A Regularity Result for a Solid-Fluid System Associated to the Compressible Navier-Stokes Equations

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www.elsevier.com/locate/anihpc

A regularity result for a solid–fluid system associated to the compressible Navier–Stokes equations

M. Boulakia, S. Guerrero

^∗

Université Pierre et Marie Curie – Paris 6, UMR 7598 Laboratoire Jacques-Louis Lions, Paris F-75005, France Received 5 July 2007; accepted 8 February 2008

Available online 21 March 2008

Abstract

In this paper we deal with a fluid-structure interaction problem for a compressible fluid and a rigid structure immersed in a regular bounded domain in dimension 3. The fluid is modelled by the compressible Navier–Stokes system in the barotropic regime with no-slip boundary conditions and the motion of the structure is described by the usual law of balance of linear and angular moment.

The main result of the paper states that, for small initial data, we have the existence and uniqueness of global smooth solutions as long as no collisions occur. This result is proved in two steps; first, we prove the existence and uniqueness of local solution and then we establish some a priori estimates independently of time.

Résumé

Dans cet article, nous considérons un problème d’interaction fluide-structure entre un fluide compressible et une structure ri- gide évoluant à l’intérieur d’un domaine borné et régulier en dimension 3. Le fluide est décrit par le système de Navier–Stokes compressible barotrope avec des conditions de non-glissement sur le bord et le mouvement de la structure est régi par les lois de conservation des moments linéaire et angulaire.

Nous montrons, pour des données initiales petites, l’existence et l’unicité de solutions globales régulières tant qu’il n’y a pas de chocs. Ce résultat est obtenu en deux temps ; tout d’abord, nous prouvons l’existence et l’unicité de solutions locales puis nous démontrons des estimations a priori indépendamment du temps.

Keywords:Fluid–structure interaction; Compressible fluid; Strong solutions; Navier–Stokes equations

* Corresponding author.

E-mail addresses:boulakia@ann.jussieu.fr (M. Boulakia), guerrero@ann.jussieu.fr (S. Guerrero).

doi:10.1016/j.anihpc.2008.02.004

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1. Introduction

1.1. Statement of problem

We consider a rigid structure immersed in a viscous compressible fluid. At timet, we denote byΩ_S(t )the domain occupied by the structure. The structure and the fluid are contained in a fixed bounded domainΩ⊂R³. We suppose that the boundaries ofΩ_S(0)andΩ are smooth (C⁴for instance) and that

d

ΩS(0), ∂Ω

>0. (1)

For anyt >0, we denote byΩ_F(t )=Ω\Ω_S(t )the region occupied by the fluid. The time evolution of the Eulerian velocityuand the densityρin the fluid are governed by the compressible Navier–Stokes equations and the continuity equation:∀t >0, ∀x∈Ω_F(t )

(ρ_t+ ∇ ·(ρu))(t, x)=0,

(ρu_t+ρ(u· ∇)u)(t, x)− ∇ ·(2μ(u)+μ(∇ ·u)Id)(t, x)+ ∇p(t, x)=0, (2) where(u)=¹₂(∇u+ ∇u^t)denotes the symmetric part of the gradient. The viscosity coefficientsμandμare real constants which are supposed to satisfy

μ >0, μ+μ0. (3)

We suppose that we are in a barotropic regime where a constitutive law gives the relation between the pressure of the fluidpand the densityρ. Thus, we suppose that

p=P (ρ) whereP ∈C^∞ R^∗₊

, P (ρ) >0 andP(ρ) >0, ∀ρ >0. (4) For instance,P (ρ):=ρ^γ withγ >0 is admissible.

Concerning the compressible fluids, a local in time result of existence and uniqueness of a smooth solution was proved in [21]. In [18], the authors proved the existence and uniqueness of a regular solution for small initial data and external forces.

Next, for isentropic fluids (P (ρ)=ρ^γ, γ >0), the global existence of a weak solution for small initial data was proved in [13] (forγ =1) and in [14] (forγ >1). Also for an isentropic fluid, the first global result for large data was proved in [15] (see also [16]) (withγ9/5 for dimensionN=3 and withγ > N/2 forN4). Finally, this last result was improved in [9] (see also [11]) (withγ > N/2 forN3).

At timet, the motion of the rigid structure is given by the positiona(t )∈R³of the center of mass and by a rotation (orthogonal) matrixQ(t )∈M3×3(R). Without loss of generality, we can suppose that

a(0)=0 and Q(0)=Id. (5)

At timet, the domain occupied by the structureΩ_S(t )is defined by Ω_S(t )=χ_S

t, Ω_S(0)

, (6)

whereχ_Sdenotes the flow associated to the motion of the structure:

χ_S(t, y)=a(t )+Q(t )y, ∀y∈Ω_S(0),∀t >0. (7)

We notice that, for eacht >0,χS(t,·):ΩS(0)→ΩS(t )is invertible and χS(t,·)⁻¹(x)=Q(t )⁻¹

x−a(t )

, ∀x∈ΩS(t ).

