Jérôme Lohéac
email: loheac@iecn.u-nancy.fr

Jean-François Scheid
email: scheid@iecn.u-nancy.fr

Marius Tucsnak
email: tucsnak@iecn.u-nancy.fr

Controllability and time optimal control for low Reynolds numbers swimmers

Keywords: Stokes equations, fluid-structure interaction, controllability, time optimal control, low Reynolds number swimming AMS subject classifications. 93C10, 93C20, 93C15, 93C73, 93C95, 76D07

The aim of this paper is to tackle the self-propelling at low Reynolds number by using tools coming from control theory. More precisely we first address the controllability problem: "Given two arbitrary positions, does it exist "controls" such that the body can swim from one position to another, with null initial and final deformations?". We consider a spherical object surrounded by a viscous incompressible fluid filling the remaining part of the three dimensional space. The object is undergoing radial and axi-symmetric deformations in order to propel itself in the fluid. Since we assume that the motion takes place at low Reynolds number, the fluid is governed by the Stokes equations. In this case, the governing equations can be reduced to a finite dimensional control system. By combining perturbation arguments and Lie brackets computations, we establish the controllability property. Finally we study the time optimal control problem for a simplified system. We derive the necessary optimality conditions by using the Pontryagin maximum principle. In several particular cases we are able to compute the explicit form of the time optimal control and to investigate the variation of optimal solutions with respect to the number of inputs.

Introduction

Understanding the motion of solids (such as aquatic microorganisms and micro or nano swimming robots) is a challenging issue since the propelling mechanism should be adapted to very low Reynolds numbers, i.e., it should be essentially based on friction forces, with no role of inertia (which is essential for the swimming mechanism of macroscopic objects, such as fish like swimming). Among the early contributions to the modeling and analysis of these phenomena, we mention the seminal works by Taylor [START_REF] Taylor | Analysis of the swimming of microscopic organisms[END_REF], Lighthill [START_REF] Lighthill | Mathematical biofluiddynamics[END_REF][START_REF] Lighthill | On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers[END_REF], and Childress [START_REF] Childress | Mechanics of swimming and flying[END_REF]. In several relatively more recent papers (see, for instance, Shapere and Wilczek [START_REF] Shapere | Efficiencies of self-propulsion at low Reynolds number[END_REF], San Martin Takahashi and Tucsnak [START_REF] San Martín | A control theoretic approach to the swimming of microscopic organisms[END_REF], Alouges, Desimone and Lefebvre [START_REF] Alouges | Numerical strategies for stroke optimization of axisymmetric microswimmers[END_REF], Alouges, Desimone and Heltai [START_REF] Alouges | Optimal strokes for low Reynolds number swimmers: an example[END_REF] and Lauga and Michelin [START_REF] Michelin | Efficiency optimization and symmetry-breaking in a model of ciliary locomotion[END_REF]) the self-propelling at low Reynolds number has been investigated by using tools coming from optimization and control theory.

The aim of this work is to give a rigorous mathematical approach to the analysis and the control of a system modeling the low Reynolds number swimming of spherical object. In order to propel itself in the fluid, the swimmer is assumed to perform radial deformations. Roughly speaking, the main problems tackled in this work can be stated as follows:

1. Controllability: given an initial and a final position, prove the existence of a sequence of deformations steering the mass center of the swimmer from the first position to the second, such that at the beginning and the end of the process the body is a unit ball.

2. Time optimal control: among the sequences of deformations of given amplitude steering the mass center from one positions to another, determine a sequence accomplishing this task in minimal time.

The controllability based approach to swimming at low Reynolds numbers has been used, as far as we know in [START_REF] San Martín | A control theoretic approach to the swimming of microscopic organisms[END_REF] and [START_REF] Alouges | Numerical strategies for stroke optimization of axisymmetric microswimmers[END_REF]. In [START_REF] San Martín | A control theoretic approach to the swimming of microscopic organisms[END_REF] the swimmer is a ball and the control is a tangential velocity field on the boundary (so that the body is not deforming), whereas in [START_REF] Alouges | Numerical strategies for stroke optimization of axisymmetric microswimmers[END_REF] the authors consider a three balls swimmer. The main novelty we bring in this direction is that we consider a one piece (connected) body which advances by undergoing appropriate deformations. As far as we know, the associated time optimal control problems have not been tackled in the literature but related optimal control problems, with the efficiency cost function have been investigated (see, for instance [START_REF] Shapere | Efficiencies of self-propulsion at low Reynolds number[END_REF][START_REF] Alouges | Optimal strokes for low Reynolds number swimmers: an example[END_REF][START_REF] Michelin | Efficiency optimization and symmetry-breaking in a model of ciliary locomotion[END_REF]). We give below some notation which will be used throughout this work. Firstly, we denote by S 0 the unit ball in R 3 and by S(t) the domain occupied by the swimmer at instant t. The corresponding density field is denoted by ρ, so that, for every positive t, ρ(t, •) maps S(t) into (0, ∞). The fluid domain is denoted by Ω(t) = R 3 \S(t), see Figure 1. We assume that the motion takes places at Γ(t) n swimmer S(t)

fluid Ω(t) = R 3 \S(t) zero Reynolds number so that, at each time t 0 the velocity field u(t, •) : Ω(t) → R 3 and the corresponding pressure field p(t, •) : Ω(t) → R of the fluid satisfy the Stokes equations:

-µ∆u(t, •) + ∇p(t, •) = 0 in Ω(t), div u(t, •) = 0 in Ω(t), (1.1) where µ ∈ R * + is the viscosity. We assume that the fluid is at rest at infinity and that it sticks the swimmer, so that we impose the conditions lim |x|→∞ u(t, x) = 0 (t 0), (1.2)

u(t, •) = v S (t, •) on ∂S(t), (1.3)
where v S the velocity of the swimmer. The needed well-posedness results for the system (1.1)-(1.3) will be recalled in Section 2.

We assume that the motion of the swimmer can be decomposed into a rigid part, which is unknown, and a motion not affecting its mass center and its global orientation, which will be the input of our problem. The rigid part of the motion is characterized, at each instant t, by the position h(t) of the mass center and a rotation matrix R(t). More precisely, the motion of the swimmer is characterized by a map X : S 0 × [0, ∞) → R 3 which can be written

X(t, y) = h(t) + R(t)X ⋆ (t, y) (y ∈ S 0 , t 0) .
where X ⋆ : [0, ∞) × S 0 → R 3 is the input function (see Fig. 1). The current domain S(t) occupied by the swimmer is thus given by S

(t) = X(t, •)(S 0 ) (t 0) .
Since swimming is generally a periodic action, we constrain the input X * to satisfy

S ⋆ (t) S 0 X(t, .) X ⋆ (t, .) 0 x h(t) S(t) h(t) + R(t)I d 0 x ⋆ y 0
X ⋆ (t, •) = X ⋆ (t + T, •) (t 0), (1.4)
for some T > 0. The corresponding Eulerian velocity field v S of the swimmer can thus be written

v S (t, x) = ḣ(t)+ω(t)×(x-h(t))+R(t) ∂X ⋆ ∂ t t, X ⋆ (t, •) -1 R ⊤ (t)(x -h(t)) (t 0, x ∈ S(t)), (1.5
) where R ⊤ (t) stands for the transposed matrix of R(t) and ω is the angular velocity vector, defined by Ṙ = R ⊤ A(ω), with

A(ω) =   0 -ω 3 ω 2 ω 3 0 -ω 1 -ω 2 ω 1 0   (ω = (ω 1 , ω 2 , ω 3 ) ⊤ ) . Using (1.5), condition (1.3) becomes u(t, x) = ḣ(t)+ω(t)×(x-h(t))+R(t) ∂X ⋆ ∂ t t, X ⋆ (t, •) -1 R ⊤ (t)(x -h(t)) (t 0, x ∈ ∂S(t)).
(1.6) To be consistent with the assumption of low Reynolds number flow, we assume that at each instant t the swimmer is in equilibrium under the action of forces exerted by the fluid. This means that we have

           0 = ∂S(t) σn dΓ 0 = ∂S(t) (x -h) × σn dΓ (t 0), (1.7)
where

σ = σ(u, p) = µ ∇u + ∇u T -pI 3 , (1.8)
is the classical Cauchy stress tensor in a viscous incompressible fluid.

The full system under investigation, of unknowns u, p, σ, h and R, with the given input X * is formed by the equations (1.1), (1.2), (1.6), (1.7) and (1.8). The unknown σ can be substituted, using (1.8), into (1.7) so that it suffices to consider the unknowns u, p, h and R. Moreover, solving, for each positive t the stationary Stokes equations (1.1), the full system reduces to a system of ordinary differential equations for h and R (see Section 3).

The main theoretical results concern controllability issues. They imply that for any h 0 , h 1 ∈ R 3 there exists T > 0 and an input X * satisfying (1.4) such that the mass center of the swimmer is steered from h 0 to h 1 in time T , that is, we have

h(0) = h 0 , h(T ) = h 1 .
A special attention is devoted to the case in which X * is axi-symmetric, so that the trajectory of the mass center is a straight line. In this situation we provide a simplified model and we apply Pontryagin's principle to give an explicit solution of the associated time optimal control problem.

