MI Preprints|九州大学 マス・フォア・インダストリ研究所 math 5ae2774857bb1

MI Preprint Series

Mathematics for Industry

Kyushu University

Global existence of solutions of the

compressible viscoelastic fluid

around a parallel flow

Yusuke Ishigaki

MI 2018-3

( Received April 26, 2018 )

Institute of Mathematics for Industry

Graduate School of Mathematics

Global existence of solutions of the compressible

viscoelastic fluid around a parallel flow

Yusuke Ishigaki

Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

Abstract

We consider a system of equations for a motion of a compressible viscoelastic fluid in an infinite layer. When the external force has a suitable form, the system has a solution of parallel flow type. It is shown that the solution of the system exists globally in time if the initial data is sufficiently close to the one of the parallel flow, provided that the Reynolds number and the Mach number are sufficiently small, and that the speed of the propagation of shear wave is sufficiently large.

Keywords: Compressible viscoelastic fluid; parallel flow; global existence.

1

Introduction

This paper is concerned with the stability of parallel flows of the compressible viscoelastic fluid system of the Oldroyd model

∂tρ+ div(ρv) = 0, (1.1)

ρ(∂tv+v· ∇v)−µ∆v−(µ+µ′)∇divv+∇P(ρ) =αdiv(ρF⊤F) +ρg, (1.2)

∂tF+v· ∇F=∇vF, (1.3)

in an infinite layer Ωl=Rn−1×(0, l):

Ωl={x= (x′, xn); x′= (x1, . . . , xn−1)∈Rn−1, 0< xn< l} (n= 2,3).

Here ρ = ρ(x, t), v = v(x, t) and F = F(x, t) are the unknown density, the velocity field and the deformation tensor, respectivity, at the timet >0 and positionx∈Ωl;P =P(ρ) is the pressure;µand

µ′ _{are the viscosity coefficients satisfyingµ >0 and} 2

nµ+µ

′_{>0;α >0 is the constant called the speed}

of propagation of shear wave;gis an external force which has the form

g=⊤(g1(xn, t),0,· · ·,0), g1(0, t) =g1(l, t) = 0, (1.4) whereg1 _{is a given smooth function of (x}

n, t) converging tog∞1 =g1∞(xn)̸= 0 astgoes to infinity. Here

We assume thatP is a smooth function ofρand satisfies

P′(ρ∗)>0

for a given positive numberρ∗.

The system (1.1)–(1.3) is one of the basic model of the hydrodynamical system with the elastic effect ([3, 9, 4, 16]). The first equation (1.1) and the second equation (1.2) are the compressible Navier-Stokes equations including the elastic forceαdiv(ρF⊤_{F). The third equation (1.3) is derived by differentiating}

F = ∂x

∂X in t([9, 16]). HereX denotes the material coordinate andx=x(X, t) is defined under the flow map

 



dx

dt =v(x(X, t), t), x(X,0) =X.

Whenghas the form of (1.4) and is suitably smooth, the system (1.1)–(1.3) has a solution representing a parallel flow, more precisely, a solution of the form ¯u=⊤_{(¯ρ,v,¯ F¯) with ¯ρ=ρ}

∗ and ¯v = ¯v1(xn, t)e1,

wheree1=⊤(1,0,· · ·,0)∈Rn. In this paper we are interested in the stability of the parallel flow ¯u.

Parallel flows have been widely studied as a good example in the hydrodynamic stability theory. As for the mathematical study of the stability of parallel flows of viscoelastic fluid, Endo, Giga, G¨otz and Liu [1] considered the stability of parallel flows under incompressible perturbations. In [1], parallel flows induced by pressure gradient were considered and it was proved that if the pressure gradient is sufficiently small, then the parallel flow is exponetially stable under sufficiently small initial perturbations. As for the compressible case, there seems no mathematical study of the stability of parallel flows, while the stability of the motionless state has been studied so far. See [4, 5, 6, 15, 18, 19] for the stability of the motionless state in the whole space and [14] for the problems on the half space, bounded domains and exterior domains. We also refer to [10, 11, 12] for the stability of the motionless state in the incompressible case. In this paper we show that the parallel flow is exponentially stable under sufficiently small perturba-tions, ifµ,P′_(ρ

∗) andαare sufficiently large compared tog1. We briefly present the main result of this

paper in a more precise way. We introduce the following non-dimensional variables:

˜

x=1

lx, ˜t= V

l t, ˜v=

1

Vv, ρ˜=

1

ρ∗

ρ, F˜ =F, g˜= l

V2g, P˜=

1

ρ∗P′(ρ∗)

P, V = ρ∗∥g

1_∥

L∞l2

µ .

The system (1.1)–(1.3) is then rewritten into the following dimensionless one on the layer Ω = Ω1 = Rn−1×(0,1):

∂_t˜ρ˜+ divx(˜˜ ρv˜) = 0, (1.5)

˜

ρ(∂˜tv˜+ ˜v· ∇˜x˜v)−ν∆˜x˜v−(ν+ν′)∇xdiv˜ ˜x˜v+∇˜xP˜(˜ρ) =β2div˜x(˜ρF˜⊤F˜) + ˜ρg,˜ (1.6)

∂t˜F˜+ ˜v· ∇x˜F˜ =∇x˜˜vF .˜ (1.7)

Hereν,ν′_{,γ andβ are the nondimensional parameters defined by}

ν= µ

ρ∗lV

, ν′₌ µ′

ρ∗lV

, γ=

√ P′_(ρ

∗)

V , β=

√

We note that Re= 1

ν andM a =

1

γ are the Reynolds number and the Mach number. We consider the system (1.5)–(1.7) under the non-slip boundary condition

˜

v|xn=0,1= 0 (1.8)

and the initial condition

˜

ρ|t=0= ˜ρ0, v˜|t=0= ˜v0, F˜|t=0= ˜F0. (1.9)

We also impose the periodic boundary condition in ˜x′_:

˜

ρ,˜v,F˜ : 2π

αj−

periodic in ˜xj, j= 1, . . . , n−1.

In what follows we abbreviate ˜x, ˜t, ρ,˜ v,˜ F ,˜ and ˜g asx, t, ρ, v, F,andg, respectively.

Under a suitable condition on g, we see that there exists a parallel flow (¯ρ,¯v,F¯) of (1.5)–(1.7) with the following properties:

¯

ρ= 1, v¯= ¯v1(xn, t)e1, F¯= (∇(x−ψ¯1e1))−1,

∥¯v(t)∥2H5≤Ce−c0κt

(

∥v¯0∥2H5+O

( ₁ ν2

)

+O

( ₁ κν2

)) ,

∥∂t¯v(t)∥2H3 ≤Ce−c0κt

( β4

ν2∥¯v0∥ 2

H5+O(1) +O

(

1

κ ))

,

∥F¯(t)−F¯∞∥2H4 ≤Ce−c0κt

(

1

ν2∥v¯0∥ 2

H5+O

(

1

β4

)

+O

(

1

κβ4

)) ,

whereκ= min{ν,β_ν2}, ¯ψ1_(x

n, t) =

∫t

0¯v 1_(x

n, s)ds, ¯ψ1∞=β−2(−∂x2n)

−1_g1

∞, and ¯F∞= (∇(x−ψ¯1∞e1))−1.

Here (−∂2

xn)

−1_{is the inverse of −∂}2

xn with domainD(−∂ 2

xn) =H

2_(0,1)∩H1 0(0,1).

We denote the periodic cell byD:

D={x= (x′, xn);x′= (x1,· · ·, xn−1)∈Πnj=1−1T2π

αj,0< xn <1}.

HereT2π αj =R/

(

2π αj

)

Z,αj>0,j = 1, . . . , n−1.

We will show that if ν ≫1,γ ≫1,β ≫1 and∥¯v0∥2_H5_(0,1)≪1, then (1.5)–(1.9) has a unique global

solution (ρ, v, F) such that (ρ, v, F) ∈C([0,∞), H2_{(D)) and ∥(ρ(t), v(t), F(t))−(1,v¯(t),F¯(t))∥}

H2 → 0

exponentially as t → ∞, provided that (ρ0−1, v0−v¯0, F0−F¯0) ∈ H2(D) is sufficiently small. As a

result, we have ∥(ρ(t), v(t), F(t))−(1,0,F¯∞)∥H2 → 0 exponentially as t → ∞. We thus see that if

g∞=g1∞e1̸= 0, then the viscoelastic compressible flow converges to the motionless state with nontrivial

deformation ¯F∞ due to the elastic force. This is quite in contrast to the case of the usual viscous

compressible fluid where nontrivial flows are in general observed when external forces are nontrivial. In fact, in the case of the usual viscous compressible fluid, under the action ofg∞ =g∞1 e1̸= 0, a parallel

flow with non-zero velocity field is stable for sufficiently small perturbations whenν ≫1 andγ≫1; see [7].