Thus, the Eulerian velocity of the structure is given by (χ_S)_t(t,·)◦χ_S(t,·)⁻¹(x)= ˙a(t )+ ˙Q(t )Q(t )⁻¹

x−a(t )

, ∀x∈Ω_S(t ).

SinceQ(t )Q(t )˙ ⁻¹is skew-symmetric, for eacht >0, we can represent this matrix by a unique vectorω(t )∈R³such that

Q(t )Q(t )˙ ⁻¹y=ω(t )∧y, ∀y∈R³.

(3)

Reciprocally, ifωbelongs toL²(0, T ), then there exists a unique matrixQ∈H¹(0, T )such thatQ(0)=Id and which satisfies this formula.

Thus, the Eulerian velocityu_Sof the structure is given by uS(t, x)= ˙a(t )+ω(t )∧

x−a(t )

, ∀x∈ΩS(t ). (8)

For the equations of the structure, we denote bym >0 the mass of the rigid structure andJ (t )∈M3×3(R)its tensor of inertia at timet. This tensor is given by

J (t )b· ˜b=

Ω_S(0)

ρ0,S(y)

b∧Q(t )y

·b˜∧Q(t )y

dy, ∀b,b˜∈R³, (9)

whereρ0,S>0 is the initial density of the structure. One can prove that

J (t )b·bC_J|b|²>0 for allb∈R³\ {0}, (10)

whereCJ is independent oft >0. The equations of the structure motion are given by the balance of linear and angular momentum. We have, for allt∈(0, T )

ma¨=

∂Ω_S(t )(2μ(u)+μ(∇ ·u)Id−pId)n dγ , Jω˙=(J ω)∧ω+

∂Ω_S(t )(x−a)∧((2μ(u)+μ(∇ ·u)Id−pId)n) dγ . (11) In these equations,nis the outward unit normal to∂Ω_S(t ). On the boundary of the fluid, the Eulerian velocity has to satisfy a no-slip boundary condition. Therefore, we have, for allt >0

u(t, x)=0, ∀x∈∂Ω, u(t, x)= ˙a(t )+ω(t )∧

x−a(t )

, ∀x∈∂Ω_S(t ). (12)

The system is completed by the following initial conditions:

u(0,·)=u₀ inΩ_F(0), ρ(0,·)=ρ₀ inΩ_F(0), a(0)=0, a(0)˙ =a₀, ω(0)=ω₀, (13) which satisfy

a₀, ω₀∈R³, ρ₀, u₀∈H³ Ω_F(0)

, ρ₀(x) >0, ∀x∈Ω_F(0). (14)

Since we will deal with smooth solutions, we will also need some compatibility conditions to be satisfied:

u₀=a₀+ω₀∧x on∂Ω_S(0), u₀=0 on∂Ω (15)

and 1 ρ₀∇ ·

2μ(u0)+μ(∇ ·u0)Id

− 1

ρ₀∇P (ρ0)

= 1 m

∂ΩS(0)

2μ(u0)+μ(∇ ·u0)Id−P (ρ0)Id

n dγ+ J (0)⁻¹

J (0)ω0

∧ω0

∧x

+J (0)⁻¹

∂Ω_S(0)

x∧

2μ(u0)+μ(∇ ·u₀)Id−P (ρ₀)Id n

dγ ∧x+ω₀∧(ω₀∧x) on∂Ω_S(0),

∇ ·

2μ(u0)+μ(∇ ·u₀)Id

− ∇P (ρ₀)=0 on∂Ω. (16)

These two conditions are formally obtained by differentiating system (12) with respect to time and takingt=0. To do this, we consider the second equation of (12) on a fix domain by settingx=χ_S(t, y), withy∈∂Ω_S(0).

Let us now recall some of the most relevant results in problems of fluid–structure interaction. In the below lines, when we refer to a global result, we mean before collision (in the case of a rigid solid) or before interpenetration of the structure (in the case of an elastic solid).

•Incompressible fluids:

As long as rigid solids are concerned, a local result was proved in [12], while the existence of global weak solutions is proved in [5] and [7] (with variable density) and [19] (2D, with variable density); in this last paper, the existence of

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a solution is proved even beyond collisions. Later, the existence and uniqueness of strong global solutions in 2D was proved in [20] as well as the local in time existence and uniqueness of strong solutions in 3D.