Some background on the exterior Stokes problem

In this section we first recall some known results on the exterior (i.e., posed in R 3 \ S, where S is a bounded obstacle) Dirichlet boundary value problem for the stationary Stokes system. We next introduce the force operator, which associates to a given Dirichlet condition on the boundary of S the total force exerted by the fluid on that obstacle. The main results in this section concern the smoothness of this operator with respect to the shape of the obstacle. In order to apply the standard control theoretic results we need this dependence to be quite regular, say C ∞ . Let us note that the same problem, with more general deformations and with less regular dependence on the shape, has been considered in Dal Maso, DeSimone and Morandotti [START_REF] Maso | An existence and uniqueness result for the motion of self-propelled microswimmers[END_REF]. Since the smoothness with respect to the shape results needed in this work do not seem to follow directly from [START_REF] Maso | An existence and uniqueness result for the motion of self-propelled microswimmers[END_REF], we give below a quick derivation of the needed smoothness results, using a classical approach introduced in Murat and Simon [START_REF] Bello | The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow[END_REF].

Throughout this section, S denotes a bounded domain of R 3 with locally Lipschitz boundary and we set Ω = R 3 \ S. To state some well-posedness results for the exterior Stokes problem we recall the definition of some function spaces, which are borrowed from [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF].

Definition 2.1. The homogeneous Sobolev space D 1,2 0 (Ω) is the closure of C ∞ 0 (Ω) with respect to the norm | • | 1,2 defined by |ϕ| 1,2 = ∇ϕ L 2 (Ω) (ϕ ∈ C ∞ 0 (Ω)) . The homogeneous Sobolev space D 1,2 (Ω) is defined by D 1,2 (Ω) = ϕ ∈ L 1 loc (Ω), ∇ϕ ∈ L 2 (Ω) .
We note that D 1,2 0 (Ω) ⊂ D 1,2 (Ω) and that | • | 1,2 is a semi-norm on D 1,2 (Ω). Moreover, we denote by

D -1,2 0 (Ω) completion of C ∞ 0 (Ω) with respect to the norm |f | -1,2 = sup u∈D 1,2 (Ω), |u| 1,2 =1
f, u L 2 (Ω) .

The limit statement lim |x|→∞ u(x) = 0 for u ∈ D 1,2 (Ω) has the meaning:

lim r→+∞ S(0,r) |u(x)| dΓ = 0 , (2.1)
with S(0, r) the sphere in R 3 of radius r centered in 0.

At this stage we need a well-posedness result for the exterior Stokes system. This result is a direct consequence of Galdi, [8, chap. 5, Theorem 2.1, p. 251] and of a lifting result concerning the divergence operator, sometimes called Bogovski's Lemma, see, for instance, [START_REF] Galdi | An introduction to the mathematical theory of the Navier-Stokes equations[END_REF]chap. 3,Theorem 3.4,p. 142].

Theorem 2.2. Let f ∈ D -1,2 0 (Ω), g ∈ L 2 (Ω) and v 0 ∈ H 1 2 (∂Ω). Then there exists a unique (v, p) ∈ D 1,2 (Ω) × L 2 (Ω) weak solution of:              -µ∆v + ∇p = f , in Ω , div v = g , in Ω , v = v 0 , on ∂Ω , lim |x|→∞ v = 0 , (2.2) such that: |v| 1,2 + p L 2 (Ω) c(Ω) |f | -1,2 + g L 2 (Ω) + v 0 H 1 2 (∂Ω)
.

We next define an operator playing a central role in the remaining part of this work. More precisely, we introduce the force operator, associating to each Dirichlet boundary condition v 0 in (2.2) the resulting force exerted by the fluid on S Let e z is the unitary vector having the direction Oz and let (u z , q z ) ∈ D 1,2 (Ω) × L 2 (Ω) be the weak solution of:

             -µ∆u z + ∇q z = 0 , in Ω , div u z = 0 , in Ω , u z = e z , on ∂Ω , lim |x|→∞ u z = 0 (2.3)
An important role in this work is played by the bounded linear operator F(S) :

H 1 2 (∂S) → R defined by F(S)(v 0 ) = Ω σ(v, p) : ∇u z dx (v 0 ∈ H 1 2 (∂S)), (2.4)
where (v, p) is the solution of (2.2) with f = 0, g = 0 and

σ(v, p) = 2µD(v) -pI 3 ,
is the Cauchy stress tensor in the fluid. Note that, for (v, p) smooth enough, we have

F(S)(v 0 ) = ∂Ω σ(v, p)n • e z dΓ (v 0 ∈ H 1 2 (∂S)), (2.5)
where n is the unitary vector field normal to ∂S and oriented towards the exterior of S. In other words, for (v, p) smooth enough, F(S)(v 0 ) is the force on the direction Oz exerted by the fluid on the solid S due to the velocity field v 0 at the interface.