The proof of the main result of this paper is based on the Matsumura-Nishida energy method [13] which gives an appropriate a priori estimate of exponential decay type. To establish the a priori estimate, we introduce the displacement vectorψ as in [12, 14]:

It then follows thatF is written in terms of ψas

F = ¯F−F¯∇(

ψ−(−ψ¯1e1)

)_¯ F+h(

∇(ψ−(−ψ¯1e1))

) ,

where h satisfies h(∇(ψ−(−ψ¯1_(t)e

1)) = O(|∇(ψ−(−ψ¯1(t)e1))|2). By using ψ, the problem for the

perturbation is reduced to the one foru(t) = (ϕ(t), w(t), ζ(t)) =(

ρ(t)−1, v(t)−v¯(t), ψ(t)−(−ψ¯1_(t)e 1))

which takes the following form:

     

    

∂tϕ+ divw=f1,

∂tw−ν∆w−ν˜∇divw+γ2∇ϕ+β2(∆ζ+K∞ζ) =f2,

∂tζ+w−w· ∇ψ¯∞=f3,

w|xn=0,1= 0, ζ|xn=0,1= 0, (ϕ, w, ζ)|t=0= (ϕ0, w0, ζ0).

(1.10)

Here ˜ν = ν +ν′ _{and ¯ψ}

∞ = ¯ψ1∞e1; K∞ζ is a linear term of ζ satisfying ∥K∞ζ∥L2 ≤ C

β2∥∇ζ∥H1;

fi_{, i = 1,2,3 are written in a sum of nonlinear terms and linear terms with coefficients including} ¯

v,ψ¯1_−ψ¯1

∞which decay exponentially int. Applying a variant of the Matsumura-Nishida energy method

given in [14] to (1.10) and estimating the interaction between the parallel flow and the perturbation, we establish the estimate:

∥u(t)∥2H2_×H2_×H3+

∫ t

0

e−c1(t−s)

∥u(s)∥2H2_×H3_×H3 ds≤Ce−c1t∥u₀∥_H22_×H2_×H3,

provided thatν≫1,γ≫1,β ≫1, and the initial perturbation is sufficiently small.

We finally mention one remark. Wheng(xn, t)≡g∞1 (xn)e1, g∞1 ̸= 0, the parallel flow in this paper

is a stationary solution ¯u∞ = (1,0,F¯∞(xn)), which represents the motionless state with nontrivial

deformation given by ¯F∞. When β = 0, we formally obtain the usual compressible Navier-Stokes

equations. In this case, the system (1.5)–(1.7) with β = 0 has a parallel flow ¯us = (1,¯vs(xn)) with ¯

vs(xn)̸= 0; and it was shown by Kagei [7] that the parallel flow ¯usis stable ifν ≫1 andγ≫1. On the other hand, the main result of this paper shows that the parallel flow ¯uin the viscoelastic compressible fluid is stable ifν≫1,γ≫1 and β≫1. Namely, the motionless state ¯u∞ with nontrivial deformation

is stable ifβ≫1, while the parallel flow with non-zero velocity field is stable ifβ= 0. This leads to an interesting question what happens whenβ decreases; it should occur some transition to nontrivial flows at some value ofβ. We will investigate this issue in the future work.

This paper is organized as follows. In Section 2 we introduce some notations and function spaces. In Section 3 we show the existence of the parallel flow and then state the main result of this paper on the stability of the parallel flow. In Section 4 we establish the a priori estimate which ensures the global existence of the perturbation and its exponential decay ast → ∞. In the appendix, we give a proof of the existence of the parallel flow and its estimates.  

2

Notation

In this section we introduce notations and function spaces which will be used throughout this paper. In this paper, we consider functions 2π

αj-periodic inxj, j= 1, . . . , n−1, and therefore, we set

D= Πn−1

j=1T2π

whereT2π αj =R/

(

2π αj

)

Z,αj >0,j= 1, . . . , n−1.

For 1 ≤p ≤ ∞, we denote by Lp_{(D) the Lebesgue space on D and its norm is denoted by ∥ · ∥} Lp.

Let m be a nonnegative integer. We denote by Hm_{(D) the m-th order L}2 _{Sobolev space on D with}

norm∥ · ∥Hm. In paticular, H0(D) is same asL2(D). We denote by H₀1(D) the completion ofC∞

c (D) inH1_{(D). HereC}∞

c (D) is the set of allC∞functions with compact support inD.

We simply denote byLp_{(D) (resp.,H}m_{(D)) the set of all vector fieldsw=}⊤_(w1_{, . . . , w}n_{) onD with}

wj_∈Lp_{(D) (resp. H}m_{(D)),j= 1, . . . , n, and its norm is also denoted by ∥ · ∥}

Lp (resp. ∥ · ∥_Hm).

The inner product ofL2_{(D) is denoted by}

(f, g) =

∫

D

f(x)g(x) dx, f, g∈L2(D).

Partial derivatives of a functionuinxi, (i= 1, . . . , n) andtare denoted by∂xiuand∂tu, respectivity.

We write m-th order partial derivatives of uas ∇m_{u = {∂}α

xu | |α| = m}. We also write m-th order tangential derivatives of u as ∂m_{u= {∂}α

xu| |α| =m, αn = 0} and we abbriviate ∂ =∂1. We denote div, ∇ and ∆ by the usual divergence, gradient and Laplacian with respect to x, respectivity. For a vector fieldw=⊤_(w1_{, . . . , w}n_{) and a functionu, we denote ∇}′_·w′ _{and ∆}′_uby

∇′·w′= n−1

∑

j=1

∂xjw

j

and

∆′u= n−1

∑

j=1

∂x2ju,

respectivity. For w = ⊤_(w1_{, . . . , w}n_{), we denote by ∇w the matrix whose (i, j) component (∇w)ij is} given by∂xjw

i_{. Divergence of a matrix-valued functionF = (F}ij₎

1≤i,j≤n is denoted by (divF)i=

n

∑

j=1

∂xjF

ij_.

In paticular, let I be the n×n identity matrix. Then the following identities hold for a scalar-valued functionuand a vector fieldw:

div(uI) =∇u, div(∇w) = ∆w, div(⊤(∇w)) =∇divw.

For matrix-valued functionsF = (Fij₎

1≤i,j≤n andG= (Gij)1≤i,j≤n, we denoteF∇Gby

(F∇G)i= (div(G⊤F)−G(div⊤F))i = n

∑

k,l=1

Flk∂xlG

ik_.

3

Main Result

In this paper, we impose the following conditions forρ0, F0:

div(ρ0⊤F0) = 0, (3.1)

ρ0detF0= 1. (3.2)

It then follows from (1.5) that these quantities are conserved.

Lemma 3.1 If(ρ, v, F)is solution of the problem(1.5)–(1.9), then the following identities hold fort≥0:

div(ρ⊤F) = 0, (3.3)

ρdetF = 1. (3.4)

See [15, Proposition 1] for a proof of Lemma 3.1.

We introduce the parallel flow. We show the existence of a parallel flow (¯ρ,¯v,F¯) of (1.5)–(1.7), as in [1], satisfying

¯

ρ= ¯ρ(xn, t), ρ¯|t=0= 1

¯

v= ¯v1(xn, t)e1, v¯1|xn=0,1= 0, v¯1|t=0= ¯v01,

¯

F = ¯F(xn, t), F¯|t=0=I.

Letx(X, t) =⊤_(x1_{(X, t),· · ·, x}n_{(X, t)) be the flow map given by}

 



dx

dt(X, t) = ¯v(x

n_{(X, t), t),}

x(X,0) =X.

This yields

x(X, t) =X+

(∫ t

0

¯

v1(xn, s)ds

) e1.

We set

¯

ψ1(xn, t) =

∫ t

0

¯

v1(xn, s)ds. (3.5)

By using the flow mapx(X, t), deformation tensor ¯F is written by

¯

F = ∂x

∂X = (_∂x

i

∂Xj

)

1≤i,j≤n

.

It is easy to see that

¯

F =∇(x+ ¯ψ1e1) =

(

∇(x−ψ¯1e1)

)−1

=



     

1 0 . . . 0 ∂xnψ¯ 1

0 1 . . . 0 0 ..

. ... . .. ... ... 0 0 . . . 1 0 0 0 . . . 0 1



      .

We set ¯ρ= 1. Inserting (¯ρ,¯v,F¯) into (1.6), we see that and ¯ψ1_satisfies

{ ∂2

tψ¯1−β2∂x2nψ¯ 1_−ν∂

t∂x2nψ¯ 1_=g1_,

¯

ψ1_(x

n,0) = 0, ∂tψ¯1(xn,0) = ¯v1(xn,0).

We also assume the compatibility conditions forg1_:

g1_{(0, t) =g}1_{(1, t) = 0, t≥0,}

∂2

xng

1_{(0, t) =∂}2

xng

1_{(1, t) = 0, t≥0.} (3.7)

The existence of the parallel flow is now stated as follows.

Proposition 3.2 Let κ = min{ν,β_ν2}, Assume that g1 _{∈ H}1

loc([0,∞);H3(0,1)) satisfies the

compati-bility conditions(3.7), and thatg1

∞∈H3(0,1). Ifg1 satisfies ec0κt∂tg1∈L2((0,∞);H3(0,1)) for some

positive constantc0, then the following assertions hold.

If(¯ρ,¯v,F¯)|t=0= (1,v¯0, I)andv¯0∈H5(0,1), then there exist a parallel flow (¯ρ,¯v,F¯)of (3.1) that has

the following properties:

(i) ¯ρ= 1.

(ii)There existsψ¯1_{= ¯ψ}1_(x

n, t)∈Rsuch that

¯

ψ1|t=0= 0, F¯=∇(x+ ¯ψ1e1) =

(

∇(x−ψ¯1e1)

)−1

, ∂tψ¯1= ¯v1.