When talking about elastic solids, a first existence result of weak solution was proved in [8], when the elastic deformation is given by a finite sum of modes. A local existence result of a strong solution for an elastic plate was proved in [1] (2D). The local existence of a strong solution is proved in [6]. In [4] and [3] (with variable density), the authors proved the global existence of a weak solution.

•Compressible fluids:

Concerning rigid solids, the global existence of a weak solution was proved in [7] for γ 2 and in [10] for γ > N/2. For elastic solids, in [2] the author proved the global existence of a weak solution in 3Dforγ >3/2.

In this paper, we will prove the existence and uniqueness of smooth global solutions for small initial data (Theo- rem 3). We can also prove the same result for initial data close to a stationary solution(ρ, u, a, ω)=(ρ_e,0,0,0)and for special right-hand sides (see Remark 6 for more details).

We give a lemma which allows to extend the flowχ_Sby a flowχdefined on the global domainΩ.

Lemma 1.LetT ∈(0,+∞)and(a, ω)∈(H³(0, T )∩W^2,^∞(0, T ))×(H²(0, T )∩W^1,^∞(0, T ))be given. We suppose that(a, Q)satisfies(5)whereQ∈H³(0, T )is the rotation matrix associated toω. We consider the associated flowχS, the Eulerian velocityuS and the domain defined by(6)to(8). We suppose that there existsα >0such that

∀t∈ [0, T], d

Ω_S(t ), ∂Ω

α >0. (17)

Then, we can extend the flowχS by a flowχ∈H³(0, T;C^∞(Ω))such that

• χ (t, y)=a(t )+Q(t )yfor everyysuch thatdist(y, ∂ΩS(0)) < α/4and everyt∈(0, T ).

• χ (t, y)=yfor everyysuch thatdist(y, ∂Ω) < α/4and for everyt∈(0, T ).

• For allt∈(0, T ),χ (t,·)is invertible fromΩontoΩand fromΩ_F(0)ontoΩ_F(t )and (t, x)∈(0, T )×Ω→χ (t,·)⁻¹(x)

belongs toH³(0, T;C^∞(Ω)).

• The Eulerian velocityv∈H²(0, T;C^∞(Ω))associated toχ satisfies v(t, x)=u_S(t, x), ∀x∈Ω_S(t ), ∀t∈(0, T ),

v(t, x)=0, ∀x∈∂Ω,∀t∈(0, T ).

• Forp∈Nthere exists a constantC₁>0just depending onpandΩsuch that χ_t(t, x)

H^p(Ω)C₁a(t )˙ ²+ω(t )²1/2

, ∀t∈(0, T ). (18)

Proof. We consider a cut-off function which satisfies

ξ∈C^∞(Ω), ξ(y)=1, dist(y, ∂Ω) > α/2, ξ(y)=0, dist(y, ∂Ω) < α/4. (19) Let define the Eulerian velocity in the whole(0, T )×Ω by

v(t, y):=ξ(y)

˙

a(t )+ω(t )∧

y−a(t )

, t∈(0, T ), y∈Ω. (20)

Next, we can define the flow χ associated to v. For ally∈Ω, χ (·, y)is defined on(0, T )as the solution of the ordinary differential equation

χ_t(t, y)=v

t, χ (t, y) ,

χ (0, y)=y. (21)

In{y: dist(y, ∂Ω) < α/4}, sincev=0, we have according to the uniqueness of the flow thatχ (t, y)=y. Analogously, if we take a point in the set {y: dist(y, ∂ΩS(0)) < α/4},χ (t, y)=χ_s(t, y)=a(t )+Q(t )y. The last point comes from (19), (20) and (21). 2

We define now a flow which transforms the moving domains into the initial one:

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Corollary 2.Under the same hypotheses as in the previous lemma, one can define an inverse flowψextending χ_S⁻¹(t, x)=Q(t )⁻¹

x−a(t )

, t∈(0, T ), x∈Ω_S(t ), to the whole domainΩand which satisfies:

• ψ (t, x)=Q(t )⁻¹(x−a(t )),∀x∈ΩS(t ),∀t∈(0, T ).

• ψ (t, x)=x,∀x∈∂Ω,∀t∈(0, T ).

• The velocityw:=ψt associated toψbelongs toH²(0, T;C^∞(Ω))and satisfies w(t, x)= −Q(t )⁻¹(a(t )˙ +ω∧(x−a(t ))), ∀x∈Ω_S(t ),∀t∈(0, T ),

w(t, x)=0, ∀x∈∂Ω,∀t∈(0, T ).

