutions when theinitial data belong to “large sets”. The definition of a large set given in [RS1,2,3] is ratherinvolved; roughly speaking: the initial data belong to a ball in Vǫ with radius going toinfinity when ǫ goes to zero.

In the same spirit, using the smallness of the domain, by imposing conditions relating thefirst eigenvalue of the Laplacian, the viscosity ν and the size of the initial data, J. Avrinestablished similar global existence results for the 3D-NSE with the Dirichlet boundarycondition [Av]. The results obtained in [Av] are based on a contraction principle argument.

Many other studies on partial differential equations in thin domains appear in the workof Ph. Ciarlet and his collaborators; see e.g. Ph. Ciarlet [Ci], H. L. Le Dret [LD], and thereferences therein. The purpose in this work is to systematically derive the equations forplates and shells by passing to the limit, ǫ → 0, in the the equations of linear or nonlinearelasticity in three dimensional domains.

In the present article, we develop a general study of the Navier-Stokes equations onthin domains and obtain global existence (in time) of the strong solutions for initial databelonging to large sets, for which we give a very simple characterization. The study givenin this article treats systematically various boundary conditions (listed below); as we willsee at the end of this article, the behavior of the solutions depend strongly on the boundaryconditions. A major difference between our work and that of [RS 1,2,3] is that we founduseful to avoid the dilation of the domain and work instead in the actual domain Ωǫ. Theprice to pay for that is that all constants appearing in the various functional inequalitiesdepend now on ǫ. The first part of our work (Section 2) is to determine precisely the

NAVIER-STOKES EQUATIONS IN THIN DOMAINS 3

dependence on the thickness ǫ of the constants appearing in the classical Sobolev-typeinequalities, often used in the study of the Navier-Stokes equations, namely the Poincaré,Ladyzhenskaya and Agmon inequalities; we also determine the dependence on ǫ of theconstant appearing in the Cattabriga-Solonnikov regularity inequality; we did not find allthese constants available in the literature and a number of them are new to the best of ourknowledge.

A useful tool in obtaining the dependence of the constants in the functional inequalitiesis the average operator in the thin direction which allows us to use the Poincaré inequalityin the thin direction. The definition of the average operator for vector functions dependson the type of boundary conditions; roughly speaking, if a component does not satisfy theDirichlet condition in the thin direction then we take its average, otherwise we set it to bezero (see Section 1 for more details).

The boundary conditions of interest to us are combinations of the usual boundary con-ditions, namely the Dirichlet (D), periodic (P) and free boundary (F) conditions. We willcombine these boundary conditions on Γt ∪ Γb = ω × {0, ǫ} and Γl = ∂ω × (0, 1), consider-ing (FP), (FD), (FF), (PP), (DD) and (DP) (see below). We could also consider differentcombinations of boundary conditions on Γt, Γb and Γl (which is not always straightforwardas it will appear in Section 4) but we refrained to do so to avoid lengthly developments.

The size of the large sets of initial data for which we obtain the global existence intime of the strong solution depends on the boundary condition. We divide the boundaryconditions above into three types (see below for the notations):

Type I. It contains the boundary conditions (FF) and (FP); i.e. the free boundary con-dition in the thin direction and either the periodic or the free boundary condition on thelateral boundary. In this type of boundary conditions, we obtain that whenever the initialdata satisfy

limǫ→0

ǫq(∣

∣A12ǫ u0

∣

∣

2

ǫ+

∣

∣f∣

∣

2

ǫ) = 0, for some q < 1,

then, there exists ǫ0(ν) such that for ǫ ≤ ǫ0, the maximal time of existence T (ǫ) of thestrong solution of the 3D Navier-Stokes equations with one of these boundary conditionssatisfies

T (ǫ) = +∞.

Type II. It contains the boundary conditions (FD): the free boundary condition in thethin direction and the Dirichlet boundary condition on the lateral boundary, and (PP): thepurely periodic boundary condition. For this type of boundary conditions, we obtain thatwhenever the initial data satisfy (see Sections 2 and 3 for the notations)

If for some arbitrarily fixed constants K1 > 0, and K2 > 0,

|A12ǫ M̃ǫu

ǫ0|2ǫ + |M̃ǫf ǫ|2ǫ ≤ K1ǫ ln

∣

∣ln ǫ∣

∣, and |A12ǫ Ñǫu

ǫ0|2ǫ + |Ñǫf ǫ|2ǫ ≤ K2 ln

∣

∣ln ǫ∣

∣,


then the same conclusions as for Type I hold.

Type III. It contains the boundary conditions (DD) and (DP); i.e. the Dirichlet boundarycondition in the thin direction and either the Dirichlet or the periodic condition on thelateral boundary. For this type of boundary conditions, we obtain the same conclusionswhenever the initial data satisfy

limǫ→0

ǫ12 (

∣

∣A12ǫ u

ǫ0

∣

∣

2

ǫ+

∣

∣f ǫ∣

∣

2

ǫ) = 0.

We also give an asymptotic expansion of the solution uǫ for ǫ small in the case ofthe channel flow. The asymptotic expansion in more general cases will be given in aforthcoming article [TZ1]. The results obtained for the averages are described at the endof the Introduction.

This work was motivated by the study of the Navier-Stokes equations in thin sphericalshells in order to justify the Navier-Stokes equations on the sphere in view of applicationsto geophysical flows. The results obtained in the spherical case will be given elsewhere[TZ2].

The global existence results obtained in thin domains allows us to prove the existenceof the attractors, to give a characterization of them and also to determine the dependenceof their fractal and Hausdorff dimensions on the thickness ǫ; all these results will appearin [Z3].

The setting of the problem.

The nondimensionalized form of the Naiver-Stokes equations (NSE) is

∂u

∂t− ν∆u + (u · ∇)u + ∇p = f in Ωǫ × (0,∞),(0.1)

div u = 0 in Ωǫ × (0,∞),(0.2)u(·, 0) = u0(·) in Ωǫ.(0.3)

Here u = (u1, u2, u3) is the velocity vector at point x and time t, and p(x, t) is the pressure.

Equations (0.1)-(0.3) are supplemented with boundary conditions. We denote the bound-ary of Ωǫ by ∂Ωǫ = Γt ∪ Γb ∪ Γl, where

(0.4) Γt = ω × {ǫ}, Γb = ω × {0}, and Γl = ∂ω × (0, ǫ).

The boundary conditions under consideration.

The boundary conditions of interest to us are combinations of the usual boundary con-ditions, namely the Dirichlet (D), periodic (P) and free boundary (F) conditions. Moreprecisely, we consider the following combinations:


(FP) The Free boundary condition on Γt ∪ Γb and the periodic condition on Γl; i.e. hereω = (0, l1) × (0, l2) and

u3 = 0 and∂uα

∂x3= 0, α = 1, 2 on Γt ∪ Γb; and

ui, i = 1, 2, 3 are periodic in the directions x1, x2 with periods l1, l2 respectively, and

∫

Ωǫ

(u0)αdx =

∫

Ωǫ

fαdx = 0, α = 1, 2.

(FD) The Free boundary condition on Γt ∪ Γb and the Dirichlet boundary condition onΓl; i.e. here ω is a C

2- bounded domain in R2, and

u3 = 0 and∂uα

∂x3= 0, α = 1, 2 on Γt ∪ Γb, and u = 0 on Γl.

(FF) The Free boundary condition on ∂Ωǫ, i.e.

u · ~n = 0, curl u × ~n = 0 on ∂Ωǫ,

where ~n is the outward unit normal vector to ∂Ωǫ.

(PP) The Periodic boundary condition on ∂Ωǫ, i.e. here ω = (0, l1) × (0, l2) and u areΩǫ-periodic, and, for the data

∫

Ωǫ

u0dx =

∫

Ωǫ

fdx = 0.

(DD) The Dirichlet boundary condition on ∂Ωǫ,

u = 0 on ∂Ωǫ.

(DP) The Dirichlet boundary condition on Γt ∪ Γb and the periodic condition on Γl.

The Mathematical Setting of the problem.

We denote by Hs(Ωǫ), s ∈ R, the Sobolev space constructed on L2(Ωǫ) and L2(Ωǫ) =(L2(Ωǫ))

3, Hs(Ωǫ) = (Hs(Ωǫ))

3. We also denote by Hs0(Ωǫ) the closure in the space Hs(Ωǫ)

of C∞0 (Ωǫ), the space of infinitely differentiable functions with compact support in Ωǫ. Weneed also the following spaces:

(0.5) Ḣm(Ωǫ) =

{

u ∈ Hm(Ωǫ);∫

Ωǫ

udx = 0

}

,


and the spaces Hmper(Ωǫ), which are defined with the help of Fourier series; we write

(0.6) u(x) =∑

k∈Z3

ukexp(2ik ·x

L),

with ūk = u−k (so that u is real valued) and

x

L=

(x1

l1,x2

l2,x3

ǫ

)

; k · xL

= k1x1

l1+ k2

x2

l2+ k3

x3

ǫ.

Then, u is said to be in L2(Ωǫ) if and only if

|u|2L2(Ωǫ) = ǫl1l2∑

k∈Z3

|uk|2 < ∞,

and u is in Hsper(Ωǫ), s ∈ R+, if and only if∑

k∈Z3

(1 + |k|2)s|uk|2 < ∞.

For the mathematical setting of the Navier-Stokes equations, we consider a Hilbert spaceHǫ, which is a closed subspace of L

2(Ωǫ) (see [T1]). Depending on the boundary condition,we define the following:

HFP =

{

u ∈ L2(Ωǫ); div u = 0;∫

Ωǫ

uαdx = 0, u3 = 0 on Γt ∪ Γb

and u∣

∣Γα = u∣

∣Γα+3; α = 1, 2

}

,

where Γα and Γα+3 are the faces xα = 0 and xα = lα of ∂Ωǫ. The condition uα|Γα =uα|Γα+3 expresses the periodicity of uα in the direction xα.

HFD ={

u ∈ L2(Ωǫ); div u = 0; u · ~n = 0 on ∂Ωǫ}

;

HPP =

{

u ∈ L2(Ωǫ); div u = 0;∫

Ωǫ

udx = 0, uj |Γj = uj |Γj+3; j = 1, 2, 3}

,

HDD = HFF = HFD and HDP = HFP .

Another useful space is Vǫ, a closed subspace of H1(Ωǫ), which is defined depending on

the boundary condition as follows:

VFP = {u ∈ H1(Ωǫ) ∩ HFP ; u|Γα = u|Γα+3},VFD = {u ∈ H1(Ωǫ) ∩ HFD; u = 0 on Γl},VPP = {u ∈ Ḣ1per(Ωǫ); div u = 0},VDD = {u ∈ H10(Ωǫ); div u = 0}, andVFF = {u ∈ H1(Ωǫ) ∩ HFF ; u · ~n = 0 on ∂Ωǫ}.


In the remainder of this paper, unless there are differences in the proofs, we will omit thereference to the boundary condition; we denote, for instance, anyone of the spaces definedabove by Hǫ or Vǫ.

The scalar product on Hǫ is denoted by (·, ·)ǫ, the one on Vǫ is denoted by ((·, ·))ǫ, andthe associated norms are denoted by | · |ǫ and || · ||ǫ respectively. We denote by Aǫ theStokes operator defined as an isomorphism from Vǫ onto the dual V

′ǫ of Vǫ, by

(0.7) ∀v ∈ Vǫ; < Aǫu, v >V ′ǫ ,Vǫ= ((u, v))ǫ.