(iii) ¯v∈C1_([0,∞);H3_(0,1)).

(iv)There hold the following estimates uniformly fort≥0:

∥¯v1(t)∥2H5_(0,1)≤Ce−c0κt

(

∥v¯0∥2H3_(0,1)+

1

ν2∥g 1₍₀₎

∥2H1_(0,1)+

1

κν2∥e

c0κt_∂

tg1∥2L2_(0,∞;H1_(0,1))

)

, (3.8)

∥∂t¯v1_(t)∥2

H3_(0,1)≤Ce−c0κt

(_β4

ν2∥v¯0∥ 2

H5_(0,1)+∥g1(0)∥2_H1_(0,1)+

1

κ∥e

c0κt_∂

tg1∥2L2_(0,∞;H1_(0,1))

)

, (3.9)

∥ψ¯1(t)−ψ¯∞1 ∥2H5_(0,1)≤Ce−c0κt

(

1

ν2∥v¯0∥ 2

H5_(0,1)+

1

β4∥g 1₍₀₎

∥2H3_(0,1)+

1

κβ4∥e

c0κt_∂

tg1∥2L2_(0,∞;H3_(0,1))

) .

(3.10)

Hereψ¯1

∞ satisfies

−β2∂x2nψ¯ 1

∞=g1∞,ψ¯1∞|xn=0,1= 0,F¯(xn, t)→F¯∞(xn) =∇(x+ ¯ψ1∞e1) =(∇(x−ψ¯∞1 e1))

−1

,

and the following estimate

∥ψ¯1∞∥H5_(0,1)≤

C β2∥g

1

∞∥H3_(0,1). (3.11)

The proof of Proposition 3.2 will be given in the Appendix.  

We next consider the stability of the parallel flow ¯u = ⊤_{(1,v,¯ F¯). We will show that under some}

assumptions on ν, γ, and β, the perturbation of ¯u exists globally in time and decay exponentially as

t→ ∞. To this end, we first state the local time existence of the solution of the problem (1.5)–(1.9). By a similar argument to that in [8, 14, 17], one can prove the following local existence of solutions.

Proposition 3.3 If (ρ0, v0, F0) ∈ H2(D) satisfies (3.1)–(3.2) and ρ0 ≥ 1₂, then there exists

posi-tive numbers T and C such that the following assertion holds. The problem (1.5)–(1.9) has a unique

solution (ρ, v, F) ∈ C([0, T];H2_{(D)) satisfying ∂}

tρ, ∂tF ∈ C([0, T];L2(D)), v ∈ L2([0, T];H3(D)),

∥(ρ(t), v(t), F(t))∥H2 ≤C∥(ρ₀, v₀, F₀)∥_H2

for0≤t≤T.

As for the global existence, we have the following result.

Theorem 3.4 In addition to the assumptions of the Propositions 3.2 and 3.3, there are positive numbers

ν0,γ0 andβ0 such that if ν ≥ν0, γ

2

ν+ν′ ≥γ₀2 and β2

γ2 ≥β

2

0, then the following assertion holds. There is

a positive numberϵ0 such that if (ρ0, v0, F0)∈H2(D)andv¯0∈H5(0,1)satisfies ∥(ρ0−1, v0−v¯0, F0−

¯

F0)∥2_H2_(D)+∥v¯0∥2_H5_(0,1) ≤ ϵ0,

∫

D(ρ0−1)dx= 0, then there exists a unique global solution (ρ, v, F) ∈

C([0,∞);H2_{(D))of the problem(1.5)–(1.9), and the perturbationU(t) = (ρ(t)−1, v(t)−v¯(t), F(t)−F¯(t))}

satisfies

∥U(t)∥2H2+

∫ t

0

e−c1(t−s)

∥U(s)∥2H2_×H3_×H2 ds≤Ce−c1t∥U0∥2_H2

fort≥0.

4

A priori estimate

Theorem 3.4 is proved by combining Proposition 3.3 and the following a priori estimate.

Proposition 4.1 There exist positive numbersν0,γ0andβ0such that ifν ≥ν0, γ

2

ν+ν′ ≥γ₀2and β

2

γ2 ≥β₀2,

then the following assertion holds. Let T be an arbitrarily given positive number. Then there exists a

positive constant δ such that if ∥¯v0∥2_H5_(0,1)+ ˜E(t) ≤ δ uniformly for t ∈ [0, T], it holds the following

estimate:

˜

E(t) +

∫ t

0

e−c1(t−s)_D˜_(s)ds≤C

(

e−c1t_E˜_{(0) +}

∫ t

0

e−c1(t−s)_R˜_(s)ds

)

uniformly fort∈[0, T], whereC is a positive constant independent ofT. HereE˜(t)and D˜(t) are some quantities equivalent to

∥U(t)∥2H2+∥∂_tU(t)∥2_L2

and

∥U(t)∥2

H2_×H3_×H2+∥∂tU(t)∥2L2_×H1_×L2,

respectivity;_R˜(t)is a function satisfying

˜

R(t)≤C (₁

ν +

1

β2 +

√_ν

β + γ β

)

˜

D(t) + ( ˜E(t)21 + ˜E(t)) ˜D(t)

uniformly fort∈[0, T]with a positive constantC independent of T .

By a standard argument, one can show that Propositions 3.3 and 4.1 imply Theorem 4.2 if∥U0∥H2+ ∥v¯0∥2_H5_(0,1) is small enough andν ≥ν0,

γ2

ν+ν′ ≥γ₀2, β

2

To prove Proposition 5.1, we introduce the displacement vector and write the perturbation equation by using the displacement vector in place ofF. We denote the displacement vector byψ:

ψ(x, t) =X−1(x, t)−x.

We see thatψsatisfies

∂tψ+v=−v· ∇ψ,

ψ|{xn=0,1} = 0.

See [12, 14]. We also note thatF has its inverseF−1_{fort≥0 by Lemma 3.1, andF}−1 _{is written as}

G=F−1= ∂X

−1

∂x .

We assume thatGandX−1 _satisfy

{

G(x,0) =∇X−1_{(x,0) forx∈D,}

X−1=xon{xn= 0,1} fort≥0.

(4.1)

Lemma 4.2 IfG=F−1 _{satisfies the condition(4.1), then}

G=∇X−1 _(4.2)

forx∈D andt≥0.

Proof. A direct computation shows thatGis a solution of the following transport equation:

∂tG+v· ∇G+G∇v= 0. (4.3)

We also see that∇X−1 _{satisfies the same equation asG. By (4.1), Gand∇X}−1 _{have the same initial}

value, and therefore, the uniqueness of solutions of (4.3) implies that G=∇X−1_{. This completes the}

proof. ■

In terms ofψ, we see from (4.2) thatF is written as

F = (I+∇ψ)−1=I− ∇ψ+h(∇ψ).

Here

h(∇ψ) = (I+∇ψ)−1_−I+∇ψ,

h(∇( ¯ψ1_e 1)) = 0.

It then follows that (ρ, v, ψ) satisfies

∂tρ+ div(ρv) = 0,

∂tv+v· ∇v−

ν ρ∆v−

ν+ν′

ρ ∇divv+

∇p(ρ)

ρ =β

2₍

−∆ψ+N(∇ψ)) +g,

∂tψ+v=−v· ∇ψ,

⊤_{(I− ∇ψ)∇ρ=ρ∇divψ−div(ρ}⊤_h),

where

N(∇ψ) =∇(h(∇ψ)) + (∇ψ)∇(∇ψ)−(∇ψ)∇(h(∇ψ))−(h(∇ψ))∇(∇ψ) + (h(∇ψ))∇(h(∇ψ)).

We setρ= 1 +ϕ, v= ¯v+w andψ=−ψ¯1_e

1+ζ. Since

F−F¯=−F¯∇ζF¯+h1_(∇ζ).

with

h1=h( ¯F∇ζ) ¯F ,

|h1|=O(|∇ζ|2) for|∇ζ| ≪1,

we see thatu= (ψ, w, ζ) is a solution of the following initial boundary problem:

∂tϕ+ divw=f1, (4.4)

∂tw−ν∆w−ν˜∇divw+γ2∇ϕ+β2(∆ζ+K∞ζ) =f2, (4.5)

∂tζ+w−w· ∇ψ¯∞=f3, (4.6)

∇ϕ=∇divζ+M∞ζ+f4, (4.7)

w|xn=0,1= 0, ζ|xn=0,1= 0, (ϕ, w, ζ)|t=0= (ϕ0, w0, ζ0). (4.8)