Remark 2.3. It can be easily checked that F is essentially invariant with respect to translations of S. More precisely, for every h ∈ R 3 and for every v 0 ∈ H 1 2 (∂(S + h)), we have :

F(S + h)(v 0 ) = F(S)( ṽ0 ) , with ṽ0 ∈ H 1 2 (
∂S) defined, for every x ∈ ∂S, by ṽ0 (x) = v 0 (x + h).

In the remaining part of this section we study the regularity of F with respect to shape of S. We first recall a classical result on the robustness of the locally Lipschitz property of the boundary ∂S with respect to small geometric perturbations (see, for instance, Bello, Fernandez-Cara, Lemoine and Simon [START_REF] Bello | The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow[END_REF]).

Lemma 2.4. There exists a positive constant c(S) such that for every

θ ∈ W 1,∞ (R 3 , R 3 ) with θ W 1,∞ (R 3 ,R 3 ) < c(S), the set (I + θ)(S)
is an open bounded domain with locally Lipschitz boundary in R 3 .

We then define the set of deformations for S:

Θ(S) = θ ∈ W 1,∞ (R 3 , R 3 ), θ W 1,∞ (R 3 ,R 3 ) < min c(S), 1 (2.6)
Then for every θ ∈ Θ(S), the mapping

I + θ is a diffeomorphism of R 3 , (I + θ)(S)
is an open bounded domain of R 3 with a locally Lipschitz boundary and

(I + θ)(Ω) = R 3 \ (I + θ)(S).
We next give a differentiability (with respect to the shape) result. We skip its proof since it can be obtained by a slight variation of the proof of Theorem 5 from [START_REF] Bello | The differentiability of the drag with respect to the variations of a Lipschitz domain in a Navier-Stokes flow[END_REF], combined with Theorem 2.2.

Theorem 2.5. Let v 0 ∈ H 1 2 (∂Ω). The map associating to every τ ∈ Θ(S) the couple

v(τ ) • (I + τ ), p(τ ) • (I + τ ) ∈ D 1,2 (Ω) × L 2 (Ω),
where v(τ ), p(τ ) is the solution of

             -µ∆v(τ ) + ∇p(τ ) = 0 , in (I + τ )(Ω) , div v(τ ) = 0 , in (I + τ )(Ω) , v(τ ) = v 0 • (I + τ ) -1 , on ∂(I + τ )(Ω) , lim |x|→∞ v(τ ) = 0 (2.7)
is infinitely differentiable in a neighborhood of 0.

We are now in a position to prove the smoothness of the real valued function F, defined in (2.4), with respect to the shape of the solid.

Theorem 2.6. For every v 0 ∈ H 1 2 (∂Ω) the function mapping Θ(S) into R defined by θ → F((I + θ)(S))(v 0 • (I + θ) -1 ) (θ ∈ Θ(S)) , is of class C ∞ around 0.
Proof. For τ ∈ Θ(S), let v(τ ), p(τ ) be the solution of (2.7) and let u z (τ ),

q z (τ ) , ∈ D 1,2 (I + τ )(Ω) × L 2 (I + τ )(Ω) satisfy:              -µ∆u z (τ ) + ∇q z (τ ) = 0 , in (I + τ )(Ω) , div u z (τ ) = 0 , in (I + τ )(Ω) , u z (τ ) = e z , on ∂(I + τ )(Ω) , lim |x|→∞ u z (τ ) = 0. Define (U z (τ ), Q z (τ )) = u z (τ ), q z (τ ) • (I + τ ) and (V (τ ), P (τ )) = v(τ ), p(τ ) • (I + τ ).
Then, using (2.4) and the change of variables x = (I + τ )y, it follows that :

F (I + τ )(S) v 0 • (I + τ ) -1 = (I+τ )(Ω) σ v(τ ), p(τ ) : ∇u z (τ ) dx = Ω µ (I 3 + ∇τ ) -1 ∇V (τ ) + ∇V (τ ) T (I 3 + ∇τ T ) -1 -P (τ )I 3 : (I 3 + ∇τ ) -1 ∇U (τ ) Jac(I + τ ) dy
The conclusion follows from the above formula by applying Theorem 2.5.

The case of radial deformations

In this section and in the remaining part of this work we assume that the swimmer's body occupies at rest the unit sphere S 0 of R 3 and its mass density is constant (ρ 0 = 1).