The operator Aǫ is extended to Hǫ as a linear unbounded operator. The domain of Aǫ inHǫ is denoted by D(Aǫ). The space D(Aǫ) can be fully characterized using the regularitytheory. We refer for the study of the regularity of the Stokes operator to [Ca], [Da 1,2], [Gh],[T1,2] and [Z2]. Here we give the characterization of the domain of the Stokes operator:

D(AFP ) =

{

u ∈ H2(Ωǫ) ∩ VFP ;∂uα

∂x3= 0,

∂ui

∂xα

∣

∣Γα = −∂ui

∂xα

∣

∣Γα+3;

i = 1, 2, 3, α = 1, 2

}

,

D(AFD) =

{

u ∈ H2(Ωǫ) ∩ VFD;∂uα

∂x3= 0 on Γb ∪ Γt, α = 1, 2

}

,

D(APP ) = Ḣ2per(Ωǫ) and D(ADD) = H

2(Ωǫ) ∩ H10(Ωǫ).

D(AFF ) ={

u ∈ H2(Ωǫ) ∩ VFF ; curl u × ~n = 0 andu · ~n = 0on ∂Ωǫ}

.

We should also recall the Leray’s projector Pǫ, which is the orthogonal projector ofL

2(Ωǫ) onto Hǫ. The Stokes operator can be given with the help of the Leray projector asfollows:

(0.8) Aǫu = Pǫ(−∆u), for u ∈ D(Aǫ).

Let bǫ be the continuous trilinear form on Vǫ defined by:

(0.9) bǫ(u, v, w) =3

∑

i,j=1

∫

Ωǫ

ui∂vj

∂xiwjdx, u, v, w ∈ H1(Ωǫ).

We denote by Bǫ the bilinear form defined for (u, v) ∈ Vǫ × Vǫ by

< Bǫ(u, v), w >V ′ǫ ,Vǫ= bǫ(u, v, w), ∀w ∈ Vǫ,


andBǫ(u) = Bǫ(u, u).

We assume in this work that the data ν, u0 and f satisfy

(0.10) ν > 0, u0 ∈ Hǫ (or Vǫ); f ∈ L∞(0, +∞; Hǫ).

The system of equations (0.1)–(0.3), with one of the boundary conditions listed above,can be written as a differential equation in V ′ǫ :

(0.11)

{

u′ + νAǫu + Bǫ(u) = f,

u(0) = u0,

where u′ denotes the derivative (in the distribution sense) of the function u with respectto time. We recall now the classical result of existence of solutions to problem (0.11). See[CF], [FGT], [KL], [La], [Li], [Le], [LP], [T1,2], etc....

Theorem 0.1. For u0 ∈ Hǫ, there exists a solution (not necessarily unique) u = uǫ toproblem (0.11) such that:

(0.12) uǫ ∈ L2(0, T ; Vǫ) ∩ L∞(0, T ; Hǫ), ∀T > 0.

Moreover, if u0 ∈ Vǫ, then there exists Tǫ = Tǫ(Ωǫ, ν, u0, f) > 0, and a unique solution uǫto problem (0.11) such that:

(0.13) uǫ ∈ L2(0, Tǫ; D(Aǫ)) ∩ L∞(0, Tǫ; Vǫ).

The solution uǫ which satisfies (0.13) is called the strong solution of (0.11). The studyof the global existence (in time) of the strong solution and the uniqueness of solutions toproblem (0.11) is still open in three dimensional domains; this work will give a partialanswer to this question when the three dimensional domain is thin.

The Main Results

First we state the main results concerning the global existence results. Let R(ǫ) be amonotone positive function satisfying for some q ≤ 1

limǫ→0

ǫqR2(ǫ) = 0,

and assume that the initial data satisfy

|A1/2ǫ u0|2

ǫ + |f |2ǫ ≤ R2(ǫ).


The global existence results will depend on the type of the boundary condition:

- The boundary conditions (DD) and (DP): we take q = 1.- The boundary conditions (FF) and (FP): we take q < 1.- The boundary conditions (PP) and (FD): we assume, in this case, a stronger condition onthe initial data; i.e. for arbitrary positive constants K1, K2 (independent of ǫ), we assumethat

|A1/2ǫ M̃ǫu0|2ǫ + |M̃ǫf |2ǫ ≤ K1ǫln|lnǫ| and |A1/2ǫ Ñǫu0|2ǫ + |Ñǫf |2ǫ ≤ K2ln|lnǫ|,

where M̃ǫ is the average operator in the thin direction and Ñǫu = u − M̃ǫu (see Section 1for more details on M̃ǫ and Ñǫ).

Theorem A1. With the assumptions above on the initial data, there exists ǫ0 > 0, suchthat:

∀ ǫ ≤ ǫ0, ∀ u0 ∈ Vǫ, ∀ f ∈ Hǫ, with |A1/2ǫ u0|2

ǫ + |f |2ǫ ≤ R2(ǫ)

the maximal time Tǫ of existence of the strong solution uǫ of problem (0.1)-(0.3) satisfies

Tǫ = ∞.

In order to see the improvement that this result brings to the classical small data exis-tence result [CF], [RS3], [T1,2], etc ..., we recall that if, e.g., u0 and f are independent onx3 (the thin direction variable) then

|A1/2ǫ u0|2

ǫ + |f |2ǫ ≤ C0ǫ,

where C0 is independent of ǫ, which implies that R(ǫ) goes to zero as ǫ goes to zero, while

in our results R2(ǫ) and therefore |A1/2ǫ u0|2

ǫ + |f |2ǫ can be very large for ǫ small.

Now we state the results concerning the behavior of the average, in the thin direction, ofthe strong solution uǫ of (0.1)-(0.3) when ǫ approaches zero. For this purpose, we introducethe spaces

H̃P =

{

ũ ∈ (L2(ω))2; div ′ ũ = 0;∫

ω

ũdx = 0, ũα|Γ̃α = ũα|Γ̃α+2 ; α = 1, 2}

,

where Γ̃α is the face of ∂ω in the direction xα.

H̃D ={

ũ ∈ (L2(ω))2; div′ ũ = 0; ũ · ~̃n = 0 on ∂ω}

,


where ~̃n is outward unit normal vector to ∂ω.

H̃F = H̃D

We also introduce

ṼP = H̃P ∩ (Ḣ1per(ω))2; ṼD = (H10 (ω))2, and ṼF = (H1(ω))2 ∩ H̃F .

The 2D-Navier-Stokes equations on ω are given by:

∂ṽ

∂t− ν∆′ṽ + (ṽ · ∇′)ṽ + ∇′p̃ = f̃ in ω × [0,∞),

div′ ṽ = 0 in ω × [0,∞),ṽ(x′, 0) = ṽ0(x

′) in ω.

We denote, here and henceforth, the two-dimensional operators with a prime, for example

∇′ =( ∂

∂x1,

∂

∂x2, 0

)

and x′ = (x1, x2).

The 2D Navier-Stokes equations above are supplemented with one of the followingboundary conditions: The periodic boundary condition (P̃ ), the Dirichlet boundary con-

dition (D̃) or the free boundary condition (F̃ ). Note that the study of the existence anduniqueness of solutions to the 2D Navier-Stokes equations is complete. See [CF], [FGT],[KL], [La], [Le], [LP], [T1,2], etc....

We will prove the following:

Theorem A2. In the case of one of the boundary conditions (FD), (FP), (FF) or (PP),we assume that we are given a family of functions uǫ0 ∈ Vǫ and f ǫ ∈ Hǫ defined on thedomains Ωǫ, 0 < ǫ < 1 such that:

limǫ→0

1

ǫ

∫ ǫ

0

uǫ0(·, x3)dx3 = ṽ0 in Ṽ − weak

limǫ→0

1

ǫ

∫ ǫ

0

f ǫ(·, x3)dx3 = f̃ in H̃ − weak.

Then, for all T > 0, there exists ǫ0 > 0 such that, for 0 < ǫ < ǫ0, there exists a uniquestrong solution uǫ of (0.1)-(0.3) defined on [0, T ] and

limǫ→0

1

ǫ

∫ ǫ

0

uǫ(t; x1, x2, x3)dx3 = ṽ(t; x1, x2) in C([0, T ]; H̃) ∩ L2(0, T ; Ṽ ).


where ṽ is the “unique” strong solution of the 2D Navier-Stokes equations above, with theboundary condition (P̃ ) in the case of (PP) or (FP); the boundary condition (D̃) in the

case of (FD) and the boundary condition (F̃ ) in the case of (FF).

We will also show that in the case of the Dirichlet boundary condition on either the topor the bottom of Ωǫ, we have

limǫ→0

1

ǫ

∫ ǫ

0

uǫ(t; x1, x2, x3)dx3 = 0 in C([0, T ]; H̃) ∩ L2(0, T ; Ṽ ).

We also give an asymptotic expansion of the solution uǫ for a channel flow when ǫ issmall. For more details, see Section 4.

This article is organized as follows: In Section 1 we define the average operator andstudy its properties. Section 2 is devoted to some fundamental inequalities related to theNavier-Stokes equations on thin domains. Section 3 gives the necessary a priori estimatesfor the average of the strong solution in the thin direction. In Section 4, we study thebehavior of the maximal time of existence of strong solutions when ǫ goes to zero, andfinally in Section 5, we study the behavior of the averages as the thickness goes to zero.

1. Preliminaries

For a scalar function ϕ ∈ L2(Ωǫ), we define its average in the thin direction as follows:

(1.1) (Mǫϕ)(x1, x2) =1

ǫ

∫ ǫ

0

ϕ(x1, x2, s)ds,

and we set

(1.2) Nǫϕ = ϕ − Mǫϕ, i.e. Mǫϕ + Nǫϕ = IL2(Ωǫ),

where IL2(Ωǫ) is the identity operator on L2(Ωǫ).

In order to define the average operator M̃ǫ, we will need to specify the boundary condi-tion. For u = (u1, u2, u3) ∈ L2(Ωǫ), we write

(1.3) M̃ǫu =

(Mǫu1, Mǫu2, 0) for (FF), (FP) and (FD)

(Mǫu1, Mǫu2, Mǫu3) for (PP)

0 for (DD) and (DP),

and we set

(1.4) Ñǫu = u − M̃ǫu; i.e. M̃ǫ + Ñǫu = IL2(Ωǫ).


The reason for which we gave different definitions for the average operator lies on thefact that there is no need to take the average of a function when its boundary values arezero at either the top or the bottom of the thin domain; i.e. when a function satisfies thePoincaré inequality in the thin direction.

It is useful to observe that for the boundary conditions under consideration, each com-ponent (Ñǫu)i, i = 1, 2, 3 of Ñǫu satisfy one of the conditions:

(1.5) (Ñǫu)i = 0 on Γt and Γb, or

∫ ǫ

0

(Ñǫu)i(x1, x2, x3, t)dx3 = 0.

All these operators are projectors, i.e.

(1.6) M2ǫ = Mǫ, N2ǫ = Nǫ, M̃

2ǫ = M̃ǫ, Ñ

2ǫ = Ñǫ.

Furthermore,we have the following properties which are obvious:

(i) Mǫ is an orthogonal projector from L2(Ωǫ) onto L

2(ω).

(1.7) (ii) MǫNǫ = 0, and M̃ǫÑǫ = 0

(1.8) (iii) Mǫ∇′ = ∇′Mǫ, Nǫ∇′ = ∇′Nǫ, and M̃ǫ∇′ = ∇′M̃ǫ, Ñǫ∇′ = ∇′Ñǫ,(1.9) (iv) ϕ ∈ Hk(Ωǫ) ⇒ Mǫϕ ∈ Hk(ω) and Nǫϕ ∈ Hk(Ωǫ), k ≥ 0.(1.10) (v) The boundary condition for M̃ǫ on ∂ω is the same as the one for u on ∂ω×(0, ǫ);i.e.

If u satisfies (FD) or (DD), then M̃ǫu is zero on ∂ω.

If u satisfies (FP) or (PP), then M̃ǫu is periodic.

If u satisfies (FF) , then M̃ǫu satisfies (FF).

In the following lemma, we give the basic properties of the operators Mǫ and M̃ǫ. Theseproperties hold for all the boundary conditions listed above; therefore, we omit the indicesof the boundary conditions.