• Forp∈Nthere exists a constantC₂>0just depending onpandΩ such that ψt(t, x)

H^p(Ω)C2a(t )˙ ²+ω(t )²1/2

, ∀t∈(0, T ). (22)

1.2. Main result

Let us now introduce some notation which we will employ all along the paper. Leth∈(0,+∞]and (a, ω)˙ ∈ (H²(0, h)∩W^1,^∞(0, h))²be given. First we define

Z_h:=

(s, x): s∈(0, h), x∈Ω_F(s)

, Σ_h:=

(s, x): s∈(0, h), x∈∂Ω_S(s)

. (23)

Then, forr, p0 natural numbers, we introduce

L²_h L²

:=L²(Z_h)=

umeasurable : h 0

Ω_F(s)

|u|²dx ds <+∞

,

L²_h H^p

:=

u∈L²_h

L² :

h 0

u²H^p(Ω_F(s))ds <+∞

,

H_h^r H^p

:=

u∈L²_h

L² :

h 0

r β=0

∂_t^βu²

H^p(ΩF(s))ds <+∞

, with associated norms given by the definition. On the other hand, we define

C_h⁰ L²

:=

usuch thatu(s, x)ˆ :=u(s, x)1Ω_F(s)∈C⁰

[0, h];L²(Ω) and

C_h^r H^p

:=

u: ∂_t^β∂_x^αu∈C_h⁰ L²

, ∀0βr, ∀0|α|p with associated norms given by

u_C⁰

h(L²):= max

t∈(0,h)

u(t )

L²(ΩF(t ))= max

t∈(0,h)

u(t )ˆ

L²(Ω)

and

uC_h^r(H^p):=

r β=0

tmax∈(0,h)

∂_t^βu(t )

H^p(Ω_F(t )).

It is clear that for regular functions, the norm · C^r_h(H^p)coincides with the norm · _W^r,^∞

h (H^p). In the sequel, we will always use the second notation.

The main goal of our paper is to prove the following theorem:

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Theorem 3.Letρ¯be the mean-value ofρ₀inΩ_F(0). We suppose that(1), (3)and(4)are satisfied. We suppose that the initial conditions satisfy(5), (14)and the compatibility conditions(15)–(16). Then there exists a constantδ >0 such that, if

ρ₀− ¯ρ_H³_(Ω_F₍₀₎₎+ u₀_H³_(Ω_F₍₀₎₎+ |a₀| + |ω₀|< δ, (24) the system of Eqs.(2), (11), (12)and(13)admits a unique solution(ρ, u, a, ω)defined on(0, T )for allT such that (17)is satisfied. Moreover, this solution belongs to the following space

ρ∈L²_T H³

∩C_T⁰ H³

∩H_T¹ H²

∩C_T¹ H²

∩H_T² L²

, u∈L²_T

H⁴

∩C_T⁰ H³

∩C_T¹ H¹

∩H_T² L²

, a˙∈H²(0, T )∩C¹ [0, T]

, ω∈H²(0, T )∩C¹ [0, T] and there exists a positive constantC₁independent ofT such that

ρ− ¯ρ_L²

T(H³)+ ρ− ¯ρ_L∞

T(H³)+ ρ− ¯ρ_W^1,∞

T (H²)+ ρ− ¯ρ_H²

T(L²)+ ρ− ¯ρ_H¹

T(H²)

+ u_L²

T(H⁴)+ uL^∞_T(H³)+ u_W^1,∞

T (H¹)+ u_H²

T(L²)+ ˙a_W^1,∞

T + ˙a_H²

T + ω_W^1,∞

T + ω_H²

T

C₁

ρ₀− ¯ρ_H³_(Ω_F(0))+ u₀_H³_(Ω_F(0))+ |a₀| + |ω₀|

. (25)

Remark 4.In Theorem 3, we prove that forδ >0 small enough depending on the initial distanced(Ω_S(0), ∂Ω), there exists a unique regular global in time solution of the system of Eqs. (2), (11), (12) and (13) provided that the structure does not touch∂Ω (see (17)). Observe that this condition will always be satisfied on an interval(0, Tmin), whereT_min>0 only depends onC₁,δandd(∂Ω, Ω_S(0)), as easily seen from estimate (25).

Remark 5.This condition onT is natural for fluid-structure interaction problems (see, for instance, [7], [5] and [20]).

For results concerning the existence of weak solutions of this type of systems after a collision has occurred, we refer to [19] and [10].

Remark 6.As proved in [18], there exists a unique stationary solution(ρ_e(x),0,0,0)of (2), (12), where, on the fluid, the compressible Navier–Stokes equation is completed with a right-hand side of the formf_i(x)=∂_iφ(x) (i=1,2,3) satisfying (24) for the H³-norm. Then, we can prove that Theorem 3 also holds withρ¯ replaced byρ_e and for a right-hand side of the above kind.