Here ˜ν =ν+ν′_{, ¯ψ}

∞= ¯ψ1∞e1;K∞ζandM∞ζ are given by

K∞ζ= div( ¯E∞∇ζ+ ¯E∞∇ζ+ ¯E∞∇ζE¯∞) + ( ¯F∞∇ζF¯∞)∇E¯∞+ ¯E∞∇( ¯F∞∇ζF¯∞),

M∞ζ= div(⊤( ¯E∞∇ζ+∇ζE¯∞+ ¯E∞∇ζE¯∞))−E¯∞⊤div(⊤( ¯F∞∇ζF¯∞)),

and fi_{, i = 1,2,3,4, denote the sum of nonlinear terms and linear terms with coefficients including} ¯

v, ψ¯exp= ¯ψ−ψ¯∞;

f1_=f1

L+fN1;

f1

L=−¯v· ∇ϕ, fN1 =−div(ϕw),

f2_{= (f}2,1_{, . . . , f}2,n_{) =f}2

L+fN2;

f2

L=−¯v· ∇w−w· ∇¯v−β2Kexpζ,

f_N2 =−w· ∇w+ νϕ

1 +ϕ(−∆w+ ∆¯v)−

˜

νϕ

1 +ϕ∇divw− γ2_ϕ

1 +ϕ∇ϕ− γ2

1 +ϕ∇Q(ϕ) +β

2_h2_,

f3_{= (f}3,1_{, . . . , f}3,n_{) =f}3

L+fN3;

f3

L=−w· ∇ψ¯exp−v¯· ∇ζ, fN3 =−w· ∇ζ,

f4_{= (f}4,1_{, . . . , f}4,n_{) =f}4

L+fN4;

f4

where ¯

E∞= ¯F∞−I=∇( ¯ψ1∞e1), E¯exp= ¯F−F¯∞=∇( ¯ψexp1 e1),

Q(ϕ) =ϕ2 ∫ 1

0

P′′(1 +sϕ)ds, ∇Q=O(ϕ)∇ϕfor|ϕ| ≪1, h2_{= ¯F∇h}1_{+ ( ¯F∇ζF¯)∇( ¯F∇ζF¯−h}1_{) +h}1_{∇( ¯F−F¯∇ζF¯+h}1_),

Kexpζ= ( ¯F∇ζF¯)∇E¯exp+ ( ¯F∞∇ζE¯exp+ ¯Eexp∇ζF¯∞+ ¯Eexp∇ζE¯exp)∇E¯∞

+ ¯Eexp∇( ¯F∇ζF¯) + ¯F∞∇( ¯F∞∇ζE¯exp+ ¯Eexp∇ζF¯∞+ ¯Eexp∇ζE¯exp),

Mexpζ=⊤F¯∞−1div⊤( ¯Eexp∇ζF¯∞+ ¯F∞∇ζE¯exp+ ¯Eexp∇ζE¯exp)−⊤E¯expdiv(⊤( ¯F∇ζF¯)).

Since ρ0 = 1 +ϕ0 andF0 =I− ∇ζ0+h(∇ζ0), (3.1), (3.2), and ∫_D(ρ0−1)dx= 0 yield the following

relations;

⊤_(I

− ∇ζ0+h(∇ζ0))∇ϕ0= (1 +ϕ0)(∇divζ0−div(⊤(h(∇ζ0))), (4.9)

(1 +ϕ0) det(I− ∇ζ0+h(∇ζ0)) = 1, (4.10)

∫

D

ϕ0dx= 0. (4.11)

We then have the following a priori estimate for the perturbationu= (ϕ, w, ζ).

Proposition 4.3 Under the assumption of Proposition 4.1, the following assertion holds. There exists

a positive constantδwithδ <1such that if _∥v¯0∥2H5_(0,1)+E(t)≤δuniformly for t∈[0, T], then it holds

the following estimate:

E(t) +

∫ t

0

e−c1(t−s)_D(s)ds ≤C

(

e−c1t_{E(0) +}

∫ t

0

e−c1(t−s) R(s)ds

)

(4.12)

uniformly fort∈[0, T]with a positive constantC independent of T. HereE(t)andD(t)are equivalent to

∥u(t)∥2H2_×H2_×H3+∥∂_tu(t)∥2_L2_×L2_×H1,

and

∥u(t)∥2H2_×H3_×H3+∥∂_tu(t)∥2_L2_×H1_×H1

respectivity;R(t)is a function satisfying

R(t)≤C (₁

ν +

1

β2 +

√

ν β +

γ β

)

D(t) + (E(t)21 +E(t))D(t)

uniformly fort∈[0, T]with a positive constantC independent of T .

Proposition 4.1 immediately follows from Proposition 4.3. In the remaining of this paper, we will give a proof of Proposition 4.3.

5

Proof of Proposition 4.3

Let T be an arbitrarily positive given number. Throughout this section, we assume that u(t) = (ϕ(t), w(t), ζ(t)) is a solution of (4.4)–(4.11) on [0, T].

Proposition 5.1 Let j and k be nonnegative integers satisfying 0 ≤ 2j+k ≤ 2. Then it holds the estimate:

1 2

d dt(γ

2

∥∂jt∂kϕ∥2L2+∥∂j_t∂kw∥2_L2+β2∥∇∂_tj∂kζ∥2_L2) +ν∥∇∂_tj∂kw∥_L22+ ˜ν∥div∂_tj∂kw∥2_L2

≤β2(_|(K

∞∂tj∂kζ, ∂ j

t∂kw)|+|(∇(∂ j

t∂kw· ∇ψ¯∞),∇∂tj∂kζ)|

)

+N1

j,k,

(5.1)

where

Nj,k1 =γ2|(∂ j t∂kf1, ∂

j

t∂kϕ)|+|(∂ j t∂kf2, ∂

j

t∂kw)|+β2|(∂ j

t∂kf3,∆∂ j t∂kζ)|.

Proof. We consider the case j =k = 0 only. The other cases can be treated similarly. We take the

inner product of (4.4) withϕto obtain

1 2

d dt∥ϕ∥

2

L2+ (divw, ϕ) = (f1, ϕ).

By integration by parts, we have (divw, ϕ) =−(w,∇ϕ), and therefore,

1 2

d dt∥ϕ∥

2

L2−(w,∇ϕ) = (f1, ϕ). (5.2)

We take the inner product of (4.5) withwto obtain

1 2

d dt∥w∥

2

L2−ν(∆w, w)−ν˜(∇divw, w) +γ2(∇ϕ, w) +β2(∆ζ, w) +β2(K∞ζ, w) = (f2, w)

By integration by parts, we obtain

1 2

d dt∥w∥

2

L2+ν∥∇w∥2_L2+ ˜ν∥divw∥2_L2+γ2(∇ϕ, w) +β2(∆ζ, w) +β2(K∞ζ, w) = (f2, w). (5.3)

We take the inner product of (4.6) with−∆ζ to obtain

−(∂tζ,∆ζ)−(w,∆ζ) + (w· ∇ψ¯∞,∆ζ) =−(f3,∆ζ).

By integration by parts, we have

−(∂tζ,∆ζ) = (∂t∇ζ,∇ζ) = 1 2

d dt∥∇ζ∥

2

L2,

and

(w· ∇ψ¯∞,∆ζ) =−(∇(w· ∇ψ¯∞),∇ζ).

We thus obtain

1 2

d dt∥∇ζ∥

2

L2−(w,∆ζ)−(∇(w· ∇ψ¯∞),∇ζ) =−(f3,∆ζ). (5.4)

It then follows fromγ2_{×(5.2) + (5.3) +β}2_{×(5.4) that}

1 2

d dt(γ

2

∥ϕ∥2L2+∥w∥2_L2+β2∥∇ζ∥2_L2) +ν∥∇w∥2_L2+ ˜ν∥divw∥2_L2

=β2(

(K∞ζ, w) + (∇(w· ∇ψ¯∞),∇ζ))+γ2(f1, ϕ) + (f2, w)−β2(f3,∆ζ).

(5.5)

Proposition 5.2 Letk be a nonnegative integer satisfying0≤k≤2. Then it holds the estimate:

−d

dt(∂

k_{w, ∂}k_{ζ) +}β2 2 ∥∇∂

k_ζ∥2

L2+

γ2

2 ∥div∂ k_ζ∥2

L2

≤

(

1 + ν

2

2β2

)

∥∇∂k_w∥2

L2+

˜

ν2

2γ2∥div∂

k_w∥2

L2

+γ2_|(M

∞∂kζ, ∂kζ)|+β2|(K∞∂kζ, ∂kζ)|+|(∂kw· ∇ψ¯∞, ∂kw)|+Nk2,

(5.6)

where

Nk2=|(∂kf2, ∂kζ)|+|(∂kf3, ∂kw)|+γ2|(∂kf4, ∂kζ)|.

Proof. We consider the casek= 0 only. We take the inner product of (4.5) with−ζ to obtain

−(∂tw, ζ) +ν(∆w, ζ) + ˜ν(∇divw, ζ)−γ2(∇ϕ, ζ)−β2(∆ζ, ζ)−β2(K∞ζ, ζ) =−(f2, ζ). (5.7)

The first term on the left-hand side of (5.7) is written as

−(∂tw, ζ) =− d

dt(w, ζ) + (w, ∂tζ).

By integration by parts, the fifth term of (5.7) is written as −(∆ζ, ζ) =∥∇ζ∥2L2. It then follows from

(5.7) that

−_dd_t(w, ζ) + (w, ∂tζ) +β2∥∇ζ∥2L2−γ2(∇ϕ, ζ) =−ν(∆w, ζ)−ν˜(∇divw, ζ) +β2(K∞ζ, ζ)−(f2, ζ).

(5.8)

We take the inner product of (4.6) with−wto obtain

−(w, ∂tζ) =∥w∥2L2−(w· ∇ψ¯∞, w)−(f3, w). (5.9)

By (5.8) + (5.9), we obtain

−_dd_t(w, ζ) +β2∥∇ζ∥2L2−γ2(∇ϕ, ζ)

=−ν(∆w, ζ)−ν˜(∇divw, ζ) +∥w∥2

L2 −β2_(K

∞ζ, ζ)−(w· ∇ψ¯∞, w)−(f2, ζ)−(f3, w).