Lemma 1.1. For all u, v ∈ H1(Ωǫ), we have

(1.11)

∫

Ωǫ

∇Ñǫu · ∇M̃ǫvdx = 0.

(1.12) |u|2ǫ = |M̃ǫu|2ǫ + |Ñǫu|2ǫ and ‖u‖2ǫ = ‖M̃ǫu‖2ǫ + ‖Ñǫu‖2ǫ , ∀u ∈ H1(Ωǫ).

(1.13) If v ∈ Vǫ, then M̃ǫv ∈ Vǫ and Ñǫv ∈ Vǫ.


(1.14) bǫ(u, u, M̃ǫv) = bǫ(M̃ǫu, M̃ǫu, M̃ǫv) + bǫ(Ñǫu, Ñǫu, M̃ǫv), for u, v ∈ Vǫ.

(1.15) bǫ(u, u, Ñǫv) = bǫ(Ñǫu, M̃ǫu, Ñǫv) + bǫ(M̃ǫu, Ñǫu, Ñǫv)

+ bǫ(Ñǫu, Ñǫu, Ñǫv), for u, v ∈ Vǫ.

For all u ∈ D(Aǫ), we have

(1.16) ∆Ñǫu = Ñǫ∆u, ∆M̃ǫu = M̃ǫ∆u.

Proof. (i) Let u, v ∈ H1(

Ωǫ), we have immediately

(1.17)

∫

Ωǫ

∇Ñǫu · ∇M̃ǫvdx =∫

Ωǫ

∇′Ñǫu · ∇′M̃ǫvdx =∫

Ωǫ

Ñǫ(∇′u) · M̃ǫ(∇′v)dx = 0.

Hence, we have (1.11).

(ii) The first Pythagorean identity in (1.12) is a consequence of the fact that M̃ǫ is anorthogonal projector in L2(Ωǫ), while the second one is a consequence of (1.7), (1.8) and(1.11).

(iii) Let v ∈ Vǫ, it is clear that M̃ǫv ∈ H1(Ωǫ) and satisfies the same boundary conditionas v on ∂Ωǫ. The only point that remains to be checked is div M̃ǫv = 0 : In the case of theboundary conditions (DD) and (DP), we have M̃ǫv = 0. Hence, we need only to considerthe boundary conditions (FF), (FP), (FD)and (PP). We integrate div v = 0 between 0 andǫ and obtain

(1.18) div′ M̃ǫv =

∫ ǫ

0

∂v3

∂x3dx3 = 0.

The second term in the left hand side of (1.18) vanishes in all the cases (FF), (FP), (FD)and(PP). Hence,

(1.19) div′ M̃ǫv = 0.

(iv) Let u, v ∈ Vǫ, we have

(1.20) bǫ(u, u, M̃ǫv) =

∫

Ωǫ

(M̃ǫu · ∇)M̃ǫu · M̃ǫvdx +∫

Ωǫ

(Ñǫu · ∇)M̃ǫu · M̃ǫvdx

+

∫

Ωǫ

(Ñǫu · ∇)Ñǫu · M̃ǫvdx +∫

Ωǫ

(M̃ǫu · ∇)Ñǫu · M̃ǫvdx,


Now, since

∫ ǫ

0

Ñǫudx3 = ǫM̃ǫ(

Ñǫu)

= 0, we have

∫

Ωǫ

(Ñǫu · ∇)M̃ǫu · M̃ǫvdx =∫

ω

(∫ ǫ

0

Ñǫudx3 · ∇)

M̃ǫu · M̃ǫvdx′ = 0.

For the integral

∫

Ωǫ

(M̃ǫu · ∇)Ñǫu · M̃ǫvdx, we observe that

(1.21)

3∑

j=1

∫

Ωǫ

Mǫui∂Nǫuj

∂xiMǫvjdx =

3∑

j=1

∫

Ωǫ

MǫuiMǫvjNǫ(∂uj

∂xi

)

dx = 0,

due to (1.8), for i = 1, 2, while for i = 3, the similar integral vanishes because of (1.5)Hence (1.14).

(v) Similarly, for u, v ∈ Vǫ, we have

(1.22) bǫ(u, u, Ñǫv) =

∫

Ωǫ

(M̃ǫu · ∇)M̃ǫu · Ñǫvdx +∫

Ωǫ

(M̃ǫu · ∇)Ñǫu · Ñǫvdx

+

∫

Ωǫ

(Ñǫu · ∇)M̃ǫu · Ñǫvdx +∫

Ωǫ

(Ñǫu · ∇)Ñǫu · Ñǫvdx,

but

(1.23)

∫

Ωǫ

(M̃ǫu · ∇)M̃ǫu · Ñǫvdx =∫

ω

M̃ǫ[

(M̃ǫu · ∇′)M̃ǫu] ·(

∫ ǫ

0

Ñǫvdx3

)

dx′ = 0.

Hence (1.15) is established.

(vi) In the case of the boundary condition (DP) or (DD), the identities (1.16) are obvious,

since M̃ǫu = 0 and Ñǫu = u. The case of the purely periodic condition (PP) is also obvious,

since∂

∂xiM̃ǫ = M̃ǫ

∂

∂xi, i = 1, 2, 3. We need to prove (1.16) when the boundary condition

is (FF), (FP) or (FD). We will only treat (FD); the two other cases can be treated similarly.

Let v ∈ (C∞0 (ω))2, we have by integration by parts

(1.24)

∫

ω

∆M̃ǫu · vdx′ =∫

ω

∆′M̃ǫu · vdx′ = −∫

ω

∇′M̃ǫu · ∇′vdx′

= −∫

ω

M̃ǫ(∇′u) · ∇′vdx′ = −∫

Ωǫ

∇′u · ∇′vdx =∫

Ωǫ

∆′u · vdx

=

∫

ω

M̃ǫ(∆′u) · vdx′ =

∫

ω

M̃ǫ(∆u) · vdx′.


The last equality is obtained, thanks to∂uα

∂x3= 0 on Γt ∪ Γb, α = 1, 2. Hence ∆M̃ǫu =

M̃ǫ∆u. Then, we have on the one hand

(1.25) ∆u = ∆M̃ǫu + ∆Ñǫu = M̃ǫ(∆u) + ∆Ñǫu,

and on the other hand

(1.26) ∆u = M̃ǫ(∆u) + Ñǫ(∆u);

therefore, ∆Ñǫu = Ñǫ(∆u). �

In addition to the Lemma above, we need to establish the commutativity of the operatorsM̃ǫ and Ñǫ with the Stokes operator Aǫ. In the case of one of the boundary conditions (FF),(FP) and (PP), we have ∆u = Aǫu for u ∈ D(Aǫ) (see [Z1]) and the commutativity followsfrom (1.16). In the case of the boundary conditions (DP) and (DD), we have M̃ǫ = 0.Hence the commutativity follows. It remains the case of the boundary condition (FD).First we note that

∆u|Γt∪Γb = 0, for u ∈ D(Aǫ).Indeed, since div u = 0, we have

∂2u3

∂x23= − ∂

2u1

∂x3x1− ∂

2u2

∂x3x2in Ωǫ

and since ∂u1∂x3 =∂u2∂x3

= 0 on Γt ∪ Γb, we have

∂2u3

∂x23= 0 on Γt ∪ Γb.

Moreover, since u3 = 0 on Γt ∪ Γb, it is clear that

∂2u3

∂x21=

∂2u3

∂x22= 0 on Γt ∪ Γb.

Now from the characterization of the Leray projector (see [CF] and [T1]), we have

Aǫu = −∆u + ∇p,

where

∆p = 0 in Ωǫ,

∂p

∂nω= ∆′u · nω on ∂ω × (0, ǫ),

∂p

∂x3= 0 on ω × {0, ǫ}.


Applying the operator M̃ǫ and using (1.8) and (1.16), we obtain

M̃ǫAǫu = −∆M̃ǫu + ∇′Mp

and

∆′Mp = 0 in ω,

∂Mp

∂nω= (∆′Mu) · nω,

which is the characterization of the 2D Leray projector for ∆′M̃ǫu. Therefore

AǫM̃ǫu = M̃ǫAǫu.

Finally, we recall an important orthogonal property related to the trilinear form. Theorthogonal property reads

(1.27) b̃(v, v, Ãv) = 0, ∀v ∈ D(Ã),

where b̃ is the 2D-trilinear form, and Ã is to the 2D-Stokes operator with either the periodicboundary condition (see [CF], [T2,T3]) or the free boundary condition (see [Z1]). The iden-tity (1.27) will be used in our work in order to obtain better estimates of the strong solutionin thin three-dimensional domains, and will apply in the cases of the boundary conditions(FF) and (FP). We state also the following lemma, which is obtained by integrating byparts several times (see [Z1]):

Lemma 1.2. In the case of either boundary conditions (PP), (FF) or (FP), we have foru ∈ D(Aǫ)

(1.28) bǫ(M̃ǫu, M̃ǫu, AǫM̃ǫu) = b(M̃ǫu, M̃ǫu, ÃM̃ǫu) = 0.

2. Fundamental inequalities in thin domains

2.1 Sobolev-type inequalities.

One of the basic tools in the study of nonlinear partial differential equations in thindomains is the knowledge of the exact dependence, with respect to the thickness of thedomain, of the constants appearing in the Sobolev and related inequalities. In the classicalform of Sobolev inequalities, the dilation invariance of the constants is emphasized. How-ever, in thin domains, the shape of the domain is important. Therefore, we break away


from isotropy and derive a version of the inequalities emphasizing the dependence of theconstants on the thickness ǫ of the domain.

In our work, we will appropriate versions of the Poincaré inequality, Agmon’s inequalityand Ladyzhenskaya’s inequalities. The Poincaré inequality in thin domains was obtainedin [HR1,2] and [RS1,2,3]. The proof is classical (see [GT] and [N]). Also, Ladyzhenskaya’sinequalities were obtained in [RS1,2,3] when the boundary condition is purely periodic andindependently in [ABC] for the Dirichlet boundary condition.

We should also mention the work of Solonnikov [S] in which he obtained anisotropicinequalities for functions in Wm,p(Rn), the space of distributions which are in Lp(Rn)along with their derivatives of order ≤ m. He showed that

If b =n

∑

j=1

βj < 1 and, either p, q, τ > 1 or

µ = 1 −n

∑

j=1

αj

mj+

(1

q− 1

p

)

n∑

j=1

1

mj> 0,

with q ≥ p ≥ 1 and τ ≥ 1, then there exists a positive constant c such that:

(*)∣

∣Dαu∣

∣

Lq(Rn)≤ c

( n∏

j=1

∣

∣

∂mju

∂xjmj

∣

∣

βj

Lp(Rn)

)

∣

∣u∣

∣

1−Pn

j=1 βj

Lτ (Rn) ,

where Dα =∂α1

∂xα11. . .

∂αn

∂xαnnfor α = (α1, . . . , αn).

The method that we develop below will imply the validity of the inequality (*) in boundedparallelepipeds (see the proof of Proposition 2.2).

The following result is easy and classical (see [GT], [N], [HR1,2]), etc...); we incorporateit here for the sake of completeness.

Proposition 2.1. (Poincaré’s inequality) For u ∈ H1(Ωǫ) satisfying one of the followingconditions:

(2.1)

(i) u = 0 on Γt,

(ii) u = 0 on Γb,

(iii)

∫ ǫ

0

u(x1, x2, x3)dx3 = 0 a.e. in ω,

we have

(2.2) |u|L2(Ωǫ) ≤ ǫ∣

∣

∂u

∂x3

∣

∣

L2(Ωǫ).


We recall that (2.1) (i), (ii) or (iii) is valid for each component of a function Ñǫu, whereu satisfies any of the boundary conditions under consideration.