2. Intermediate results

Let us recall thatρ¯is the mean-value ofρ₀:

¯

ρ= 1

V (Ω_F(0))

Ω_F(0)

ρ₀, (26)

whereV (Ω_F(0))stands for the volume ofΩ_F(0).

Now, for 0h+∞, we define the space X(0, h)=

(ρ, u, a, ω): (ρ, u)∈C_h⁰ H³

∩L²_h H³

∩C_h¹ H²

∩H_h¹ H²

∩H_h² L²

× L²_h

H⁴

∩C_h⁰ H³

∩H_h¹ H²

∩C_h¹ H¹

∩H_h² L²

, (a, ω)˙ ∈

C_h¹∩H_h²2 endowed with the following norm:

N_0,h(ρ, u, a, ω)= ρ²_L∞

h(H³)+ ρ²_L2

h(H³)+ ρ²_W1,∞

h (H²)+ ρ²_H1

h(H²)+ ρ²_H2

h(L²)+ u²_L2 h(H⁴)

+ u²_L∞

h(H³)+ u²_H1

h(H²)+ u²_W1,∞

h (H¹)+ u²_H2

h(L²)+ ˙a²_W1,∞ h

+ ˙a²_H2 h

+ ω²_W1,∞

h

+ ω²_H2 h

1/2

. (27)

By abuse of notation, we will denote for each 0h+∞

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X(h, h)=

(ρ_h, u_h, a_h, ω_h): ρ_h∈H³ Ω_F(h)

, u_h∈H³ Ω_F(h)

,

× L²_h

H⁴

∩C_h⁰ H³

∩H_h¹ H²

∩C_h¹ H¹

∩H_h² L²

, (a, ω)˙ ∈

C_h¹∩H_h²2 .

The proof of Theorem 3 is divided in two steps: first, a local existence result and next a priori estimates for the system. A suitable combination of both will yield the desired global result.

Thus, we first formulate a local existence and uniqueness result for small initial data.

Proposition 7.Let(1)be satisfied and leth0:

•Forh=0, let(ρ₀, u₀, a₀, ω₀)satisfy(14)–(16).

•Forh >0, we suppose that(2), (11), (12)and(13)admits a unique solution(ρ, u, a, ω)inX(0, h)satisfying (17)for someα >0independent ofhand forT replaced byh.

Then, there exist constantsδ₀,τ >0andC₂>0independent ofh(forh=0, we mean independently of the initial condition)such that, ifN_h,h(ρ− ¯ρ, u, a, ω)δ₀, problem(61)has a unique solution(ρ, u, a, ω)defined on(h, h+τ ) satisfying

(ρ, u, a, ω)∈X(h, h+τ ), N_h,h₊_τ(ρ− ¯ρ, u, a, ω)C₂N_h,h(ρ− ¯ρ, u, a, ω).

We present now the a priori estimates:

Proposition 8. We suppose that there exists T >0 such that the problem (2), (11), (12) and (13) admits a so- lution (ρ, u, a, ω) in X(0, T ). Then there exist constants 0< δ₁δ₀ and C₃>0 independent of T such that if N_0,T(ρ− ¯ρ, u, a, ω)δ₁then

N_0,T(ρ− ¯ρ, u, a, ω)C₃N_0,0(ρ− ¯ρ, u, a, ω).

These two propositions will be proved in the next sections. They allow to prove Theorem 3 in the following way:

Proof of Theorem 3. We will apply iteratively Propositions 7 and 8. We suppose that N_0,0(ρ− ¯ρ, u, a, ω)min

δ₀, δ₁

C₂, δ₁ C₃

1+C₂²

.

According to Proposition 7 forh=0, one can define on(0, τ )a solution(ρ^∗, u, a, ω)∈X(0, τ )such that N0,τ(ρ− ¯ρ, u, a, ω)C2N0,0(ρ− ¯ρ, u, a, ω)δ1δ0.

SinceNτ,τ(ρ− ¯ρ, u, a, ω)N0,τ(ρ− ¯ρ, u, a, ω), we can apply again Proposition 7. Our solution can be extended on(τ,2τ )andNτ,2τ(ρ− ¯ρ, u, a, ω)C2N0,τ(ρ− ¯ρ, u, a, ω). Thus

N_0,2τ² (ρ− ¯ρ, u, a, ω)=

N_0,τ² +N_τ,2τ²

(ρ− ¯ρ, u, a, ω) 1+C₂²

N_0,τ² (ρ− ¯ρ, u, a, ω).