Moreover, we consider only radial axi-symmetric deformations X ⋆ (t, •) of the unit sphere S 0 . This means, using appropriate spherical coordinates (r, θ, φ) ∈ R + × [0, π] × [0, 2π[ (see Fig. 3), that X ⋆ can be written in the form

X ⋆ (t, y) = 1 + r ⋆ t, cos θ(y) y (t ∈ [0, T ], y ∈ S 0 ) , (3.1)
where

r ⋆ ∈ C ∞ [0, T ] × [-1, 1]
, R and θ is the function which maps x ∈ R 3 to the angle θ(x) ∈ [0, π] (see Fig. 3). We denote by Jac X ⋆ the Jacobian of the mapping X ⋆ and by ρ ⋆ the mass density field on S ⋆ (t) = X ⋆ (t, •)(S 0 ) which is defined by

ρ ⋆ (t, x ⋆ ) = 1 Jac (X ⋆ (t, •)) X ⋆ (t, •) -1 (x ⋆ ) (t ∈ [0, T ], x ⋆ ∈ S ⋆ (t)) . (3.2)
Let (P i ) i 0 be the Legendre polynomials which are known to form an orthogonal basis in

L 2 [-1, 1].
The result below shows that, under mild assumption on r * the deformations defined by (3.1) satisfy the so-called self-propelling conditions. This result is given without proof.

Proposition 3.1. Assume that for every t ∈ [0, T ] we have r ⋆ (t, •) is orthogonal to {P 0 , P 1 } (in the L 2 sense) and that r ⋆ (t, x) > -1 (t ∈ [0, T ], x ∈ [-1, 1]). (3.3)
Then we have

• For every t > 0 the mapping X ⋆ (t, •) is a diffeomorphism from S 0 onto X ⋆ (t, •)(S 0 ) and 0 < Jac X ⋆ (t, y) (t ∈ [0, T ], y ∈ S 0 ) . (3.4) Moreover, X ⋆ (t, •)(S 0 ) is a bounded domain with Lipschitz boundary for every t ∈ [0, T ].
• the mass center is fixed,

0 = S ⋆ (t) ρ ⋆ (t, x ⋆ )x ⋆ dx ⋆ (t ∈ [0, T ]) , (3.5)
• the angular momentum is constant,

0 = S ⋆ (t) ρ ⋆ (t, x ⋆ ) Ẋ⋆ (t, X ⋆ (t, •) -1 (x ⋆ )) × x ⋆ dx ⋆ (t ∈ [0, T ]). (3.6)
Formulas (3.5) and (3.6) correspond to the self-propelling conditions which are natural requirements in understanding swimming from a mathematical view point (at least for large Reynolds numbers). In the precise control problem which is considered in this work (i.e., at low Reynolds numbers and with X * time periodic), the reachable space by general axi-symmetric deformations X * coincides with the reachable space obtained with the smaller input space defined corresponding to the assumptions in Proposition 3.1. This means, that in the particular case considered in this work, the self-propelling conditions play no role, as shown in Proposition 3.4 below.

We next show that, under the assumptions of Proposition 3.1, the considered system reduces to a quite simple system of ordinary differential equations. Indeed, using the notations introduced by (2.4) and assuming that the deformation X ⋆ is defined by (3.1), we can rewrite problem (1.1)-(1.3), (1.7) in the form:

0 = F X ⋆ (t, •)(S 0 ) + h(t)e z ḣ(t)e z + Ẋ⋆ (t, • -h(t)e z ) , (3.7)
h(0) = 0 , (3.8)
Using the fact that for every t ∈ [0, T ], F X ⋆ (t, •)(S 0 )+h(t)e z is a linear map from H 1 2 ∂(X ⋆ (t)(S)+ h(t)e z ) to R and using Remark 2.3, equation (3.7) becomes:

F (X ⋆ (t, •)(S 0 )) (e z ) ḣ(t) = -F (X ⋆ (t, •)(S 0 )) Ẋ⋆ (t, •) (3.9)
We also need a result, often in the literature, asserting that the drag acting on a solid moving at constant velocity in Stokes fluid is not vanishing. For the sake of completeness, we provide a short proof. Lemma 3.2. For every bounded and open domain S ⊂ R 3 with Lipschitz boundary we have:

F(S)(e z ) = 0 .
Proof. Let (u z , q z ) be the solution of the exterior homogeneous Stokes problem with boundary condition e z (see (2.3)). Using the Green formulae, we have:

F(S)(e z ) = R 3 \S µ(∇u z + ∇u T z ) -q z I 3 : ∇u z dx = µ 2 R 3 \S (∇u z + ∇u T z ) : (∇u z + ∇u T z ) dx+ + µ 2 R 3 \S (∇u z + ∇u T z ) : (∇u z -∇u T z ) dx - R 3 \S q z div u z dx = µ 2 R 3 \S (∇u z + ∇u T z ) : (∇u z + ∇u T z ) dx .
We argue by a contradiction argument. Assuming that F(S)(e z ) = 0, we have that ∇u z + ∇u T z , i.e., the symmetric strain rate tensor, is equal to zero. Then it is well known that Proof. Using Proposition 3.1, Lemma 3.2 and Theorem 2.6, we obtain that the map

u z (x) = a + b × x, with a, b ∈ R 3 . But due to the fact that ∇u z ∈ L 2 (R 3 \ S),
t → F (X ⋆ (t, •)(S 0 )) Ẋ⋆ (t, •) F (X ⋆ (t, •)(S 0 )) (e z ) is C ∞ on [0, T ].
Hence, the unique solution of (3.7)