Proof. First note that

{

u ∈ C1(Ω̄ǫ); u = 0 on Γt (resp. Γb)}

is dense in{

u ∈ H1(Ωǫ); u = 0 on Γt (resp. Γb)}

,

and{

u ∈ C1(Ω̄ǫ);∫ ǫ

0

u(x1, x2, x3)dx3 = 0, ∀x1, x2}

is dense in

{

u ∈ H1(Ωǫ);∫ ǫ

0

u(x1, x2, x3)dx3 = 0, a.e. in ω

}

.

Thanks to a density argument, we assume that u ∈ C1(Ωǫ). We have for any ζ, η in [0, ǫ]

(2.3) u2(x′, ζ) + u2(x′, η) = 2u(x′, ζ)u(x′, η) +(

u(x′, ζ) − u(x′, η))2

= 2u(x′, ζ)u(x′, η) +

(∫ ζ

η

∂u

∂x3(x′, s)ds

)2

,

with x′ = (x1, x2). We fix ζ and integrate with respect to η to obtain

(2.4) ǫu2(x′, ζ) +

∫ ǫ

0

u2(x′, η)dη = 2u(x′, ζ)

∫ ǫ

0

u(x′, η)dη +

∫ ǫ

0

(∫ ζ

η

∂u

∂x3(x′, s)ds

)2

dη.

Now if u = 0 on Γt (resp. Γb,) we take ζ = ǫ (resp. ζ = 0) and obtain from (2.4)

(2.5)

∫ ǫ

0

u2(x′, η)dη ≤∫ ǫ

0

|ζ − η|(

∫ ǫ

0

∣

∣

∂u

∂x3(x′, s)

∣

∣

2)

ds,

≤ (ǫ)2(

∫ ǫ

0

∣

∣

∂u

∂x3(x′, s)

∣

∣

2)

ds,

and (2.2) follows promptly.

If

∫ ǫ

0

u(x′, η)dη = 0, we infer from (2.4)

∫ ǫ

0

u2(x′, η)dη ≤∫ ǫ

0

(

∫ ζ

η

∂u

∂x3(x′, s)ds

)2dη ≤

∫ ǫ

0

|ζ − η|(

∫ ǫ

0

∣

∣

∂u

∂x3(x′, s)

∣

∣

2)ds,

≤ (ǫ)2(∫ ǫ

0

∣

∣

∂u

∂x3(x′, s)

∣

∣

2)ds.

The proof is complete. �

It is easy to see that for each boundary condition listed above, each component of Ñǫv,where v ∈ Vǫ, satisfies one of the conditions (2.1). Therefore, we have the following:


Corollary 2.1. Under any of the boundary conditions under consideration, we have forall v ∈ Vǫ

(2.6) |Ñǫv|ǫ ≤ ǫ∣

∣

∂Ñǫv

∂x3

∣

∣

ǫ.

Proposition 2.2. (Anisotropic Agmon’s inequality) Let Ω0 = (0, 1)3; then there exists an

absolute constant c such that

(2.7)∣

∣u∣

∣

L∞(Ω0)≤ c

∣

∣u∣

∣

14

L2(Ω0)

3∏

i=1

(

∣

∣

∂2u

∂x2i

∣

∣

L2(Ω0)+

∣

∣

∂u

∂xi

∣

∣

L2(Ω0)+

∣

∣u∣

∣

L2(Ω0)

)14

,

for all u ∈ H2(Ω0).

Proof. We prove the lemma in three steps.Step 1. We replace Ω0 by R

3 and assume that u ∈ C20(R3). We write, using the Sobolevinclusion H20 (R

3) ⊂ L∞(R3),

(2.8)∣

∣u∣

∣

L∞(R3)≤ c0

3∑

i=1

∣

∣

∂2u

∂x2i

∣

∣

L2(R3)+c0

∣

∣u∣

∣

L2(R3).

Let λ = (λ1, λ2, λ3) ∈ R3, λi > 0, i = 1, 2, 3 and set yi = λixi, i = 1, 2, 3. We definethe function vλ ∈ C20(R3) as follows:

(2.9) vλ(y1, y2, y3) = u(y1

λ1,y2

λ2,y3

λ3);

we have immediately

(2.10)

∣

∣vλ∣

∣

L∞(R3)=

∣

∣u∣

∣

L∞(R3),

∣

∣vλ∣

∣

L2(R3)= (λ1λ2λ3)

12

∣

∣u∣

∣

L2(R3)

∣

∣

∂2vλ

∂y2i

∣

∣

L2(R3)=

(λ1λ2λ3)12

λ2i

∣

∣

∂2u

∂x2i

∣

∣

L2(R3).

Inequality (2.8) applied to vλ yields

(2.11)∣

∣u∣

∣

L∞(R3)=

∣

∣vλ∣

∣

L∞(R3)≤ c0

4(λ1λ2λ3)

12

3∑

i=1

1

λ2i

∣

∣

∂2u

∂x2i

∣

∣

L2(R3)

+c0

4(λ1λ2λ3)

12

∣

∣u∣

∣

L2(R3),


and since (2.11) is valid for all choices of λ1, λ2, λ3, we can take

(2.12) λi =

∣

∣

∂2u∂x2

i

∣

∣

12

L2(R3)∣

∣u∣

∣

12

L2(R3)

, i = 1, 2, 3.

Hence,

∣

∣u∣

∣

L∞(R3)≤ 4c0

∣

∣u∣

∣

14

L2(R3)

3∏

i=1

∣

∣

∂2u

∂x2i

∣

∣

14

L2(R3).

Step2. We will show that there exists an absolute constant c1 > 0, such that if u ∈ C2(Ω̄0),then there exists ū ∈ C2(Ω̄1), where Ω1 = (−13 , 43 )3 such that

(2.13)∣

∣

∂kū

∂xki

∣

∣

L2(Ω1)≤ c1

∣

∣

∂kū

∂xki

∣

∣

L2(Ω0), k = 0, 1, 2; i = 1, 2, 3.

We use the classical Babitch extension operators (see [A], [LM], [N]); we first extend uto the domain (−13 , 43) × (0, 1)2 using

(2.14) E1ku(x) =

u(x) for x1 ∈ [0, 1],3

∑

j=1

(−j)kαju(−jx1, x2, x3) for x1 ∈ [−1

3, 0),

3∑

j=1

(−j)kαju(1 − j(x1 − 1), x2, x3) for x1 ∈ (1,4

3),

where (α1, α2, α3) is the unique solution to the system

(2.15)

3∑

j=1

(−j)kαj = 1, k = 0, 1, 2.

We have immediately

(2.16)∂kE10u

∂xki= E1k

∂ku

∂xki, k = 0, 1, 2; i = 1, 2, 3,

and (2.15) implies that

E10u ∈ C2([−1

3,4

3] × [0, 1]2) if u ∈ C2(Ω̄0).


Moreover,

(2.17)

∫

(−1/3,4/3)×(0,1)2

∣

∣

∂kE10u

∂xki

∣

∣

2dx =

∫

(− 13, 43)×(0,1)2

∣

∣E1k∂ku

∂xki

∣

∣

2dx

=

∫

Ω0

∣

∣

∂ku

∂xki

∣

∣

2dx +

∫

(− 13,0)×(0,1)2

∣

∣

3∑

j=1

(−j)kαj∂ku

∂xki(−jx1, x2, x3)

∣

∣

2dx

+

∫

(1, 43)×(0,1)2

∣

∣

3∑

j=1

(−j)kαj∂ku

∂xki(1 − j(x1 − 1), x2, x3)

∣

∣

2dx ≤ c21

∫

Ω0

∣

∣

∂ku

∂xki

∣

∣

2dx.

The extension is complete in the direction x1.

The same argument can be applied in the directions x2 and x3, but instead of applyingit on the domain Ω0, we apply it to extend the function E

10u defined in (−13 , 43) × (0, 1)2

to the domain (−13 , 43)2 × (0, 1). The extended function is denoted by E20u and satisfy

(2.18)∣

∣

∂kE20u

∂xki

∣

∣

2

L2((− 13, 43)2×(0,1))

≤ c1∣

∣

∂kE10u

∂xki

∣

∣

2

L2((− 13, 43)×(0,1)2)

≤ c1∣

∣

∂ku

∂xki

∣

∣

2

L2(Ω0), k = 0, 1, 2; i = 1, 2, 3.

Finally the extension of E20u in the direction x3 yields a function ū ∈ C2(Ω̄1), Ω1 = (−13 , 43 )3with

(2.19)∣

∣

∂kū

∂xki

∣

∣

2

L2(Ω1)≤ c31

∣

∣

∂ku

∂xki

∣

∣

2

L2(Ω0), k = 0, 1, 2; i = 1, 2, , 3.

Step 3. We will show that we can localize without “mixing the directions”. Let ϕ ∈ C∞0 (R)be such that

supp ϕ ⊂ (−13,4

3), 0 ≤ ϕ ≤ 1 and ϕ = 1 on [0, 1].

Set

(2.20) Φ(x1, x2, x3) =3

∏

i=1

ϕ(xi).

We have immediately (with ū being the extension of u defined in Step2)

(2.21) Φū ∈ C20(R3),∣

∣u∣

∣

L∞(Ω0)=

∣

∣Φū∣

∣

L∞(R3),

∣

∣Φū∣

∣

L2(R3)≤ c2

∣

∣u∣

∣

L2(Ω0),


where c2 is an absolute constant. Moreover,

(2.22)∂2Φū

∂x2i= Φ

∂2ū

∂x2i+ 2

∂Φ

∂xi

∂ū

∂xi+ ū

∂2Φ

∂x2i.

Let K =[

1 + supR(2

∣

∣ϕ′∣

∣+∣

∣ϕ′′∣

∣)]; we have (with Ω1 = (−13 , 43)3)

(2.23)∣

∣

∂2Φū

∂x2i

∣

∣

L2(R3)≤ K(

∣

∣

∂2ū

∂x2i

∣

∣

L2(Ω1)+

∣

∣

∂ū

∂xi

∣

∣

L2(Ω1)+

∣

∣ū∣

∣

L2(Ω1))

≤ K1(∣

∣

∂2u

∂x2i

∣

∣

L2(Ω0)+

∣

∣

∂u

∂xi

∣

∣

L2(Ω0)+

∣

∣u∣

∣

L2(Ω0)).

We conclude the lemma by combining (2.13), (2.21) and (2.23). �

Thanks to Proposition 2.2, it is easy to obtain

Corollary 2.2. (Agmon’s inequality in thin domains) There exists a positive constantc0(ω), independent of ǫ, such that: ∀u ∈ H2(Ωǫ)

(2.24)∣

∣u∣

∣

L∞(Ωǫ)≤ c0

∣

∣u∣

∣

14

L2(Ωǫ)

(

∣

∣

∂2u

∂x23

∣

∣

L2(Ωǫ)+

1

ǫ

∣

∣

∂u

∂x3

∣

∣

L2(Ωǫ)+

1

ǫ2

∣

∣u∣

∣

L2(Ωǫ)

)14

×

×2

∏

i=1

2∑

j=1

(∣

∣

∂2u

∂xi∂xj

∣

∣

L2(Ωǫ)+

∣

∣

∂u

∂xj

∣

∣

L2(Ωǫ)+

∣

∣u∣

∣

L2(Ωǫ)

14

.

Proof. First assume that the domain Ωǫ is of the form (0, 1)2 × (0, ǫ). We use (2.9) with

λ1 = λ2 = 1 and λ3 =1ǫ. This allows us to work in the domain (0, 1)3 and to use (2.7).

Thanks to (2.10), we have

∣

∣u∣

∣

L∞(Ωǫ)≤ c0

∣

∣u∣

∣

14

L2(Ωǫ)

(

∣

∣

∂2u

∂x23

∣

∣

L2(Ωǫ)+

1

ǫ

∣

∣

∂u

∂x3

∣

∣

L2(Ωǫ)+

1

ǫ2

∣

∣u∣

∣

L2(Ωǫ)

)14

×

×2

∏

i=1

(

∣

∣

∂2u

∂x2i

∣

∣

L2(Ωǫ)+

∣

∣

∂u

∂xi

∣

∣

L2(Ωǫ)+

∣

∣u∣

∣

L2(Ωǫ)

)14

.