(5.10)

We take the inner product of (4.7) withζto obtain

(∇ϕ, ζ) = (∇divζ, ζ) + (M∞ζ, ζ) + (f4, ζ).

By integration by parts, we have (∇divζ, ζ) =−∥divζ∥2

L2.We thus obtain

(∇ϕ, ζ) +∥divζ∥2L2 = (M∞ζ, ζ) + (f4, ζ). (5.11)

By (5.10) +γ2_{×(5.11), we have}

−d

dt(w, ζ) +β

2

∥∇ζ∥2L2+γ2∥divζ∥2_L2

=−ν(∆w, ζ)−ν˜(∇divw, ζ) +∥w∥2

L2

+β2_(K

∞ζ, ζ) +γ2(M∞ζ, ζ)−(w· ∇ψ¯∞, w)−(f2, ζ)−(f3, w) +γ2(f4, ζ).

By integration by parts, we have

−ν(∆w, ζ) =ν(∇w,∇ζ)≤ β

2

2 ∥∇ζ∥

2

L2+

ν2

2β2∥∇w∥ 2

L2,

−˜ν(∇divw, ζ) = ˜ν(divw,divζ)≤ γ

2

2 ∥divζ∥

2

L2+

˜

ν2

2γ2∥divw∥ 2

L2.

It then follows from (5.12) that

−_dd_t(w, ζ) +β

2

2 ∥∇ζ∥

2

L2+

γ2

2 ∥divζ∥

2

L2

≤

(

1 + ν

2

2β2

)

∥∇w∥2L2+

˜

ν2

2γ2∥divw∥ 2

L2

+β2_|(K

∞ζ, ζ)|+γ2|(M∞ζ, ζ)|+|(w· ∇ψ¯∞, w)|+|(f2, ζ)|+|(f3, w)|+γ2|(f4, ζ)|.

(5.13)

This proves (5.6) fork= 0. The case 0< k≤2 can be proved similarly by applying∂k _{to (4.5), (4.6)} and (4.7). This completes the proof. ■

Proposition 5.3 Letk be a nonnegative integers satisfying 0≤k≤1. Then it holds the estimate:

1 2

d dt(ν∥∇∂

k_w(t)∥2

L2+ ˜ν∥div∂kw(t)∥2_L2 −2γ2(∂kϕ,div∂kw)−2β2(∇∂kζ,∇∂kw)) +1

2∥∂t∂ k_w

∥2L2 ≤β2∥∇∂k_w∥2

L2+γ2∥div∂kw∥2_L2+β2

(

|(K∞∂kζ, ∂t∂kw)|+|(∇(∂kw· ∇ψ¯∞),∇∂kw)|

)

+Nk3,

(5.14)

where

N_k3=γ2|(∂k_f1_,div∂k_w)|+1 2∥∂

k_f2

∥2L2+β2|(∇∂kf3,∇∂kw)|.

Proof. We consider the casek= 0. We take the inner product of (4.4) with−divwto obtain

−(∂tϕ,divw)− ∥divw∥2L2 =−(f1,divw).

Since

−(∂tϕ,divw) =− d

dt(ϕ,divw) + (ϕ,div∂tw),

we obtain

−_dd_t(ϕ,divw) + (ϕ,div∂tw) =∥divw∥2L2−(f1,divw). (5.15)

We take the inner product of (4.5) with∂twto obtain

∥∂tw∥2L2−ν(∆w, ∂_tw)−ν˜(∇divw, ∂_tw) +γ2(∇ϕ, ∂_tw) +β2(∆ζ, ∂_tw) +β2(K_∞ζ, ∂_tw) = (f2, ∂_tw)

By integration by parts, we have

−(∆w, ∂tw) = (∂t∇w,∇w) = 1 2

d dt∥∇w∥

2

L2,

−(∇divw, ∂tw) = (∂tdivw,divw) = 1 2

d dt∥divw∥

2

L2,

(∆ζ, ∂tw) =−(∇ζ, ∂t∇w).

We thus obtain 1 2

d dt

(

ν∥∇w∥2L2+ ˜ν∥divw∥2_L2

)

+∥∂tw∥2L2−γ2(ϕ,div∂_tw)−β2(∇ζ,∇∂_tw)

=−β2_(K

∞ζ, ∂tw) + (f2, ∂tw).

(5.16)

We take the inner product of (4.6) with ∆wto obtain

(∂tζ,∆w) + (w,∆w)−(w· ∇ψ¯∞,∆w) = (f3,∆w).

By integration by parts, we have

(∂tζ,∆w) =−(∂t∇ζ,∇w) =− d

dt(∇ζ,∇w) + (∇ζ, ∂t∇w),

(w,∆w) =−∥∇w∥2L2,

−(w· ∇ψ¯∞,∆w) = (∇(w· ∇ψ¯∞),∇w).

We thus obtain

−d

dt(∇ζ,∇w) + (∇ζ, ∂t∇w) =∥∇w∥

2

L2−(∇(w· ∇ψ¯∞),∇w)−(∇f3,∇w). (5.17)

Byγ2_{×(5.15) + (5.16) +β}2_{×(5.17), we have}

1 2

d

dt(ν∥∇w∥

2

L2+ ˜ν∥divw∥_L22−2γ2(ϕ,divw)−2β2(∇ζ,∇w)) +∥∂_tw∥2_L2

=β2∥∇w∥2L2+γ2∥divw∥2_L2−β2

(

(K∞ζ, ∂tw) + (∇(w· ∇ψ¯∞),∇w))

−γ2_(f1_{,divw) + (f}2_{, ∂}

tw) +β2(∇f3,∇ζ).

(5.18)

This, together with the inequality|(f2_{, ∂}

tw)| ≤ 1₂∥∂tw∥2_L2+1₂∥f2∥2_L2, proves (5.14) fork= 0. The case

k= 1 can be proved similarly by applying∂ to (4.4), (4.5) and (4.6). This completes the proof. ■

We next estimatexn-derivatives ofϕ. We introduce the following quantities :

˙

ϕ=∂tϕ+ (¯v+w)· ∇ϕ, (5.19)

q=νw−β2ζ. (5.20)

Note that

˙

ϕ=−divw−ϕdivw. (5.21)

Proposition 5.4 Let kand l be nonnegative integers satisfyingl ≥1,0≤k+l−1≤1. Then it holds the estimate:

1 2

d dt∥∂

k_∂l xnϕ∥

2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂

k_∂l xnϕ∥

2

L2+b0 ν+ ˜ν

β2_+γ2∥∂

k_∂l xnϕ˙∥

2

L2

≤ C

(β2_+γ2_{)(ν+ ˜ν)}

{

∥∂t∇l−1∂kw∥2L2+∥∇l∂k+1q∥2_L2+β4(∥∇l−1K_∞∂kζ)∥2_L2

+∥∇l−1_M

∞∂kζ∥2L2)

}

+CNk,l4 .

(5.22)

Here b0>0 is a constant independent ofν, ν, γ˜ 2 andβ2, and

N4

k,l=

1

(β2_+γ2_{)(ν+ ˜ν)}(∥∂

k_∇l−1_f2_∥2

L2+β4∥∂k∇l−1f4∥2_L2)

+ ν+ ˜ν

β2_+γ2∥∂

k

∇l(ϕdivw)∥L22+|(∂k∂_xl

n((¯v+w)· ∇ϕ), ∂

Proof. We consider the casek= 0 andl= 1 only. We see from then-th equation of (4.5)

∂twn−ν∆wn−ν∂˜ xndivw+γ 2_∂

xnϕ+β

2_(∆ζn_{+ (K}

∞ζ)n) =f2,n,

which is rewritten by using (5.20) as

−∂x2nq

n

−ν∂˜ x2nw

n_+γ2_∂

xnϕ=−∂tw

n_{+ ∆}′_qn_{+ ˜ν∂} xn∇

′

·w′−β2(K∞ζ)n+f2,n. (5.23)

By applying∂xn to (5.21), we have

∂x2nw

n_+∂

xnϕ˙=−∂xn∇

′

·w′−∂xn(ϕdivw). (5.24)

By (5.23) + ˜ν×(5.24), we obtain

−∂x2nq

n_{+ ˜ν∂} xnϕ˙+γ

2_∂

xnϕ=−∂tw

n_{+ ∆}′_qn

−β2(K∞ζ)n+f2,n−ν∂˜ xn(ϕdivw). (5.25)

We see from then-th equation of (4.7) that

−∂x2nζ

n_+∂

xnϕ=∂xn∇

′

·ζ′+ (M∞ζ)n+f4,n. (5.26)

Byν×(5.24) +β2_{×(5.26), we obtain}

∂x2nq

n_+ν∂

xnϕ˙+β 2_∂

xnϕ=−∂xn∇

′

·q′+β2(M∞ζ)n+β2f4,n−ν∂xn(ϕdivw). (5.27)

By (5.25) + (5.27), we obtain

(ν+ ˜ν)∂xnϕ˙+ (β

2_+γ2_)∂

xnϕ

=−∂twn+ ∆′qn−∂xn∇

′_·q′_−β2_((K

∞ζ)n+ (M∞ζ)n) +β2f4,n−(ν+ ˜ν)∂xn(ϕdivw).