Thanks to Proposition 8 forT =τ and the choice ofN_0,0(ρ− ¯ρ, u, a, ω), we deduce N0,2τ(ρ− ¯ρ, u, a, ω)C3

1+C₂²N0,0(ρ− ¯ρ, u, a, ω)δ1.

This allows to repeat this process and obtain the existence of a regular solution as long as (17) is satisfied. 2 Now, we state a result which will be useful in our analysis:

Lemma 9. Let t > 0. There exists C >0 independent of t such that for all u with (u)∈ L²(Ω_F(t )) and u∈L²(Ω_F(t )), we have thatu∈H¹(Ω_F(t ))and

uH¹(Ω_F(t ))C(u)

L²(Ω_F(t ))+ uL²(Ω_F(t ))

. (28) Consequently, ifu_|_∂Ω=0, we have that

uH¹(Ω_F(t ))C(u)

L²(Ω_F(t )), (29)

forC >0independent oft.

(8)

Proof. All along this proof,Cwill stand for a positive constant independent oft. For the first inequality, we use that

∂x_jx_ku(t, x)=1 2

∂x_j(u)ik+∂x_k(u)ij−∂x_i(u)j k

. First, observe that

vL²(Ω_F(t ))C∇vH−1(Ω_F(t )),

where C only depends on the size of Ω_F(t )⊂Ω. We apply this inequality for v:= ∇u and so we deduce that

∇u∈L²(Ω_F(t ))and

∇uL²(ΩF(t ))C(u)

L²(ΩF(t )).

Then, inequality (28) readily follows. In order to prove (29) we observe that, thanks tou_|_∂Ω=0, we have that uL²(ΩF(t ))C∇uL²(ΩF(t )).

Here,Cdepends on the regularity ofΩF(t )(which is that ofΩF(0)) and the size ofΩ. 2 3. A local existence result: proof of Proposition 7

This section is devoted to the proof of Proposition 7. We only consider the case h=0; the caseh >0 follows directly from the arguments below.

3.1. Statement of the problem in a fixed domain

The system of equations can be written on the reference domains Ω_S(0) andΩ_F(0)with the help of the flow defined by Lemma 1. We consider(ρ, u, a, ω)which satisfies the system (2)-(11)-(12) with the hypothesis (3)–(4).

We suppose that (17) is satisfied for someα >0 and that the functionsρ,u,aandωare regular enough. Let us define the functionsu,˜ ρ˜andp˜onV_T :=(0, T )×Ω_F(0)by

˜

u(t, y)=u

t, χ (t, y)

, ρ(t, y)˜ =ρ

t, χ (t, y)

− ¯ρ, p(t, y)˜ =P

˜

ρ(t, y)+ ¯ρ

, ∀(t, y)∈V_T. (30) We have the following formulas,∀(t, y)∈V_T,

˜

ρ_t(t, y)=ρ_t

t, χ (t, y) +

(∇χ )⁻¹χ_t

· ∇ ˜ρ (t, y), (∇ ˜u)_ij(t, y)=∂_y_ju˜_i=

∇u

t, χ (t, y)

∇χ (t, y)

ij. Thus we get, for the first equation of system (2),

˜ ρ_t+

(∇χ )⁻¹(u˜−χ_t)

· ∇ ˜ρ+(ρ˜+ ¯ρ)tr

∇ ˜u(∇χ )⁻¹

=0, inV_T. (31)

Next, the second equation of (2) becomes,∀i=1,2,3, inV_T (ρ˜+ ¯ρ)(u˜_i)_t+(ρ˜+ ¯ρ)(u˜_j−χ_j,t)

∇ ˜u(∇χ )⁻¹

ij−μ∂_y_l

∂_y_ku˜_i(∇χ )⁻_kj¹ (∇χ )⁻_lj¹

−(μ+μ)∂y_l

∂y_ku˜j(∇χ )⁻_kj¹

(∇χ )⁻_li¹+(∇χ )⁻_ki¹∂y_kp˜=0. (32) In this equation and in what follows, we implicitly sum over repeated indexes. Eqs. (11) become:

ma¨=

∂ΩS(0)(μ(∇ ˜uQ^t+Q(∇ ˜u)^t)+μtr(∇ ˜uQ^t)Id− ˜pId)Qn dγ , Jω˙ =(J ω)∧ω+

∂ΩS(0)(Qy)∧((μ(∇ ˜uQ^t+Q(∇ ˜u)^t)+μtr(∇ ˜uQ^t)Id− ˜pId)Qn) dγ , (33) where we have denotedA⁻^t=(A^t)⁻¹At last, the boundary conditions (12) become

u(t, y)˜ =0, ∀y∈∂Ω,

˜

u(t, y)= ˙a(t )+ω(t )∧(Qy), ∀y∈∂ΩS(0). (34)