For a general domain Ωǫ = ω× (0, ǫ), ω a C2 bounded domain of R2, we obtain similarly(2.24): We proceed in the same way in the x3 direction and, in directions x1, x2 we classi-cally proceed by localization using a partition of unity. The localization procedure inducesa mixing of the derivatives in the directions x1 and x2; hence (2.24). �


For any component (Ñǫu)i of Ñǫu, where u ∈ D(Aǫ), we observe that besides (2.2), wehave by integration by parts in x3 :

∫

Ωǫ

(Ñǫu)i∂2(Ñǫu)i

∂x23dx = −

∫

Ωǫ

(∂(Ñǫu)i

∂x3)2dx,

so that∣

∣

∂(Ñǫu)i∂x3

∣

∣

L2(Ωǫ)≤ ǫ

∣

∣

∂2(Ñǫu)i∂x23

∣

∣

L2(Ωǫ),

and

(2.25)∣

∣

∂Ñǫu

∂x3

∣

∣

L2(Ωǫ)≤ ǫ

∣

∣

∂2Ñǫu

∂x23

∣

∣

L2(Ωǫ).

Hence, combining Corollaries 2.1 and 2.2, we have

Corollary 2.3. There exists a positive constant c0(ω), independent of ǫ, such that

(2.26)∣

∣Ñǫu∣

∣

L∞(Ωǫ)≤ c0

∣

∣Ñǫu∣

∣

14

L2(Ωǫ)

( 3∑

i,j=1

∣

∣

∂2Ñǫu

∂xi∂xj

∣

∣

L2(Ωǫ)

)34

, ∀u ∈ D(Aǫ).

The use of Ñǫu in (2.26) is necessary to obtain a constant independent of ǫ, since Ñǫusatisfies (2.2) and (2.25).

Now we give the anisotropic Ladyzhenskaya inequality.

Proposition 2.3. (Anisotropic Ladyzhenskaya’s inequality). Let Ω =∏3

i=1(ai, bi). Thereexists an absolute constant c0 such that

(2.27)∣

∣u∣

∣

L6(Ω)≤ c0

3∏

i=1

(

1

bi − ai∣

∣u∣

∣

L2(Ω)+

∣

∣

∂u

∂xi

∣

∣

L2(Ω)

)13

, ∀u ∈ H1(Ω).

Proof. It is enough to establish the inequality for u ∈ C∞(Ω̄), where Ω̄ denotes the closureof Ω. Without loss of generality, we can assume that u is positive.

We write, with ω =

2∏

i=1

(ai, bi),

(2.28)

∫

Ω

u6(x)dx ≤∫ b3

a3

dx3

[∫ b1

a1

maxx2

u3(x)dx1

][∫ b2

a2

maxx1

u3(x)dx2

]

.


Now we estimate

∫ b1

a1

maxx2

u3(x)dx1 and

∫ b2

a2

maxx1

u3(x)dx2. For ai ≤ xi ≤ai + bi

2, we

have by integration by parts in t (with 1 < p < ∞)

(2.29)

∫

bi−ai2

0

up(x′, xi +bi − ai

2− t)dt = bi − ai

2up(x)

+ p

∫

bi−ai2

0

tup−1(x′, xi +bi − ai

2− t) ∂u

∂xi(x′, xi +

bi − ai2

− t)dt,

but since t ∈ [0, bi − ai2

], we have

(2.30)

bi − ai2

up(x) ≤∫ bi

ai

up(x)dxi +p

2(bi − ai)

∫ bi

ai

up−1(x)∣

∣

∂u

∂xi(x)

∣

∣dxi

≤∫ bi

ai

up−1(x)[∣

∣u(x)∣

∣+p

2(bi − ai)

∣

∣

∂u

∂xi(x)

∣

∣

]

dxi

≤ (Cauchy-Schwarz)

≤(

∫ bi

ai

u2(p−1)(x)dxi

)12(

∫ bi

ai

[

|u(x)| + p2(bi − ai)

∣

∣

∂u

∂xi(x)

∣

∣

]2dxi

)12

.

The inequality above is also valid forai + bi

2≤ xi ≤ bi. Hence, for i, j = 1, 2, i 6= j, we

have (with p = 3)(2.31)

bi − ai2

∫ bj

aj

maxxi

u3(x)dxj ≤(

∫

ω

u4(x)dx′)

12(

∫

ω

[∣

∣u(x)∣

∣+3(bi − ai)

2

∣

∣

∂u

∂xi(x)

∣

∣

]2dx′

)12

.

Therefore,

(2.32)

∫

Ω

u6(x)dx ≤∫ b3

a3

dx3

[∫

ω

u4(x)dx′][ 2

∏

i=1

∫

ω

[2

bi − ai∣

∣u(x)∣

∣+3∣

∣

∂u

∂xi

∣

∣]2dx′]

12

≤(

maxx3

∫

ω

u4(x)dx′)[ 2

∏

i=1

∫

Ω

[2

bi − ai∣

∣u(x)∣

∣+3∣

∣

∂u

∂xi

∣

∣]2dx

]12

.

Finally, we estimate maxx3

∫

ω

u4(x)dx′; we use inequality (2.30) with i = 3 and p = 4 and

obtain after integration with respect to x1 and x2

(2.33)b3 − a3

2

∫

ω

u4(x)dx′ ≤(

∫

Ω

u6(x)dx

)12(

∫

Ω

[

|u(x)| + 2(b3 − a3)∣

∣

∂u

∂x3

∣

∣

]2dx

)12

.


Hence,

(2.34) maxx3

∫

ω

u4(x)dx′ ≤ |u|3L6(Ω)(∫

Ω

[ 2

b3 − a3∣

∣u(x)∣

∣+4∣

∣

∂u

∂x3

∣

∣

]2dx′

)12

.

Combining (2.32) and (2.34), we obtain

∣

∣u∣

∣

6

L6(Ω)≤ 36

∣

∣u∣

∣

3

L6(Ω)

3∏

i=1

(

1

bi − ai∣

∣u∣

∣

L2(Ω)+

∣

∣

∂u

∂xi

∣

∣

L2(Ω)

)13

, ∀u ∈ H1(Ω).

The proof is complete (c0 =3√

36). �

Remark 2.1. Inequality (2.27) can be extended to hold for a general domain Ωǫ = ω×(0, ǫ),where ω is a C2 bounded domain in R2. We have, then, as in the case of Corollary 2.3, amixing of the derivatives in the directions x1 and x2, i.e., we have the following inequality:(2.35)

|u|L6(Ω) ≤ c0(ω)(

1

ǫ|u|L2(Ω) +

∣

∣

∂u

∂x3

∣

∣

L2(Ω)

)13(

|u|L2(Ω) +∣

∣

∂u

∂x1

∣

∣

L2(Ω)+

∣

∣

∂u

∂x2

∣

∣

L2(Ω)

)23

,

for all u in H1(Ωǫ).In order to avoid the appearance of ǫ in (2.35), we need to use the Poincaré inequality

in the thin direction; hence, we work with Ñǫu and obtain

Corollary 2.4. There exists a positive constant c0, independent of ǫ, such that

(2.36)∣

∣Ñǫu∣

∣

2

L6(Ωǫ)≤ c0‖Ñǫu‖2ǫ .

Proof. We take a3 = 0, b3 = ǫ. Then, we have by (2.2) and (2.35)

(2.37)∣

∣Ñǫu∣

∣

L6(Ωǫ)≤ c0(ω)

∣

∣

∂Ñǫu

∂x3

∣

∣

13

ǫ

[

∣

∣Ñǫu∣

∣

ǫ+

∣

∣∇′Ñǫu∣

∣

ǫ

]23

,

and (2.36) follows promptly. �

Using Proposition 2.1, Corollary 2.4 and an interpolation argument, we obtain

Lemma 2.4. For 2 ≤ q ≤ 6, there exists a positive constant c(q), independent of ǫ, suchthat

(2.38)∣

∣Ñǫu∣

∣

2

Lq(Ωǫ)≤ c(q)ǫ

6−q

q ‖Ñǫu‖2ǫ ∀ u ∈ Vǫ.


2.2. Inequalities related to the Stokes operator and the trilinear form.

Let Ω be a bounded open subset of Rn, n ≥ 2, with a C2-boundary Γ = ∂Ω. We denoteby B the second fundamental form of Γ. The quadratic form B is defined as follows (see[W]):

(2.39) BP0(ζ1, ζ2) = −∇ζ1~n · ζ2,

where ζ1 and ζ2 are tangent vectors to Γ at the point P0, the vector ~n denotes the unitoutward normal vector to Γ and ∇ζ1~n denotes the covariant derivative, with respect to ζ1,of the vector ~n. We recall that, in the case of a convex domain Ω, the second fundamentalform is non positive [Gr], [W].

For a vector field v defined on Γ, we denote its normal and tangential components by:

(2.40) vn = v · ~n and vT = v − vn~n.

Similarly, we denote the tangential gradient of a scalar function ϕ by

(2.41) ∇T ϕ = ∇ϕ − (∇ϕ · ~n)~n = ∇ϕ −∂ϕ

∂n~n.

Now we recall the following identity due to Iooss and Grisvard, see [Gr]

Theorem 2.1. Let Ω be a bounded open subset of Rn, and let v ∈ (H1(Ω))n. Then

(2.42)

∫

Ω

∣

∣div v∣

∣

2dx −

n∑

i,j=1

∫

Ω

∂vi

∂xj

∂vj

∂xidx = −2 < (γv)T ;∇T (γv · ~n) >

−∫

Γ

{B((γv)T ; (γv)T ) + tr B[(γv) · ~n]2}dσ.

Here γ is the trace operator from (H1(Ω))n onto (H12 (Γ))n, tr B is the trace of the bilinear

form B (in the “matrix-sense”) and < ·; · > is the duality bracket between (H 12 (Γ))n−1 and(H−

12 (Γ))n−1.

Assume that Ω is convex. If ϕ ∈ H2(Ω)∩H10 (Ω), then writing (2.42) with v = ∇ϕ yields

(2.43)

n∑

i,j=1

∫

Ω

∣

∣

∂2ϕ

∂xi∂xj

∣

∣

2dx =

∫

Ω

∣

∣∆ϕ∣

∣

2dx +

∫

Γ

tr B[γ(∇ϕ) · ~n]2dσ.


However, since Ω is convex, we have tr B ≤ 0. Therefore,

(2.44)n

∑

i,j=1

∫

Ω

∣

∣

∂2ϕ

∂xi∂xj

∣

∣

2dx ≤

∫

Ω

∣

∣∆ϕ∣

∣

2dx.

The inequality (2.44) is also true for any component of a vector field u ∈ H2(Ωǫ) satisfyingany of the boundary conditions under consideration in this article. For the proof we referthe reader to the Appendix in [HR2].

Inequality (2.44) is an important tool for the study of the H2-regularity of second orderstrongly elliptic operators defined in convex domains (see [Gr]). It was used in the studyof reaction-diffusion equations and damped hyperbolic equations on thin domains [HR1,2].This type of inequalities was also used in the study of Navier-Stokes equations on thindomains [RS1,2,3], in the case of the purely periodic boundary condition; the proof wasgiven with the help of Fourier series [RS3]. Note that inequality (2.44) gives a control ofthe L2-norms of “the mixed second derivatives“ of a function ϕ in terms of the L2-norm ofits Laplacian with a constant, which is independent of the shape of the domain (i.e. in ourcase independent of the thickness ǫ). Our purpose now is to obtain similar inequalities forthe Stokes operator under various boundary conditions.

First, we consider the purely periodic boundary condition (PP), the free boundary con-dition (FF) and the boundary condition (FP). For these boundary conditions, we have

(2.45) Aǫu = −∆u, ∀u ∈ D(Aǫ).