(5.28)

This gives

∂xnϕ˙+

β2_+γ2

ν+ ˜ν ∂xnϕ

= 1

ν+ ˜ν(−∂tw

n_{+ ∆}′_qn

−∂xn∇

′

·q′)

− β

2

ν+ ˜ν((K∞ζ)

n_{+ (M}

∞ζ)n) + 1

ν+ ˜νf

2,n₊ β2

ν+ ˜νf

4,n

−∂xn(ϕdivw).

(5.29)

We take the inner product of (5.29) with∂xnϕto obtain

(∂xnϕ, ∂˙ xnϕ) +

β2_+γ2

ν+ ˜ν ∥∂xnϕ∥ 2

L2

= 1

ν+ ˜ν(−∂tw

n_{+ ∆}′_qn_−∂ xn∇

′

·q′, ∂xnϕ)

− β

2

ν+ ˜ν((K∞ζ)

n_{+ (M}

∞ζ)n, ∂xnϕ) +

β2

ν+ ˜ν(f

4,n_{, ∂}

xnϕ)−(∂xn(ϕdivw), ∂xnϕ).

(5.30)

By the definition of ˙ϕ, we have

(∂xnϕ, ∂˙ xnϕ) =

1 2

d

dt∥∂xnϕ∥ 2

The right-hand side of (5.30) is estimated as

1

ν+ ˜ν(−∂tw

n_{+ ∆}′_qn_−∂ xn∇

′

·q′, ∂xnϕ)−

β2

ν+ ˜ν((K∞ζ)

n_{+ (M}

∞ζ)n, ∂xnϕ)

+ 1

ν+ ˜ν(f

2,n_{, ∂} xnϕ) +

β2

ν+ ˜ν(f

4,n_{, ∂}

xnϕ)−(∂xn(ϕdivw), ∂xnϕ)

≤ 1₂β

2_+γ2

ν+ ˜ν ∥∂xnϕ∥ 2

L2

+ 4

(ν+ ˜ν)(β2_+γ2₎(∥∂tw

n_∥2

L2+ 2∥∇∂q∥2_L2) +

4β4

(ν+ ˜ν)(β2_+γ2₎(∥K∞ζ∥ 2

L2+∥M_∞ζ∥2_L2)

+ 4

(ν+ ˜ν)(β2_+γ2₎∥f 2_∥2

L2+

4β4

(ν+ ˜ν)(β2_+γ2₎∥f 4_∥2

L2+

4(ν+ ˜ν)

β2_+γ2∥∂xn(ϕdivw)∥ 2

L2.

(5.32)

It follows from (5.30)–(5.32) that

1 2

d dt∥∂xnϕ∥

2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂xnϕ∥ 2

L2

≤ _(β2_+γC2_{)(ν+ ˜ν)}

{

∥∂tw∥2L2+∥∇∂q∥2_L2+β4(∥K_∞∂kζ∥2_L2+∥M_∞∂kζ∥2_L2)

}

+ C

(β2_+γ2_{)(ν+ ˜ν)}(∥f 2

∥2L2+β4∥f4∥2_L2)

+C ν+ ˜ν

β2_+γ2∥∂xn(ϕdivw)∥ 2

L2+C|(∂xn((¯v+w)· ∇ϕ), ∂xnϕ)|.

(5.33)

We deduce from (5.30) and (5.33) that

ν+ ˜ν

β2_+γ2∥∂xnϕ˙∥ 2

L2

≤C (

β2_+γ2

ν+ ˜ν ∥∂xnϕ∥ 2

L2+

1

(β2_+γ2_{)(ν+ ˜ν)}(∥∂tw∥ 2

L2+∥∇∂q∥2_L2)

+ β

4

(β2_+γ2_{)(ν+ ˜ν)}(∥K∞∂

k_ζ∥2

L2+∥(M_∞∂kζ)∥2_L2)

+ 1

(β2_+γ2_{)(ν+ ˜ν)}(∥f 2

∥2L2+β4∥f4∥2_L2) +

ν+ ˜ν

β2_+γ2∥∂xn(ϕdivw)∥ 2

L2

) .

(5.34)

We thus obtain (5.22) fork= 0 andl= 1 by adding (5.33) tob0×(5.34) withb0>0 satisfyingb0C≤1₄.

The casel≥1, 0< k+l−1≤1 can be proved similarly by applying∂k_∂l−1

xn to (5.29). This completes

the proof. ■

To estimate higher order derivatives ofwandζand tangential derivatives ofϕ, we prepare the following estimates for the Stokes system.

Lemma 5.5 LetQ1∈Hk(Ω)andQ2∈Hk+1(Ω) with

∫

ΩQ1dx= 0. If(u, p)satisfies the Stokes system

  

 

divu=Q1 in Ω,

−∆u+∇p=Q2 in Ω,

u= 0 on {xn= 0,1}.

Then there exists a positive constantC independent of(u, p) such that

∥∇k+2_u

Proposition 5.6 It holds the following estimate:

∥∇2_∂q∥

L2+γ2∥∇∂ϕ∥_L2

≤C(∥∂t∂w∥L2+ (ν+ ˜ν)∥∂ϕ˙∥_H1+β2∥∂ϕ∥_H1+β2∥K_∞∂ζ∥_L2+β2∥M_∞ζ∥_H1

+(ν+ ˜ν)∥∂(ϕdivw)∥H1+∥f2∥_H1+β2∥f4∥_H1+∥∇∂q∥_L2).

(5.36)

Proof. By (4.5), (5.19) and (5.20), we have

div(∂q) =Q1 in Ω,

−∆∂q+∇(γ2ϕ) =Q2in Ω,

∂q= 0 on{xn = 0,1},

where

Q1=ν∂ϕ˙−β2∂ϕ+β2(M∞ζ)′−ν∂(ϕdivw) +β2(f4)′,

Q2=−∂t∂w−˜ν∇∂ϕ˙−ν˜∇∂(ϕdivw)−β2K∞∂ζ+∂f2.

We apply Lemma 6.6 withk= 0 to obtain

∥∇2∂q∥L2+γ2∥∇∂ϕ∥_L2 ≤C(∥Q₁∥_H1+∥Q₂∥_L2+∥∇∂q∥_L2). (5.37)

This completes the proof. ■

Proposition 5.7 Let i andl be nonnegative integers satisfying i= 1, . . . , n−1, l= 0,1. Then it hold the estimates:

∥∂l+2

xn q

i_∥ L2 ≤C

(

∥∂t∂xlnw

i_∥

L2+∥∂l

xn∂ 2_q∥

L2+ ˜ν∥∂l

xn∂xiϕ˙∥L2+γ 2_∥∂l

xn∂ϕ∥L2

+β2_∥∂l

xn(K∞ζ)∥L2+∥∂

l xnf

2_∥

L2+ ˜ν∥∂l

xn∂xi(ϕdivw)∥L2

)

, (5.38)

and

∥∂l+2

xn q

n_∥ L2 ≤C

( ν∥∂l+1

xn ϕ˙∥L2+β 2_∥∂l+1

xn ϕ∥L2+∥∂

l+1

xn ∂q∥L2

+β2_∥∂l

xn(M∞ζ)∥L2+ν∥∂

l+1

xn (ϕdivw)∥L2+β 2_∥∂l

xnf 4_∥

L2).

(5.39)

Proof. We see from thei-th equation of (4.5) fori= 1, . . . , n−1 that

∂twi−∆qi+ ˜ν∂xi( ˙ϕ+ϕdivw) +γ 2_∂

xiϕ+β 2_(K

∞ζ)i=f2,i,

which is rewritten as

∂2xnq

i₌

−∂twi+ ∆′qi−ν∂˜ xiϕ˙−γ 2_∂

xiϕ−β 2_(K

∞ζ)i+f2,i−˜ν∂xi(ϕdivw)

Proposition 5.8 It hold the estimates:

∥∂kϕ∥L2≤ ∥∇∂kζ∥_L2+∥∂k−1M_∞ζ∥_L2+∥∂k−1f4∥_L2, k= 1,2, (5.40)

ν∥∂_x2_nw∥L2 ≤ ∥∂_tw∥_L2+ (ν+ ˜ν)∥∇∂w∥_L2+γ2∥∇ϕ∥_L2+β2∥∇2ζ∥_L2+β2∥K_∞ζ∥_L2+∥f2∥_L2, (5.41)

∥∂tϕ∥L2 ≤ ∥∇w∥_L2+∥f1∥_L2, (5.42)

∥∂t∇ζ∥L2 ≤ ∥∇w∥_L2+∥∇(w· ∇ψ¯_∞)∥_L2+∥∇f3∥_L2, (5.43)

1 2

d dt∥∇

k_ζ

∥2L2+

β2

2ν∥∇

k_ζ

∥2L2 ≤

1 2νβ2∥∇

k_q

∥2L2+|(∇k(w· ∇ψ¯∞),∇kζ)|+|(∇kf3,∇kζ)|, k= 2,3.

(5.44)

Proof. We see from (4.7) that

∂k_ϕ=∂k_divζ+∂k−1_(M

∞ζ)′+∂k−1(f4)′,

which gives (5.40).