(9)

We denote(m, J, μ, μ)instead of(m/ρ, J /¯ ρ, μ/¯ ρ, μ¯ /ρ)¯ and we rewrite the resulting system as follows:

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

˜

ρ_t+((∇χ )⁻¹(u˜−χ_t))· ∇ ˜ρ+ ¯ρ∇ · ˜u=g₀(ρ,˜ u, a, ω)˜ inV_T,

˜

u_t− ∇ ·σ (u,˜ ρ)˜ =g₁(ρ,˜ u, a, ω)˜ inV_T, ma¨=

∂Ω_S(0)σ (u,˜ ρ)n dγ˜ +g2(ρ,˜ u, a, ω)˜ in(0, T ), Jω˙=

∂Ω_S(0)(Qy)∧(σ (u,˜ ρ)n) dγ˜ +g3(ρ,˜ u, a, ω)˜ in(0, T ),

˜

u=0 on(0, T )×∂Ω,

˜

u= ˙a+ω∧(Qy) on(0, T )×∂Ω_S(0),

˜

ρ(0,·)=ρ₀− ¯ρ,u(0,˜ ·)=u₀ inΩ_F(0), a(0)=0, a(0)˙ =a₀, ω(0)=ω₀,

(35)

with

g₀(ρ,˜ u, a, ω)˜ = ¯ρtr

∇ ˜u

Id−(∇χ )⁻¹

− ˜ρtr

∇ ˜u(∇χ )⁻¹

, (36)

(g1)i(ρ,˜ u, a, ω)˜ = −(u˜j−χj,t)

∇ ˜u(∇χ )⁻¹

ij+μ ρ¯

˜

ρ+ ¯ρ −1 ∂y_l

∂y_ku˜i(∇χ )⁻_kj¹ (∇χ )⁻_lj¹ +μ

∂y_l

∂y_ku˜i

(∇χ )⁻_kj¹−δkj

(∇χ )⁻_lj¹+∂_y²

ly_ju˜i

(∇χ )⁻_lj¹−δlj

+(μ+μ)

ρ¯

˜

ρ+ ¯ρ−1 ∂_y_l

∂_y_ku˜_j(∇χ )⁻_kj¹ (∇χ )⁻_li¹ +(μ+μ)

∂_y_l

∂_y_ku˜_j

(∇χ )⁻_kj¹−δ_kj

(∇χ )⁻_li¹+∂_y²

lyju˜_j

(∇χ )⁻_li¹−δ_li

−

(∇χ )⁻_ki¹−δki

P(ρ˜+ ¯ρ)∂_y_kρ˜

˜

ρ+ ¯ρ −

P(ρ˜+ ¯ρ)

˜

ρ+ ¯ρ −P(ρ)¯

¯

ρ ∂y_iρ,˜ (37)

g2(ρ,˜ u, a, ω)˜ =μ

∂ΩS(0)

(Q−Id)(∇ ˜u)^tQ+(∇ ˜u)^t(Q−Id) n dγ

+μ

∂Ω_S(0)

tr

∇ ˜u

Q^t−Id

Q+(∇ · ˜u)(Q−Id) n dγ

−

∂Ω_S(0)

P (ρ˜+ ¯ρ)−P (ρ)¯

¯

ρ (Q−Id)n dγ −

∂Ω_S(0)

P (ρ˜+ ¯ρ)−P (ρ)¯ −P(ρ)¯ ρ˜n

¯

ρdγ (38) and

g₃(ρ,˜ u, a, ω)˜ =μ

∂ΩS(0)

(Qy)∧

(Q−Id)(∇ ˜u)^tQ+(∇ ˜u)^t(Q−Id) n dγ

+μ

∂ΩS(0)

(Qy)∧ tr

∇ ˜u

Q^t−Id

Q+tr(∇ ˜u)(Q−Id) n dγ

+

∂ΩS(0)

(Qy)∧

p⁰ρ(Id˜ −Q)n dγ

+

∂Ω_S(0)

P (ρ)¯ +P(ρ)¯ ρ˜−P (ρ˜+ ¯ρ)

(Qy)∧Qn

¯

ρdγ+(J ω)∧ω. (39)

Here, we have used (64). Observe that the functions g_i are (at least) quadratic functions of the quantities ρ,˜ u,˜ (∇χ )⁻¹−Id, Q−Id, χt.