We refer to [CF] and [T2] for the proof of (2.45) in the case of the boundary condition(PP), and to [Z1] in the case of the boundary conditions (FF) and (FP). Therefore, wehave

Lemma 2.5. In the case of the boundary condition (PP), (FF) or (FP), we have

(2.46)3

∑

i,j=1

∣

∣

∂2u

∂xi∂xj

∣

∣

2

L2(Ωǫ)≤

∣

∣Aǫu∣

∣

2

L2(Ωǫ), ∀u ∈ D(Aǫ).

For the other boundary conditions, we will use a symmetry argument to establish that theconstant cǫ, appearing in the classical Cattabriga-Solonnikov “H

2-regularity” inequality:

3∑

i,j=1

∣

∣

∂2u

∂xi∂xj

∣

∣

2

L2(Ωǫ)≤ cǫ

∣

∣Aǫu∣

∣

2

L2(Ωǫ), ∀u ∈ D(Aǫ)

is independent of ǫ.


Consider the following Stokes problem:

− ν∆û + ∇p̂ = f̂ in Ω̂ǫ = ω × (−ǫ, ǫ),(2.47)div û = 0 in Ω̂ǫ;(2.48)

and the boundary conditions:

(2.49)∂û1

∂x3=

∂û2

∂x3= 0 and û3 = 0 on ω × {−ǫ, ǫ},

and

(2.50) û = 0 on ∂ω × (−ǫ, ǫ),

where f̂ ∈ Ĥǫ = {v̂ ∈ L2(Ω̂ǫ); div v̂ = 0, v̂ · ~n = 0 on ∂Ω̂ǫ}.

We write f̂ = (f̂ ′, f̂3), with f̂′ = (f1, f2) and analogously, we write û = (û

′, û3). As-

sume that f̂ ′ is an even function with respect to x3 and that f̂3 is odd in x3. Then, it isstraightforward to see that û′ and p̂ are even in x3, while û3 is odd in x3.

Now we are ready to prove the following:

Lemma 2.6. Let (u, p) be a solution of the following Stokes problem:

(2.51)

− ν∆u + ∇p = f in Ωǫ,div u = 0 in Ωǫ,

∂u1

∂x3=

∂u2

∂x3= u3 = 0 on Γt ∪ Γb,

u = 0 on Γl.

Then, there exists a positive constant c, independent of ǫ, such that:

(2.52)3

∑

i,j=1

∣

∣

∂2u

∂xi∂xj

∣

∣

2

L2(Ωǫ)≤ c

∣

∣Aǫu∣

∣

2

L2(Ωǫ), ∀u ∈ D(Aǫ)

where Aǫ is the Stokes operator associated with problem (2.51).

Proof. The regularity of the Stokes operator imply the existence of a positive constant cǫ,dilation invariant, such that

3∑

i,j=1

∣

∣

∂2u

∂xi∂xj

∣

∣

2

L2(Ωǫ)≤ cǫ

∣

∣Aǫu∣

∣

2

L2(Ωǫ)∀u ∈ D(Aǫ).


The constant cǫ is chosen to be the infimum of all constants satisfying (2.52). We extend

f to Ω̂ǫ = ω × (−ǫ, ǫ) in the following way: f1 and f2 are extended to be even functions f̂1and f̂2, and f3 is extended to be an odd function f̂3.

According to the symmetry argument given above, if (û, p̂) is a solution of the Stokesproblem (2.47)-(2.50), then (û1, û2) is even in x3 and û3 is odd in x3. Hence, û

∣

∣

Ωǫ= u and

the regularity of the Stokes problem on Ω̂ǫ implies the existence of a positive constant ĉǫsuch that

(2.53)

3∑

i,j=1

∣

∣

∂2û

∂xi∂xj

∣

∣

2

L2(Ω̂ǫ)≤ ĉǫ

∣

∣Âǫû∣

∣

2

L2(Ω̂ǫ), ∀û ∈ D(Âǫ),

where Âǫ is the Stokes operator defined on Ω̂ǫ.Now according to the symmetry of û and the fact that û

∣

∣

Ωǫ= u, we infer from (2.53)

(2.54) 23

∑

i,j=1

∣

∣

∂2u

∂xi∂xj

∣

∣

2

L2(Ωǫ)≤ 2ĉǫ

∣

∣Aǫu∣

∣

2

L2(Ωǫ), ∀u ∈ D(Aǫ).

Therefore, since cǫ was chosen to be the infimum, we have

cǫ ≤ ĉǫ.

Repeating this argument k times, with k ≥ 1ǫ , we conclude the lemma using the dilationinvariance of cǫ. �

Remark 2.2. The proof of the independence on ǫ of the constant c(ǫ) appearing in theCattabriga-Solonnikov inequality for boundary condition (DP) is similar to the one givenabove and is left as an exercise.

Remark 2.3. The proof of the independence on ǫ of the constant c(ǫ) for the boundarycondition (DD) is technical and long and will be given in a separate article [Z2].

Remark 2.4. The symmetry argument given above and the classical local regularity resultsfor the Stokes operator implies the H2 regularity of the Stokes operator, with the boundarycondition (FF), (FD) or (FP) in the domain with corners Ωǫ.

According to Lemmas 2.5 and 2.6 and Remarks 2.2 and 2.3, we have

Theorem 2.2. Under one of the boundary conditions under consideration, there exists aconstant c, independent of ǫ, such that

3∑

i,j=1

∣

∣

∂2u

∂xi∂xj

∣

∣

2

L2(Ωǫ)≤ c

∣

∣Aǫu∣

∣

2

L2(Ωǫ), ∀u ∈ D(Aǫ).

We end this section with some inequalities related to the trilinear form. Using Lemmas2.5 and 2.6, we prove the following:


Lemma 2.7. Let 0 < q < 12 . There exists a positive constant c4(q), independent of ǫ, suchthat, for anyone of the boundary conditions (FF), (FD), (FP), (PP), (DD) or (DP),wehave

(2.55)∣

∣bǫ(M̃ǫu, Ñǫu, w)∣

∣≤ c4ǫq∣

∣

∣

∣M̃ǫu∣

∣

∣

∣

ǫ

∣

∣AǫÑǫu∣

∣

ǫ

∣

∣w∣

∣

ǫ∀u ∈ D(Aǫ), w ∈ L2(Ωǫ).

(2.56)∣

∣bǫ(Ñǫu, M̃ǫu, w)∣

∣≤ c4ǫ12

∣

∣

∣

∣M̃ǫu∣

∣

∣

∣

ǫ

∣

∣AǫÑǫu∣

∣

ǫ

∣

∣w∣

∣

ǫ∀u ∈ D(Aǫ), w ∈ L2(Ωǫ).

(2.57)

∣

∣bǫ(Ñǫu, Ñǫu, w)∣

∣ ≤ c4∣

∣

∣

∣Ñǫu∣

∣

∣

∣

32

ǫ

∣

∣AǫÑǫu∣

∣

12

ǫ

∣

∣w∣

∣

ǫ

≤ c4ǫ12

∣

∣

∣

∣Ñǫu∣

∣

∣

∣

ǫ

∣

∣AǫÑǫu∣

∣

ǫ

∣

∣w∣

∣

ǫ∀u ∈ D(Aǫ), w ∈ L2(Ωǫ).

Proof. (i) We only treat the boundary conditions (FF), (FD) and (FP). The purely periodiccondition (PP) is treated similarly (the only differences are in the indices of summation in(2.58)and (2.62)). The cases (DD) and (DP) are obvious. For u ∈ D(Aǫ) and w ∈ L2(Ωǫ),we have

∣


∣≤2

∑

i=1

3∑

j=1

∫

Ωǫ

∣

∣Mǫui∣

∣

∣

∣

∂Nǫuj

∂xi

∣

∣

∣

∣wj∣

∣dx

and, according to Hölder’s inequality with exponents p, p∗ and 2, with 1p +1p∗ =

12 , and

2 < p∗ ≤ 6, (p and p∗ will be chosen later in terms of q) we write

(2.58)∣


∣≤2

∑

i=1

3∑

j=1

∣

∣Mǫui∣

∣

Lp(Ωǫ)

∣

∣

∂Nǫuj

∂xi

∣

∣

Lp∗ (Ωǫ)

∣

∣wj∣

∣

L2(Ωǫ).

Since∂Nǫuj

∂xi= Nǫ

(∂uj

∂xi

)

, for i = 1, 2, we are able to apply Lemma 2.4 and obtain

(2.59)

∣

∣

∂Nǫuj

∂xi

∣

∣

Lp∗(Ωǫ)

≤ c(p∗)ǫ6−p∗

2p∗∣

∣

∣

∣

∂Nǫuj

∂xi

∣

∣

∣

∣

ǫ

≤ (Theorem 2.2)

≤ c(p∗)ǫ6−p∗

2p∗∣

∣AǫÑǫu∣

∣

ǫ.

Moreover, since H1(ω) ⊂ Lp(ω), for 2 ≤ p < ∞, we have

(2.60)∣

∣Mǫui∣

∣

Lp(Ωǫ)≤ c(p)ǫ 1p− 12

∣

∣Mǫui∣

∣

H1(Ωǫ).

Combining (2.58), (2.59) and (2.60), we obtain

(2.61)∣


∣≤ c4ǫ1p− 1

2+ 6−p

∗

2p∗∣

∣

∣

∣M̃ǫu∣

∣

∣

∣

ǫ

∣

∣AǫÑǫu∣

∣

ǫ

∣

∣w∣

∣

ǫ.


For 2 < p∗ ≤ 4, the exponent of ǫ in the inequality above q = 2p∗ − 12 , satisfies 0 < q < 12 .Note also that for any q, 0 < q < 12 , there exist p

∗, 2 < p∗ < 4 satisfying q = 2p∗ − 12 .Hence, inequality (2.55) is concluded.

(ii) Now we prove inequality (2.56). We have

(2.62)

∣


∣ ≤2

∑

i=1

2∑

j=1

∫

Ωǫ

∣

∣Nǫui∣

∣

∣

∣

∂Mǫuj

∂xi

∣

∣

∣

∣wj∣

∣dx

≤2

∑

i=1

2∑

j=1

∣

∣Nǫui∣

∣

L∞(Ωǫ)

∣

∣

∂Mǫuj

∂xi

∣

∣

L2(Ωǫ)

∣

∣wj∣

∣

ǫ,

and, according to Corollary 2.3 and Theorem 2.2, we have

(2.63)

∣


∣ ≤ c4ǫ12

2∑

i=1

2∑

j=1

∣

∣AǫÑǫu∣

∣

ǫ

∣

∣

∂Mǫuj

∂xi

∣

∣

L2(Ωǫ)

∣

∣wj∣

∣

L2(Ωǫ)

≤ c4ǫ12

∣

∣

∣

∣M̃ǫu∣

∣

∣

∣

ǫ

∣

∣AǫÑǫu∣

∣

ǫ

∣

∣w∣

∣

L2(Ωǫ).

(iii) Now we prove the inequality (2.57). We write

(2.64)∣


∣≤3

∑

i,j=1

∫

Ωǫ

∣

∣Nǫui∣

∣

∣

∣

∂Nǫuj

∂xi

∣

∣

12∣

∣Nǫui∣

∣

∣

∣

∂Nǫuj

∂xi

∣

∣

12∣

∣wj∣

∣

L2(Ωǫ)dx,

and Hölder’s inequality, with exponents 6, 4, 12 and 2, yields

∣


∣ ≤3

∑

i,j=1

∣

∣Nǫui∣

∣

L6(Ωǫ)

∣

∣

∂Nǫuj

∂xi

∣

∣

12

L2(Ωǫ)

∣

∣

∂Nǫuj

∂xi

∣

∣

12

L6(Ωǫ)

∣

∣wj∣

∣

L2(Ωǫ)

≤ c03

∑

i,j=1

∣

∣

∂Nǫui

∂xj

∣

∣

L2(Ωǫ)

∣

∣

∂Nǫuj

∂xi

∣

∣

12

L2(Ωǫ)

∣

∣

∂2Nǫuj

∂xi∂xj

∣

∣

12

L2(Ωǫ)

∣

∣wj∣

∣

L2(Ωǫ).