As for (5.41), we see from (4.5) that

−ν∂_x2_nw′ =−∂tw′+ν∆′w′+ ˜ν∂divw−γ2∂ϕ−β2(∆ζ′+ (K∞ζ)′) + (f2)′,

and

−(ν+ ˜ν)∂x2nw

n₌

−∂twn+ν∆′wn+ ˜ν∇′·w′−γ2∂xnϕ−β

2_(∆ζn_{+ (K}

∞ζ)n) +f2,n,

which gives (5.41).

The estimate (5.42) immediately follows from the equation (4.4). As for (5.43), we∇k _{to (4.6) to obtain}

∂t∇kζ=−∇kw+∇k(w· ∇ψ¯∞) +∇kf3 fork= 1,2,3, (5.45)

which gives (5.43).

We take the inner product of (5.45) with∇k_{ζ and usew=} 1

νq+ β2

ν ζ from (5.20) to obtain,

1 2

d dt∥∇

k_ζ

∥2L2+

β2

ν ∥∇

k_ζ

∥2L2+

1

ν(∇

k_q,

∇kζ)−(∇k(w· ∇ψ¯∞),∇kζ) = (∇kf3,∇kζ).

This leads to (5.44). This completes the proof. ■

We are now in a position to prove Proposition 4.3.

Proof of Proposition 4.3. By using the Poincar´e inequality and integration of parts, we have

|(K∞∂kζ, ∂kw)|+|(∇(∂kw· ∇ψ¯∞),∇∂kζ)| ≤

C β2∥∇∂

k_ζ∥2

L2∥∇∂kw∥2_L2.

From (5.1), we obtain

d dtE

k

0 +d0Dk0 ≤

C νγ2∥∇∂

k_ζ

∥2L2+

1

γ2N 1

0,k, (5.46)

where

Ek0 =∥∂kϕ∥2L2+

1

γ2∥∂

k_w

∥2L2+

β2

γ2∥∇∂

k_ζ

∥2L2,

D0k=

ν γ2∥∇∂

k_w

∥2L2+

˜

ν γ2∥div∂

k_w

By using the Poincar´e inequality and integration by parts, we have

γ2|(M∞∂kζ, ∂kζ)|+β2|(K∞∂kζ, ∂kζ)|+|(∂kw· ∇ψ¯∞, ∂kw)|

≤C (

1 + γ

2

β2

)

∥∇∂k_ζ∥2

L2+

C β2∥∇∂

k_w∥2

L2.

From (5.6), we obtain

−_γ12

d dt(∂

k_{w, ∂}k_{ζ) +} β2 2γ2∥∇∂

k_ζ

∥2L2+

1 2∥div∂

k_ζ

∥2L2

≤

( ₁ γ2+

ν2

2β2_γ2 +

C β2

)

∥∇∂k_w∥2

L2+

˜

ν2

2γ4∥div∂

k_w∥2

L2

+C ( ₁

β2 +

1

γ2

)

∥∇∂k_ζ∥2

L2+

1

γ2N 2

k.

(5.47)

It follows from (5.46) + (5.47) that

d dt ( Ek 0 − 2

γ2(∂

k_{w, ∂}k_ζ)

) + ( Dk 0+ β2

2γ2∥∇∂

k_ζ∥2

L2+

1 2∥div∂

k_ζ∥2

L2

)

≤

( ₁ γ2 +

ν2

2β2_γ2 +

C β2

)

∥∇∂kw∥2L2+

˜

ν2

2γ4∥div∂

k_w

∥2L2

+C ( ₁

β2 +

1

γ2+

1

νγ2

)

∥∇∂k_ζ∥2

L2+

1

γ2(N 1

0,k+Nk2).

(5.48)

We take ν, ν, γ˜ 2 _{and β}2 _{so that} 1

γ2 +

ν2

2β2_γ2 +

C β2 ≤

ν

4γ2,

ν+˜ν γ2 ≤

1 2 and C

(

1

β2 +

1

γ2 +

1

νγ2

)

≤ β2

4γ2. It

then follows

d dtE

k

1+d1Dk1 ≤CRk1. (5.49)

Here

E1k=E0k−

1

γ2(∂

k_{w, ∂}k_ζ),

Dk

1 =Dk0+

β2

γ2∥∇∂

k_ζ∥2

L2+∥div∂kζ∥2_L2,

Rk1 =

1

γ2(N 1

0,k+Nk2).

We observe thatEk

1 is equivalent toE0k provided that β2>1. By using the Poincar´e inequality and

integration by parts, we have

|(K∞∂kζ, ∂t∂kw)|+|(∇(∂kw· ∇ψ¯∞),∇∂kw)|

≤ _βC2(∥∇∂

k_ζ

∥H1∥∂_t∂kw∥_L2+∥∇∂kw∥2

L2)

≤ _4β12∥∂t∂

k_w∥2

L2+

C β2(∥∇ζ∥

2

L2+∥∇2ζ∥2_H1) +

C β2∥∇∂

k_w∥2

L2.

This, together with (5.14), yields

1 2

d dt(ν∥∇∂

k_w

∥2L2+ ˜ν∥div∂kw∥2_L2−2γ2(∂kϕ,div∂kw)−2β2(∇∂kζ,∇∂kw)) +

1 4∥∂t∂

k_w

∥2L2

≤C(β2+ 1)∥∇∂kw∥2L2+Cγ2∥div∂kw∥_L22+C∥∇ζ∥2_L2+C∥∇2ζ∥2_H1+CN_k3.

Letb1 be a positive number which will be determined later. We set

E2=b1

β2_γ2

ν 2 ∑ k=0 Ek 1 + 1 ∑ k=0 (

ν∥∇∂k_w∥2

L2+ ˜ν∥div∂kw∥2_L2−2γ2(∂kϕ,div∂kw)−2β2(∇∂kζ,∇∂kw)

) ,

D2=b1

β2_γ2

ν 2 ∑ k=0 Dk 1+ 1 ∑ k=0

∥∂t∂kw∥2L2,

R2=b1

β2_γ2

ν 2 ∑ k=0 Rk 1+ 1 ∑ k=0

Nk3.

Byb1β

2

γ2

ν ×

∑2

k=0(5.49) +

∑1

k=0(5.50), we obtain

d

dtE2+D2≤ C b1

(

1 + 1

β2 +

γ2

β2

)

D2+C∥∂t∇ζ∥2H1+CR2. (5.51)

We takeν, γ2 _andβ2 _{so large that} ν β2 ≤1₂,

γ2

β2 ≤1₂ and _bC 1ν

(

1 + 1

β2 +

γ2

β2

)

≤2, and then takeb1 so large

thatE2 is equivalent to

β2_γ2

ν

2

∑

k=0

Ek

1 +γ2 1

∑

k=0

Dk

0.

It then follows

1 2

d

dtE2+D2≤C∥∇

2_ζ

∥2H1+CR2. (5.52)

From (5.36)k=0,l=1, we have

1 2

d

dt∥∂xnϕ∥ 2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂xnϕ∥ 2

L2+b0

ν+ ˜ν

β2_+γ2∥∂xnϕ˙∥ 2

L2

≤_(β2_+γC2_{)(ν+ ˜ν)}

{

∥∂tw∥2L2+∥∇∂q∥2_L2+β4(∥K_∞ζ∥2_L2+∥M_∞ζ∥2_L2)

}

+CN04,1.

(5.53)

Since

∥∇∂q∥2

L2 ≤C(ν2∥∇∂w∥2_L2+β4∥∇∂ζ∥2_L2),

β4_(∥K

∞ζ∥2L2+∥M∞ζ∥2L2)≤C∥∇ζ∥2_H1,

we see from (5.53) that

1 2

d

dt∥∂xnϕ∥ 2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂xnϕ∥ 2

L2+b0 ν+ ˜ν

β2_+γ2∥∂xnϕ˙∥ 2

L2

≤ _(β2+γC_{2)(ν+ ˜ν)}(

∥∂tw∥2L2+ν2∥∇∂w∥2_L2+∥∇ζ∥2_L2+β4∥∇∂ζ∥_L22+∥∇2ζ∥2_L2

)

+CN04,1

≤ _(β2_+γC2_{)(ν+ ˜ν)}

((

1 +ν+ 1

β2 +

ν β4

)

D2+∥∇2ζ∥2_L2

)

+CN04,1.

By (5.40)k=1, we obtain

∥∂ϕ∥2L2 ≤C(∥∇∂ζ∥2_L2+∥M∞ζ∥2_L2+∥f4∥2_L2) ≤C

( ₁ β4∥∇ζ∥

2

L2+∥∇∂ζ∥2_L2+

1

β4∥∇ 2_ζ∥2

L2+∥f4∥2_L2

)

≤C ((_ν

β4 +

ν β8

) D2+

1

β4∥∇ 2_ζ∥2

L2+∥f4∥2_L2

) .

(5.55)

Furthermore, by (5.21), we obtain

∥∂ϕ˙∥2L2 ≤C(∥∇∂w∥2_L2+∥∂(ϕdivw)∥2_L2)≤C

(

1

β2D2+∥∂(ϕdivw)∥ 2

L2

)

. (5.56)

We set

E3=∥∂xnϕ∥ 2

L2,

D3=

1 2

β2_+γ2

ν+ ˜ν ∥∇ϕ∥

2

L2+b0

ν+ ˜ν β2_+γ2∥∇ϕ˙∥

2

L2,

R3=N04,1+

1 2

β2_+γ2

ν+ ˜ν ∥f

4_∥

L2+b₀

ν+ ˜ν

β2_+γ2∥∂(ϕdivw)∥ 2

L2.