-(3.8) is h ∈ C ∞ ([0, T ], R) defined by: h(t) = - t 0 F (X ⋆ (s, •)(S 0 )) Ẋ⋆ (s, •) F (X ⋆ (s, •)(S 0 )) (e z ) ds (t ∈ [0, T ]) . (3.10)
The proposition below shows that, if we assume that r ⋆ is time-periodic, then the orthogonality on P 0 and P 1 (and, consequently, the self-propelling conditions), assumed in Proposition 3.1, does not affect the set of reachable final positions h(T ) of the solid.

Proposition 3.4. Let X ⋆
1 and X ⋆ 2 be two axi-symmetric deformations defined for t ∈ [0, T ] satisfying

X ⋆ i (0, •) = X ⋆ i (T, •) = I , i ∈ {1, 2}
and which satisfy, for every t ∈ [0, T ], the condition

X ⋆ 1 (t, •) = X ⋆ 2 (t, •) + h c (t)e z , for some h c ∈ C 1 ([0, T ], R). Let h 1 (resp. h 2 ) be defined by (3.10) with X ⋆ = X ⋆ 1 (resp. X ⋆ = X ⋆ 2 ). Then h 1 (T ) = h 2 (T ).
Proof. From (3.9) it follows that

F (X ⋆ 2 (t)(S 0 )) (e z ) ḣ2 (t) = -F (X ⋆ 2 (t)(S 0 )) Ẋ⋆ 2 (t) F (X ⋆ 1 (t)(S 0 )) (e z ) ḣ2 (t) = -F (X ⋆ 1 (t)(S 0 )) Ẋ⋆ 1 (t) . Using Remark 2.3 it follows that -F (X ⋆ 2 (t)(S 0 )) Ẋ⋆ 2 (t) = -F (X ⋆ 1 (t)(S 0 )) Ẋ⋆ 1 (t) + ḣc (t)e z = -F (X ⋆ 1 (t)(S 0 )) Ẋ⋆ 1 (t) -F (X ⋆ 1 (t)(S 0 )) (e z ) ḣc (t)
and

F (X ⋆ 2 (t)(S 0 )) (e z ) = F (X ⋆ 1 (t)(S 0 )) (e z ) Hence ḣ2 + ḣc = ḣ1 . Finally, since X ⋆ i (0, •) = X ⋆ i (T, •) = I , i ∈ {1, 2} (consequently h c (0) = h c (T ) = 0), we obtain h 2 (T ) = h 1 (T ).
Denote, for the remaining part of this paper

D i (y) = P i+1 cos θ(y) y (i 1 , y ∈ S 0 ) , (3.11)
where (P i ) i∈N are the Legendre polynomials and r(y), θ(y), φ(y) ∈ R + × [0, π] × [0, 2π[ are the spherical coordinates of y (see Figure 3). The remaining part of this section deals with the first derivative with respect to the shape of a certain ratio of hydrodynamical forces. We use, in particular, asymptotic formulas derived by Lighthill (see [START_REF] Lighthill | On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers[END_REF]). The precise form in which we use these results is borrowed from Shapere and Wilczek [17, equation (2.12)].

Proposition 3.5. Let F be the force operator introduced in Section 2, let L 1 and let M SW = (M SW i,j ) i,j∈{1,...,L} ∈ M L (R) be defined by:

M SW i,j =                (i + 1) 2 (2i + 1)(2i + 3) if j = i -1, -i 2 + 2i + 5 (2i + 3)(2i + 5) if j = i + 1, 0 otherwise. (3.12)
for every i, j ∈ {1, . . . , L}. Then

F (I + ε L j=1 α j D j )(S 0 ) ε L i=1 αi D i F (I + ε L j=1 α j D j )(S 0 ) (e z ) = ε 2 M SW α, α + o(ε 2 ) (ε → 0), (3.13
)

for every α = (α 1 , . . . , α L ) T ∈ C ∞ (R + , R L )
, where (D i ) i∈{1,...,L} are given by (3.11).

As a consequence of the above proposition, we have the following result.

Proposition 3.6. Let L 1 and (D i ) i 1 be the deformations defined by (3.11). Then for every i, j ∈ {1, . . . , L}, we have:

F(S 0 )(D i ) = 0 and F ′ (D j )(D i ) F(S 0 )(e z ) i,j∈{1,...,L} = M SW ∈ M L (R)
where M SW ∈ M L (R) is the matrix defined in Proposition 3.5 by (3.12) and, for every τ ∈ Θ(S) and every

v 0 ∈ H 1 2 (∂Ω), F ′ (τ )(v 0 ) stands for the differential of θ → F((I + θ)(S))(v 0 • (I + θ) -1
) at 0 in the direction τ (which exists according to Theorem 2.6).