Hence

(2.65)∣


∣≤ c4∣

∣

∣

∣Ñǫu∣

∣

∣

∣

32

ǫ

∣

∣AǫÑǫu∣

∣

12

ǫ

∣

∣w∣

∣

ǫ.

We use (see (2.25))∣

∣

∂Nǫuj

∂xi

∣

∣

ǫ≤ ǫ

∣

∣

∂2Nǫuj

∂xi∂x3

∣

∣

ǫi, j = 1, 2, 3,

and obtain, thanks to Corollary 2.4 and Theorem 2.2

(2.66)∣


∣≤ c4ǫ12

∣

∣

∣

∣Ñǫu∣

∣

∣

∣

ǫ

∣

∣AǫÑǫu∣

∣

ǫ

∣

∣w∣

∣

ǫ.

The proof of Lemma 2.7 is now complete. �


3. A priori estimates

In this section we derive some a priori estimates for M̃ǫu and Ñǫu. These estimatesconstitute the main tool in the study of the maximal time of existence of the strong solutionsand also in the study of the behavior of the averages when the thickness goes to zero.

First we write the equations satisfied by M̃ǫu and Ñǫu. Let v ∈ Vǫ (we omit the indicesfor the boundary conditions unless they are necessary). By Lemma 1.1, M̃ǫv, Ñǫv ∈ Vǫand

(M̃ǫu, Ñǫv)ǫ = 0 and ((M̃ǫu, Ñǫv))ǫ = 0,

we obtain, thanks to Lemma 1.1, the following weak formulation for M̃ǫu and Ñǫu :

(3.1)d

dt(M̃ǫu, M̃ǫv)ǫ + ν(A

12ǫ M̃ǫu, A

12ǫ M̃ǫv)ǫ + bǫ(M̃ǫu, M̃ǫu, M̃ǫv)

+ bǫ(Ñǫu, Ñǫu, M̃ǫv) = (M̃ǫf, M̃ǫv)ǫ

and

(3.2)d

dt(Ñǫu, Ñǫv)ǫ + ν(A

12ǫ Ñǫu, A

12ǫ Ñǫv)ǫ + bǫ(M̃ǫu, Ñǫu, Ñǫv)

+ bǫ(Ñǫu, M̃ǫu, Ñǫv) + bǫ(Ñǫu, Ñǫu, Ñǫv) = (Ñǫf, Ñǫv)ǫ.

Now we introduce the following notations: we set

(3.3) a0(ǫ) =∣

∣A12ǫ M̃ǫu0

∣

∣

ǫ, b0(ǫ) =

∣

∣A12ǫ Ñǫu0

∣

∣

ǫ, α(ǫ) =

∣

∣M̃ǫf∣

∣

ǫ, β(ǫ) =

∣

∣Ñǫf∣

∣

ǫ.

We also set

(3.4) R20(ǫ) = a20(ǫ) + b

20(ǫ) + α

2(ǫ) + β2(ǫ).

We have, according to (1.12),Lemma 1.1

∣

∣A12ǫ u0

∣

∣

2

ǫ= a20(ǫ) + b

20(ǫ),

∣

∣f∣

∣

2

ǫ= α2(ǫ) + β2(ǫ)

and

(3.5) R20(ǫ) =∣

∣A12ǫ u0

∣

∣

2

ǫ+

∣

∣f∣

∣

2

ǫ.

Now we recall the following classical fact, which is a consequence of Theorem 0.1 (Seee.g. [T1,2,3]):

(3.6) ∃ Tσ(ǫ) > 0, such that∣

∣A12ǫ u(t)

∣

∣

2

ǫ≤ σR20(ǫ), ∀ 0 ≤ t < Tσ(ǫ).

Here [0, Tσ(ǫ) ) is the maximal interval on which (3.6) holds. It is clear that if Tσ(ǫ) < ∞,then

(3.7)∣

∣A12ǫ u(T

σ(ǫ))∣

∣

2

ǫ= σR20(ǫ).


3.1. Estimates for Ñǫu.

We take v = Aǫu in (3.2) and obtain with Lemma 2.7

(3.8)1

2

d

dt

∣

∣A12ǫ Ñǫu

∣

∣

2

ǫ+ν

∣

∣AǫÑǫu∣

∣

2

ǫ≤ 1

2ν

∣

∣Ñǫf∣

∣

2

ǫ+

ν

2

∣

∣AǫÑǫu∣

∣

2

ǫ

+ c4ǫ12

(∣

∣A12ǫ M̃ǫu

∣

∣

ǫ+

∣

∣A12ǫ Ñǫu

∣

∣

ǫ

)∣

∣AǫÑǫu∣

∣

2

ǫ+c4ǫ

q∣

∣A12ǫ M̃ǫu

∣

∣

ǫ

∣

∣AǫÑǫu∣

∣

2

ǫ,

and, since 0 < ǫ < 1 and 0 < q < 12 , we have

(3.9)d

dt

∣

∣A12ǫ Ñǫu

∣

∣

2

ǫ+

[

ν − 2c4ǫq(∣

∣A12ǫ M̃ǫu

∣

∣

ǫ+

∣

∣A12ǫ Ñǫu

∣

∣

ǫ)]∣

∣AǫÑǫu∣

∣

2

ǫ≤

∣

∣Ñǫf∣

∣

2

ǫ

ν.

Thanks to (3.6), we have

(3.10)∣

∣A12ǫ M̃ǫu(t)

∣

∣

ǫ+

∣

∣A12ǫ Ñǫu(t)

∣

∣

ǫ≤ 2

√σR0(ǫ), 0 ≤ t < Tσ(ǫ),

so that

(3.11)d

dt

∣

∣A12ǫ Ñǫu

∣

∣

2

ǫ+[ν − 4

√σc4ǫ

qR0(ǫ)]∣

∣AǫÑǫu∣

∣

2

ǫ≤

∣

∣Ñǫf∣

∣

2

ǫ

ν, 0 < 2q < 1.

At this stage, we need to make the following assumption:

(H0) limǫ→0

ǫ2qR20(ǫ) = 0, 0 < 2q < 1.

We note that the assumption (H0) is not restrictive, since physically, R0(ǫ) can be assumedto go to zero when ǫ goes to zero, which is the case when f and u0 are independent of ǫ.Thanks to (H0), we can choose ǫ1 = ǫ1(ν, ω, σ) > 0, such that

(3.12) 4√

σc4ǫqR0(ǫ) ≤

ν

2, ∀ǫ, 0 < ǫ ≤ ǫ1.

From (3.11) and (3.12), we can write

(3.13)d

dt

∣

∣A12ǫ Ñǫu

∣

∣

2

ǫ+

ν

2

∣

∣AǫÑǫu∣

∣

2

ǫ≤

∣

∣Ñǫf∣

∣

2

ǫ

ν, 0 < ǫ ≤ ǫ1, 0 < t < Tσ(ǫ),

and since by Cauchy Schwarz inequality

(3.14)∣

∣A12ǫ Ñǫu

∣

∣

2

ǫ≤ ǫ2

∣

∣AǫÑǫu∣

∣

2

ǫ,


we have

(3.15)d

dt

∣

∣A12ǫ Ñǫu

∣

∣

2

ǫ+

ν

2ǫ2∣

∣A12ǫ Ñǫu

∣

∣

2

ǫ≤

∣

∣Ñǫf∣

∣

2

ǫ

ν, 0 < ǫ ≤ ǫ1, 0 < t < Tσ(ǫ).

Finally, Gronwall’s lemma yields

(3.16)∣

∣A12ǫ Ñǫu(t)

∣

∣

2

ǫ≤ b20(ǫ) exp(−

νt

2ǫ2) +

2ǫ2

ν2β2(ǫ), 0 < ǫ ≤ ǫ1, 0 < t < Tσ(ǫ).

We integrate (3.13) and (3.15) to obtain

(3.17)

∫ t

0

∣

∣A12ǫ Ñǫu(s)

∣

∣

2

ǫds ≤ 2ǫ

2

ν2β2(ǫ)t +

2ǫ2

νb20(ǫ), 0 < ǫ ≤ ǫ1, 0 < t < Tσ(ǫ)

and

(3.18)

∫ t

0

∣

∣AǫÑǫu(s)∣

∣

2

ǫds ≤ 2

ν2β2(ǫ)t +

2

νb20(ǫ), 0 < ǫ ≤ ǫ1, 0 < t < Tσ(ǫ).

Now, thanks to Cauchy-Schwarz inequality, we have

(3.19) I =∫ t

0

∣

∣A12ǫ Ñǫu(s)

∣

∣

3

ǫ

∣

∣AǫÑǫu(s)∣

∣

ǫds ≤

≤ sup0≤s≤t

∣

∣A12ǫ Ñǫu(s)

∣

∣

2

ǫ

(∫ t

0

∣

∣A12ǫ Ñǫu(s)

∣

∣

2

ǫds

)

12

(∫ t

0

∣

∣AǫÑǫu(s)∣

∣

2

ǫds

)

12

.

Therefore, combining (3.16), (3.17) and (3.18), we find

(3.20) I ≤(

b20(ǫ) +2ǫ2β2(ǫ)

ν2

) (

2ǫ2β2(ǫ)t

ν2+

2ǫ2b20(ǫ)

ν

)12

(

2β2(ǫ)t

ν2+

2b20(ǫ)

ν

)12

≤ 4ǫ(

b20(ǫ) +ǫ2β2(ǫ)

ν2

) (

β2(ǫ)t

ν2+

b20(ǫ)

ν

)

≤ c5(ν)ǫ(

ǫ2β2(ǫ) + b20(ǫ))(

tβ2(ǫ) + b20(ǫ))

,

where

(3.21) c5(ν) =[

2 max(1,1

ν2,1

ν)]2

.

We collect the previous inequalities in the following lemma:


Lemma 3.1. Under one of the boundary conditions under consideration ,we assume thatfor some 0 < q < 1

2,

limǫ→0

ǫ2q(∣

∣A12ǫ u0

∣

∣

2

ǫ+

∣

∣f∣

∣

2

ǫ) = 0.

Then, there exists ǫ1 = ǫ1(ν) > 0 such that for all ǫ, 0 < ǫ ≤ ǫ1, there exists Tσ(ǫ) > 0and a positive constant c5, independent of ǫ, such that for 0 < ǫ ≤ ǫ1 and 0 < t < Tσ(ǫ):

(i)∣

∣A12ǫ Ñǫu(t)

∣

∣

2

ǫ≤ b20(ǫ) exp(−

νt

2ǫ2) +

2ǫ2

ν2β2(ǫ),

(ii)

∫ t

0

∣

∣AǫÑǫu(s)∣

∣

2

ǫds ≤ 2

ν2β2(ǫ)t +

2

νb20(ǫ),

(iii)

∫ t

0

∣

∣A12ǫ Ñǫu(s)

∣

∣

3

ǫ

∣

∣AǫÑǫu(s)∣

∣

ǫds ≤ c5(ν)ǫ

(

ǫ2β2(ǫ) + b20(ǫ)) (

tβ2(ǫ) + b20(ǫ))

.

Remark 3.1. In the case of the boundary conditions (DD) or (DP), the lemma above isstill true with q = 12 .

We have u = Ñǫu in the case of the boundary conditions (DD) or (DP). Hence, we onlyneed inequality (2.57).

3.2. Estimates for M̃ǫu.

We take v = u in (3.1) and obtain

(3.23)1

2

d

dt

∣

∣M̃ǫu∣

∣

2

ǫ+ν

∣

∣A12ǫ M̃ǫu

∣

∣

2

ǫ+ + bǫ(Ñǫu, Ñǫu, M̃ǫu) = (M̃ǫf, M̃ǫu)ǫ.