It then follows from (5.54) +1₂β_ν2+_+˜νγ2 ×(5.55) +b0_βν2+˜₊ν_γ2 ×(5.56) that

d

dtE3+D3≤C

( ₁

(β2_+γ2_{)(ν+ ˜ν)}

(

1 +ν+ 1

β2 +

ν β4

)

+β

2_+γ2

ν+ ˜ν (

ν β4 +

ν β8

)

+ ν+ ˜ν

β2_(β2_+γ2₎

) D2

+C

( ₁

(β2_+γ2_{)(ν+ ˜ν)}+

β2_+γ2

β4_{(ν+ ˜ν)}

)

∥∇2_ζ∥2

L2+CR3.

(5.57)

From (5.36)k=1,l=1, we have

1 2

d

dt∥∂∂xnϕ∥ 2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂∂xnϕ∥ 2

L2+b₀

ν+ ˜ν

β2_+γ2∥∂∂xnϕ˙∥ 2

L2

≤ _(β2_+γC2_{)(ν+ ˜ν)}

{

∥∂t∂w∥2L2+∥∇∂2q∥2_L2+β4(∥K∞∂ζ∥2L2+∥M∞∂ζ∥2L2)

}

+CN14,1.

(5.58)

Since

∥∇∂2q∥2L2 ≤C(ν2∥∇∂2w∥2_L2+β4∥∇∂2ζ∥2_L2),

β4(∥K∞∂ζ∥2L2+∥M_∞∂ζ∥2_L2)≤C∥∇∂ζ∥2_H1,

we see from (5.58) that

1 2

d

dt∥∂∂xnϕ∥ 2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂∂xnϕ∥ 2

L2+b0

ν+ ˜ν

β2_+γ2∥∂∂xnϕ˙∥ 2

L2

≤_(β2_+γC2_{)(ν+ ˜ν)}

(

∥∂t∂w∥2L2+ν2∥∇∂2w∥2_L2+∥∇∂ζ∥2_L2+β4∥∇∂2ζ∥2_L2+∥∇3ζ∥2_L2

)

+CN14,1

≤_(β2_+γC2_{)(ν+ ˜ν)}

((

1 +ν+ 1

β2+

ν β4

)

D2+∥∇3ζ∥2L2

)

+CN14,1.

By (5.40)k=2, we obtain

∥∂2ϕ∥2L2 ≤C(∥∇∂2ζ∥2_L2+∥M∞∂ζ∥2_L2+∥∂f4∥2_L2) ≤C

( ₁ β4∥∇∂ζ∥

2

L2+∥∇∂2ζ∥2_L2+

1

β4∥∇ 3_ζ∥2

L2+∥∂f4∥2_L2

)

≤C (( _ν

β4 +

ν β8

) D2+

1

β4∥∇ 3_ζ∥2

L2+∥∂f4∥2_L2

) .

(5.60)

By (5.21), we obtain

∥∂2ϕ˙∥2L2≤C(∥∇∂2w∥_L22+∥∂2(ϕdivw)∥2_L2)≤C

(

1

β2D2+∥∂

2_(ϕdivw)

∥2L2

)

. (5.61)

We set

E4=∥∂∂xnϕ∥ 2

L2,

D4=

1 2

β2_+γ2

ν+ ˜ν ∥∇∂ϕ∥

2

L2+b0

ν+ ˜ν

β2_+γ2∥∇∂ϕ˙∥ 2

L2,

R4=N14,1+

1 2

β2_+γ2

ν+ ˜ν ∥∂f

4_∥

L2+b₀

ν+ ˜ν β2_+γ2∥∂

2_(ϕdivw)∥2

L2.

It then follows from (5.58) + (5.59) +1₂β_ν2+_+˜γ_ν2 ×(5.60) +b0_βν2+˜₊ν_γ2×(5.61) that

d

dtE4+D4≤C

( ₁

(β2_+γ2_{)(ν+ ˜ν)}

(

1 +ν+ 1

β2 +

ν β4

)

+β

2_+γ2

ν+ ˜ν (

ν β4 +

ν β8

)

+ ν+ ˜ν

β2_(β2_+γ2₎

) D2

+C

( ₁

(β2_+γ2_{)(ν+ ˜ν)}+

β2_+γ2

β4_{(ν+ ˜ν)}

)

∥∇3_ζ∥2

L2+CR4.

(5.62)

Letb2 be a positive number which will be determined later. We set

E5=b2E2+β2(E3+E4),

D5=b2D2+β2(D3+D4),

R5=b2R2+β2(R3+R4).

We see fromb2×(5.60) +β2×(5.61) +β2×(5.63) that

d

dtE5+D5≤C (

1

(β2_+γ2_{)(ν+ ˜ν)}

(

β2+β2ν+ 1 + ν

β2

)

+β

2_+γ2

ν+ ˜ν ( _ν

β2 +

ν β6

)

+ ν+ ˜ν

β2_+γ2

) D2

+C (

1 + β

2

(β2_+γ2_{)(ν+ ˜ν)}+

β2_+γ2

β2_{(ν+ ˜ν)}

)

∥∇2ζ∥2H1+CR₅.

(5.63)

Takingb2 large enough, we see from (5.63) that

d

dtE5+d5D5≤C (

1 + β

2

(β2_+γ2_{)(ν+ ˜ν)}+

β2_+γ2

β2_{(ν+ ˜ν)}

)

From (5.36)k=0,l=2, we have

1 2

d dt∥∂

2

xnϕ∥ 2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂

2

xnϕ∥ 2

L2+b0 ν+ ˜ν

β2_+γ2∥∂ 2

xnϕ˙∥ 2

L2

≤ C

(β2_+γ2_{)(ν+ ˜ν)}

{

∥∂t∇w∥2L2+∥∇2∂q∥2_L2+β4(∥∇(K_∞ζ)∥2_L2+∥∇(M_∞ζ)∥2_L2)

}

+CN04,2.

(5.65)

From (5.38), we have

1

(β2_+γ2_{)(ν+ ˜ν)}(∥∇ 2_∂q

∥2L2+γ4∥∇∂ϕ∥2_L2)

≤_(β2_+γC2_{)(ν+ ˜ν)}

(

∥∂t∂w∥2L2+ (ν+ ˜ν)2∥∂ϕ˙∥2_H1+β4∥∂ϕ∥2_H1+β4∥K_∞∂ζ∥_L22+β4∥M_∞ζ∥2_H1

+(ν+ ˜ν)2_{∥∂(ϕdivw)∥}2

H1+∥f2∥2_H1+β4∥f4∥2_H1+∥∇∂q∥2_L2

)

≤C ( ₁

β2 +

1 (β2_+γ2_{)(ν+ ˜ν)}

(

1 +ν+ ν

β4 +

ν2

β2

)) D5+

C

(β2_+γ2_{)(ν+ ˜ν)}∥∇ 2_ζ

∥2H1

+ ν+ ˜ν

β2_+γ2∥∂(ϕdivw)∥ 2

H1+

1

(β2_+γ2_{)(ν+ ˜ν)}(∥f 2

∥2H1+β4∥f4∥2_H1).

(5.66)

From (5.68) and (5.66), we have

1 2

d dt∥∂

2

xnϕ∥ 2

L2+

1 2

β2_+γ2

ν+ ˜ν ∥∂

2

xnϕ∥ 2

L2+b0 ν+ ˜ν

β2_+γ2∥∂ 2

xnϕ˙∥ 2

L2

≤ C

(β2_+γ2_{)(ν+ ˜ν)}(∥∂t∇w∥ 2

L2+∥∇2ζ∥2_H1)

+C (

1

β2 +

1

(β2_+γ2_{)(ν+ ˜ν)}

(

1 +ν+ ν

β4+

ν2

β2

))

D5+CN˜04,2,

(5.67)

where

˜

N04,2=N04,2+

ν+ ˜ν

β2_+γ2∥∂(ϕdivw)∥ 2

H1+

1

(β2_+γ2_{)(ν+ ˜ν)}(∥f 2

∥2H1+β4∥f4∥2_H1).

Letb3 be a positive number to be determined later. We set

E6=b3E5+β2∥∂x2nϕ∥ 2

L2,

D6=b3D5+

1 2

β2_(β2_+γ2₎

ν+ ˜ν ∥∂

2

xnϕ∥ 2

L2+b₀

β2_{(ν+ ˜ν)}

β2_+γ2 ∥∂ 2

xnϕ˙∥ 2

L2,

R6=b3R5+β2N˜04,2.

We see fromb3×(5.64) + (5.67) that

d

dtE6+d5D6≤C (

1

(β2_+γ2_{)(ν+ ˜ν)}(∥∂t∇w∥ 2

L2+∥∇2ζ∥2_H1+β4∥∂ϕ∥2_L2+β4∥∂2ϕ∥2_L2) +R6

)

. (5.68)

Takingb3 large enough, we deduce from (5.68) that

d

dtE6+d6D6≤

Cβ2

(β2_+γ2_{)(ν+ ˜ν)}∥∂t∇w∥ 2

L2

+C (

1 + β

2

(β2_+γ2_{)(ν+ ˜ν)}+

β2_+γ2

β2_{(ν+ ˜ν)}

)

∥∇2_ζ∥2

H1+CR6.