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3.2. Definition of the fixed point mapping

Let 0<R <˜ 1 and 0< s <1 be small enough. We define the spaceY ((0, s)˜ ; ˜R):

Y˜

(0, s); ˜R

=

(ρ,˜ u˜_F, a, ω)∈ ˜X(0, s): u˜_F =0 on∂Ω_F(0), a(0)=0, Q(0)=Id, N˜_0,s(ρ,˜ u˜_F, a, ω)R˜ , where

X(0, s)˜ =

(ρ,˜ u, a, ω):˜ ρ˜∈C_s⁰ H³

Ω_F(0)

∩C_s¹ H²

Ω_F(0)

∩H_s² L²

Ω_F(0) ,

˜ u∈L²_s

H⁴

Ω_F(0)

∩H_s² L²

Ω_F(0)

, a∈H_s³, ω∈H_s²

andN˜_0,s(ρ,˜ u˜_F, a, ω)is given by the corresponding norm. In this definition,Qis the rotation matrix associated toω.

Remark 10.The definition of the spaceY ((0, s)˜ ; ˜R)coincides with Y

(0, s);R

=

˜

ρ◦χ⁻¹,u˜_F◦χ⁻¹, a, ω

∈X(0, s)/a(0)=0, Q(0)=Id, N_0,s

˜

ρ◦χ⁻¹,u˜_F◦χ⁻¹, a, ω

R

.

This comes from the identities, for all(t, y)∈(0, s)×Ω_F(0), ifx=χ (t, y),

∂tu(t, y)˜ =∂tu(t, x)+

χt(t, y)· ∇ u(t, x) and

∇ ˜u(t, y)= ∇u(t, x)∇χ (t, y).

In order to prove Proposition 7, we will perform a fixed point argument in the space Y ((0, s)˜ ; ˜R). Indeed, we consider the mapping

Λ:Y˜

(0, s); ˜R

→ ˜Y

(0, s); ˜R , (ρ,ˆ uˆF,a,ˆ ω)ˆ →(ρ,˜ u˜F, a, ω),

withu˜_F = ˜u−χ_t, where(ρ,˜ u, a, ω)˜ is the solution of

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

˜

ρt+((∇ ˆχ )⁻¹(uˆ− ˆχt))· ∇ ˜ρ=g0(ρ,ˆ u,ˆ a,ˆ ω)ˆ − ¯ρ∇ · ˜u inVT,

˜

ut− ∇ ·(2μ(u)˜ +μ(∇ · ˜u)Id)=g1(ρ,ˆ u,ˆ a,ˆ ω)ˆ −p⁰∇ ˜ρ inVT, ma¨=

∂ΩS(0)σ (u,˜ ρ)n dγ˜ +g₂(ρ,ˆ u,ˆ a,ˆ ω)ˆ in(0, T ), Jˆω˙=

∂Ω_S(0)(Qy)ˆ ∧(σ (u,˜ ρ)n) dγ˜ +g₃(ρ,ˆ u,ˆ a,ˆ ω)ˆ in(0, T ),

˜

u=0 on(0, T )×∂Ω,

˜

u= ˙a+ω∧(Qy)ˆ on(0, T )×∂Ω_S(0),

˜

ρ(0,·)=ρ0− ¯ρ, u(0,˜ ·)=u0 inΩF(0), a(0)=0, a(0)˙ =a0, ω(0)=ω0.

(40)

Here,uˆ= ˆu_F+ ˆχ_t andJˆis defined by (9) where we replaceQbyQ.ˆ

Thanks to Remark 10, if we prove the existence of a fixed point ofΛ, the proof of Proposition 7 will be achieved.

3.3. Λis well defined

Let (ρ,ˆ u,ˆ a,ˆ ω)ˆ be given inY ((0, s)˜ ; ˜R). Our goal is to prove that the solution of (40)(ρ,˜ u, a, ω)˜ belongs to Y ((0, s)˜ ; ˜R). In order to prove this, we will establish some estimates forρ˜ regarded as the solution of a transport equation (with right-hand sideg₀− ¯ρ(∇ · ˜u)) and foru˜as solution of a heat equation (with right-hand sideg₁−p⁰∇ ˜ρ).

In the sequel, we will use the following estimates, coming from Lemma 1:

(ρ,ˆ u,ˆ a,ˆ ω)ˆ _˜

X(0,s)+(∇ ˆχ )⁻¹−Id

H_s³(C∞(Ω_F(0)))+ ˆQ−Id_H_s³+ ˆχ_t_H2

s(C^∞(ΩF(0)))CR.˜

TakingR >˜ 0 small enough, we can suppose that| ˆρ|<ρ/2 and so every single term of the expression of¯ g₁(see (37)) makes sense. Then, one can prove that