According to Lemma 2.7, we have

(3.24)1

2

d

dt

∣

∣M̃ǫu∣

∣

2

ǫ+ν

∣

∣A12ǫ M̃ǫu

∣

∣

2

ǫ≤

∣

∣M̃ǫf∣

∣

ǫ

∣

∣M̃ǫu∣

∣

ǫ+c4

∣

∣A12ǫ Ñǫu

∣

∣

32

ǫ

∣

∣AǫÑǫu∣

∣

12

ǫ

∣

∣M̃ǫu∣

∣

ǫ,

and since

(3.25)∣

∣M̃ǫu∣

∣

ǫ≤ 1

λ1

∣

∣Ã12 M̃ǫu

∣

∣

ǫ,

where λ1 is the first eigenvalue of the corresponding two-dimensional Stokes operator Ã,defined on ω, we obtain

(3.26)d

dt

∣

∣M̃ǫu∣

∣

2

ǫ+ν

∣

∣A12ǫ M̃ǫu

∣

∣

2

ǫ≤

∣

∣M̃ǫf∣

∣

2

ǫ

νλ1+

c4

νλ1

∣

∣A12ǫ Ñǫu

∣

∣

3

ǫ

∣

∣AǫÑǫu∣

∣

ǫ,


and

(3.27)d

dt

∣

∣M̃ǫu∣

∣

2

ǫ+νλ1

∣

∣M̃ǫu∣

∣

2

ǫ≤

∣

∣M̃ǫf∣

∣

2

ǫ

νλ1+

c4

νλ1

∣

∣A12ǫ Ñǫu

∣

∣

3

ǫ

∣

∣AǫÑǫu∣

∣

ǫ.

Thanks to Gronwall’s lemma, we find

(3.28)∣

∣M̃ǫu(t)∣

∣

2

ǫ≤ a20(ǫ)e−νλ1t +

α2(ǫ)

νλ1+

c4

νλ1

∫ t

0

∣

∣A12ǫ Ñǫu(s)

∣

∣

3

ǫ

∣

∣AǫÑǫu(s)∣

∣

ǫds,

and Lemma 3.1 (inequality (iii)) implies that

(3.29)∣

∣M̃ǫu(t)∣

∣

2


α2(ǫ)

νλ1+ c6(ν)ǫR

4n(ǫ)(1 + t),

where c6(ν) = 2c5(ν) is independent of ǫ and

R2n(ǫ) = max(

b20(ǫ), β2(ǫ)

)

.

We infer from (3.26), after integration in t

(3.30)

∫ t

0

∣

∣A12ǫ M̃ǫu(s)

∣

∣

2

ǫds ≤ α

2(ǫ)t

ν2λ1+

a20(ǫ)

νλ1+ c6(ν)ǫR

4n(ǫ)(1 + t).

To find the H1-estimates of M̃ǫu, we take v = Aǫu in (3.1) and obtain

(3.31)1

2

d

dt

∣

∣A12ǫ M̃ǫu

∣

∣

2

ǫ+ν

∣

∣AǫM̃ǫu∣

∣

2

ǫ= (M̃ǫf, AǫM̃ǫu)ǫ

− bǫ(M̃ǫu, M̃ǫu, AǫM̃ǫu) − bǫ(Ñǫu, Ñǫu, AǫM̃ǫu).

We rewrite (3.31), using the L2-scalar product and the L2-norm on ω, as

(3.32)1

2

d

dt

∣

∣Ã12 M̃ǫu

∣

∣

2

L2(ω)+ν

∣

∣ÃM̃ǫu∣

∣

2

L2(ω)= (M̃ǫf, AǫM̃ǫu)L2(ω)

− b̃(M̃ǫu, M̃ǫu, ÃM̃ǫu) −1

ǫbǫ(Ñǫu, Ñǫu, AǫM̃ǫu),

where Ã and b̃ are the two-dimensional Stokes operator and trilinear form respectively.

Now we distinguish the three types of boundary conditions:

Type I. It contains the boundary conditions (FF) and (FP). Note that for these boundaryconditions, we have (see Lemma 1.2)

(3.33) b̃(M̃ǫu, M̃ǫu, ÃM̃ǫu) = 0.


Type II. It contains the boundary conditions (FD) and (PP). The lack of the orthogonalidentity (3.33) forces us to use the following inequality:

(3.34)

∣

∣b̃(M̃ǫu, M̃ǫu, ÃM̃ǫu)∣

∣ ≤ c7∣

∣M̃ǫu∣

∣

12

L2(ω)

∣

∣Ã12 M̃ǫu

∣

∣

L2(ω)

∣

∣ÃM̃ǫu∣

∣

32

L2(ω)

≤ (with Young’s inequality)

≤ ν4

∣

∣ÃM̃ǫu∣

∣

2

L2(ω)+

c8

ν3

∣

∣M̃ǫu∣

∣

2

L2(ω)

∣

∣Ã12 M̃ǫu

∣

∣

4

L2(ω),

where c7 and c8 are independent of ǫ.

Type III. It contains the boundary conditions (DP) and (DD). Note that, according to

our definition of M̃ǫu in these cases, there is nothing to do. M̃ǫu = 0.

First we establish the a priori estimates in the case of the boundary conditions of TypeI. We infer from (3.32), (3.33) and Young’s inequality

(3.35)d

dt

∣

∣Ã12 M̃ǫu

∣

∣

2

ǫ+ν

∣

∣ÃM̃ǫu∣

∣

2

ǫ≤

2∣

∣M̃ǫf∣

∣

2

ǫ

ν+ c9

∣

∣A12ǫ Ñǫu

∣

∣

3

ǫ

∣

∣AǫÑǫu∣

∣

ǫ,

and according to∣

∣Ã12 M̃ǫu

∣

∣

2

ǫ≤ 1

λ1

∣

∣ÃM̃ǫu∣

∣

2

ǫ,

we have

(3.36)d

dt

∣

∣Ã12 M̃ǫu

∣

∣

2

ǫ+νλ1

∣

∣Ã12 M̃ǫu

∣

∣

2

ǫ≤

2∣

∣M̃ǫf∣

∣

2

ǫ

ν+ c9

∣

∣A12ǫ Ñǫu

∣

∣

3

ǫ

∣

∣AǫÑǫu∣

∣

ǫ.

Thanks to Gronwall’s lemma and Lemma 3.1, we find

(3.37)∣

∣Ã12 M̃ǫu(t)

∣

∣

2


2∣

∣M̃ǫf∣

∣

2

ǫ

ν2λ1+ c10(ν)ǫR

4n(ǫ)(1 + t).

We collect the previous inequalities in the following lemma:

Lemma 3.2. Under one of the boundary conditions (FF) or (FP), we assume that forsome 0 < q < 12 ,

limǫ→0

ǫ2q(∣

∣A12ǫ u0

∣

∣

2

ǫ+

∣

∣f∣

∣

2

ǫ) = 0.


Then, there exists ǫ1 = ǫ1(ν) > 0 such that for all ǫ, 0 < ǫ ≤ ǫ1, there exists Tσ(ǫ) > 0 anda positive constant c10(ν), independent of ǫ, such that

(i)∣

∣M̃ǫu(t)∣

∣

2


α2(ǫ)

νλ1+ c6(ν)ǫR

4n(ǫ)(1 + t).

(ii)

∫ t

0

∣

∣A12ǫ M̃ǫu(s)

∣

∣

2

ǫds ≤ α

2(ǫ)t

ν2λ1+

a20(ǫ)

νλ1+ c6(ν)ǫR

4n(ǫ)(1 + t).

(iii)∣

∣Ã12 M̃ǫu(t)

∣

∣

2


2∣

∣M̃ǫf∣

∣

2

ǫ

ν2λ1+ c10(ν)ǫR

4n(ǫ)(1 + t).

(iv)

∫ t

0

∣

∣ÃM̃ǫu(s)∣

∣

2

ǫds ≤ 2α

2(ǫ)t

ν+

2a20(ǫ)2(ǫ)

νc10(ν)ǫR

4n(ǫ)(1 + t).

Remark 3.2. Note that the inequalities of Lemma 3.2 are similar to those of the 2D- Navier-Stokes equations with the periodic boundary condition; we have, however, the perturbationterm c10(ν)ǫR

4n(ǫ)(1 + t), which is small for ǫ small enough.

We turn now to the study of the case of the boundary conditions of Type II. We have,thanks to (3.32), (3.34) and Young’s inequality

(3.38)d

dt

∣

∣Ã12 M̃ǫu

∣

∣

2

L2(ω)+νλ1

∣

∣Ã12 M̃ǫu

∣

∣

2

L2(ω)≤

2∣

∣M̃ǫf∣

∣

2

L2(ω)

ν+

c8

ν3

∣

∣M̃ǫu∣

∣

2

L2(ω)

∣

∣Ã12 M̃ǫu

∣

∣

4

L2(ω)

+c11

ǫ

∣

∣A12 Ñǫu

∣

∣

3

ǫ

∣

∣AǫÑǫu∣

∣

ǫ.

We set

(3.39) y(t) =∣

∣Ã12 M̃ǫu(t)

∣

∣

2

L2(ω), h(t) =

2∣

∣M̃ǫf∣

∣

2

L2(ω)

ν+

c11

ǫ

∣

∣A12 Ñǫu

∣

∣

3

ǫ

∣

∣AǫÑǫu∣

∣

ǫ

and

(3.40) g(t) =c8

ν3

∣

∣M̃ǫu∣

∣

2

L2(ω)

∣

∣Ã12 M̃ǫu

∣

∣

2

L2(ω).

We infer from (3.38)

(3.41)dy

dt≤ gy + h,

and Gronwall’s lemma yields

(3.42) y(t) ≤ y(0) exp(

∫ t

0

g(τ)dτ

)

+

∫ t

0

h(s) exp

(∫ t

s

g(τ)dτ

)

ds.


First we bound

∫ t

0

g(τ)dτ using (3.28) and (3.29). We have

(3.43)

∫ t

0

g(τ)dτ ≤ c8ν3

sup0≤τ≤t

∣

∣M̃ǫu(τ)∣

∣

2

L2(ω)

∫ t

0

∣

∣Ã12 M̃ǫu(τ)

∣

∣

2

L2(ω)dτ

≤ c12(ν, λ1)[

a20(ǫ) + α2(ǫ)

ǫ+ R4n(ǫ)(1 + t)

] [

tα2(ǫ) + a20(ǫ)

ǫ+ R4n(ǫ)(1 + t)

]

≤ c12(ν, λ1)(

R2m(ǫ) + R4n(ǫ)(1 + t)

)

[

tα2(ǫ) + a20(ǫ)

ǫ+ R4n(ǫ)(1 + t)

]

.

where

(3.44) R2m(ǫ) = max(

a20(ǫ), α2(ǫ)

)

.

Now we write

(3.45)

∫ t

0

h(s) exp

(∫ t

s

g(τ)dτ

)

ds ≤(

∫ t

0

h(s)ds

)

exp

(∫ t

0

g(τ)dτ

)

≤(

2tR2m(ǫ)

ν+ c6R

4n(ǫ)(1 + t)

)

exp

(∫ t

0

g(τ)dτ

)

.

Therefore,

(3.46)∣

∣Ã12 M̃ǫu

∣

∣

2

L2(ω)≤ c13(ν)ǫ

(

R2m + R4n

)

exp[

c12(

R2m + R4n(1 + t)

)(

R2m + R4n

)

(1 + t)]

.

Here we make the following assumption on the initial data (for the boundary conditionsof Type II):

(H1) For arbitrary K1 > 0 and K2 > 0; R2m(ǫ) ≤ K1 ln

∣

∣ln ǫ∣

∣, R2n(ǫ) ≤ K2 ln∣

∣ln ǫ∣

∣.

Taking into account the assumption (H1), we c

navier-stokes equations in three-dimensional thin …navier-stokes equations in thin domains 5 (fp)...

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