Proof. We know from Theorem 2.6 that the map

ε → F (I + ε L j=1 α j D j )(S 0 ) ε L i=1 αi D i F (I + ε L j=1 α j D j )(S 0 ) (e z )
,

is, for every α ∈ C ∞ (R + , R L ) of class C ∞ in
a neighborhood of the origin. By comparing the second order Taylor expansion of this function around the origin with formula (3.13) we obtain the required result.

Controllability results

Recall from the introduction that our aim is to steer the mass center of the swimmer in a prescribed final position, with initial and final deformation equal to zero. Since we also want to control the final deformation, it seems convenient to extend the state space by including in it the variables describing the deformation (i.e., the function r ⋆ ) and to use the derivative with respect to time of r ⋆ as new input function.

Based on the above remark and under the assumptions of Proposition 3.3, the system (3.7)-(3.8) can be written:

ḣ(t) = - F (X ⋆ (t, •)(S 0 )) (w ⋆ (t, •)) F (X ⋆ (t, •)(S 0 )) (e z ) , (4.1)
ṙ⋆ (t, •) = Q ⋆ (t, •) , (4.2)
with the initial conditions r ⋆ (0,

•) = 0 , h(0) = 0 , (4.3)
where the deformation velocity w ⋆ is given by

w ⋆ (t, y) = Q ⋆ t, cos θ(y) y ((t, y) ∈ [0, T ] × S 0 ) ,
and Q ⋆ is the new input function.

With the above notation, the main result in this section states as follows:

Theorem 4.1. For every h 1 ∈ R, there exists T > 0 and

Q ⋆ ∈ C ∞ [0, T ] × [-1, 1], R such that the solution (h, r ⋆ ) of (4.1)-(4.3) satisfies: 1. h(T ) = h 1 and r ⋆ (T, •) = 0, 2. for every t ∈ [0, T ], Q ⋆ (t, •) L ∞ (-1,1) 1,

for every

t ∈ [0, T ], r ⋆ (t, •) ∈ {P 0 , P 1 } ⊥ and inf (t,x)∈[0,T ]×[-1,1] r ⋆ (t, x) > -1.
In order to prove the above result, we use Chow's Theorem (see for instance [19, chap. 5, Proposition 5.14, p. 89] or [START_REF] Jurdjevic | Geometric control theory[END_REF]): Theorem 4.2 (Chow). Let m, n ∈ N and let (f i ) i=1,n be C ∞ vector fields on R n . Consider the control system, of state trajectory q, q = m i=1 u i f i (q), (4.4)

with input function u = (u i ) i=1,m ∈ C ∞ [0, +∞[, B R m (0, r) for some r > 0 .
Let O an open and connected set of R n and assume that

Lie q {f 1 , . . . , f m } = R n q ∈ O ,
Then the system (4.4) is controllable, i.e., for every q 0 , q 1 ∈ O there exists T > 0 and u ∈ C ∞ ([0, T ], B R m (0, r)) such that q(0) = q 0 and q(T ) = q 1 and q(t) ∈ O for every t ∈ [0, T ].

In the previous Theorem, the Lie-bracket of two vector fields g 1 , g 2 is a new vector field defined by

[g 1 , g 2 ](q) = D q g 2 • g 1 (q) -D q g 1 • g 2 (q).
We recall that a Lie-algebra is a space closed for the Lie-bracket [•, •] and that Lie{f 1 , . . . ,

f m }, [•, •]
is the smallest Lie-algebra containing {f 1 , . . . , f m }. Moreover, Lie q {f 1 , . . . , f m } ⊂ R n is the subspace of R n spanned by all the values in q of the vector fields in Lie{f 1 , . . . , f m }.

We are now in position to prove our main controllability result.

Proof of Theorem 4.1. Let consider the two dimensional space of deformations X ⋆ defined by

X ⋆ (t, y) = y + α 1 (t)D 1 (y) + α 2 (t)D 2 (y) ((t, y) ∈ [0, T ] × S 0 ) , with α i ∈ C ∞ [0, T ], (-1 2 , 1 2
) and (D i ) i∈{1,2} introduced in (3.11). Due to the fact that |P i (ζ)| 1 for every ζ ∈ [-1, 1] and i ∈ N, the above defined deformations X ⋆ satisfy the assumptions in Proposition 3.1. Moreover, if we assume that α i (0) = α i (T ) = 0 then X ⋆ (0, •) = X ⋆ (T, •) = I. With the above notation, it is easily seen that (3.9) writes:

ḣ = - 2 i=1 β i -F X ⋆ (t, •)(S 0 ) (D i ) F X ⋆ (t, •)(S 0 ) (e z ) , α1 = β 1 , α2 = β 2 .
The above system can be written in the condensed form:

q = L i=1 β i f i (q) , (4.5)
where q = (h,

α 1 , α 2 ) T ∈ R × (-1 2 , 1 2 ) 2 and ( f i ) i∈{1,2} are the C ∞ vector field on R × (-1 2 , 1
2 ) 2 defined by:

f 1 (h, α 1 , α 2 ) ⊤ =     -F (I+α 1 D 1 +α 2 D 2 )(S 0 ) (D 1 ) F (I+α 1 D 1 +α 2 D 2 )(S 0 ) (ez) 1 0     f 2 (h, α 1 , α 2 ) ⊤ =     -F (I+α 1 D 1 +α 2 D 2 )(S 0 ) (D 2 ) F (I+α 1 D 1 +α 2 D 2 )(S 0 ) (ez ) 0 1     , for every (h, α 1 , α 2 ) ⊤ ∈ R × (-1 2 , 1 2 ) 2 . According to Proposition 3.6, D f i q=0
• 0 e j = M SW e i , e j , 0, . . . , 0 ⊤ .

where the matrix M SW =∈ M 2 (R) is given by :

M SW = 3 35 0 2 3 0 . Hence [ f 1 , f 2 ] q=0 = M SW e 2 , e 1 -M SW e 1 , e 2 , 0, 0 ⊤ = 3 35 (1, 0, 0) ⊤ , so that dim Lie q=0 { f 1 , f 2 } = 3 .
Since f 1 , f 2 and q → [ f 1 , f 2 ] q are C ∞ functions (see Theorem 2.6), there exists ε > 0 such that for every

q ∈ {0}×] -ε, ε[ 2 , dim Lie q { f 1 , f 2 } = 3.
Using the fact that f 1 , f 2 and [ f 1 , f 2 ] do not depend on h (i.e., on the first component of q), it follows that

dim Lie q { f 1 , f 2 } = 3 (q ∈ R×] -ε, ε[ 2 ).
We conclude by using Theorem 4.2 with O a neighborhood of R × {(0, 0) ⊤ }.

In the end of this Section we study the time optimal control associated to the controllability problem introduced in Theorem 4.1, by restricting the inputs to a vector space of dimension L 2. Let first give some notations. Given α ∈ R L , we denote by S ⋆ (α) the deformed sphere,

S ⋆ (α) = (I + L i=1 α i D i )(S 0 ) (α = (α 1 , . . . , α L ) ⊤ ∈ R L , |α| 2 c) , (4.6)
where D i are defined in (3.11) and c > 0 is small enough in order to ensure that Jac(I + L i=1 α i D i ) > 0 on S 0 . We also introduce, for every α ∈ R L small enough, the vector

f (α) = (f i (α)) i∈{1,...,L} ∈ R L with f i (α) = F S ⋆ (α) (D i ) F S ⋆ (α) (e z ) (α = (α 1 , . . . , α L ) ∈ R L , |α| 2 c , i ∈ {1, . . . , L}) . (4.7)
With the above notation ad assumptions, the control system (4.1)-( 4.3) writes

ḣ = f (α), β (4.8) α = β (4.9) h(0) = 0, α(0) = 0 . (4.10)
Looking to the proof of Theorem 4.1, we easily see that the following Corollary holds:

Corollary 4.3. For every L 2 and h 1 ∈ R, there exists T > 0 and β ∈ C ∞ [0, T ], R L such that the solution (h, α) of (4.8)-(4.10) satisfies:

h(T ) = h 1 , α(T ) = 0 , (4.11)
with the constraint on the control variable:

|β(t)| 2 1 (t ∈ [0, T ]) (4.
β chosen in L ∞ (0, T ), R L .
Proof. For L 2, the existence of a solution (T , h, α, β) satisfying (4.8)-(4.13) is ensured according to the controllability result given in Corollary 4.3. To prove the existence of a minimal time, we shall make use of the classical Filippov Theorem (see for instance [START_REF] Hartl | A survey of the maximum principles for optimal control problems with state constraints[END_REF], [START_REF] Cesari | Optimization-theory and applications[END_REF] or [START_REF] Agrachev | Control theory from the geometric viewpoint[END_REF]). First, due to the linearity of the optimal problem with respect to the control variables β, the set

A(α, t) = ( f (α), β , β) ∈ R × R L with |β(t)| 2 1
is convex for any state variables α with |α(t)| 2 c and for all t ∈ [0, T ].

Moreover, using Theorem 2.6 we deduce that the functions f i are continuous with respect to α. Therefore, there exists a constant C > 0 such that, for all t ∈ [0, T ],

|f (α(t))| 2 C,
for any |α(t)| 2 c and |β(t)| 2 1. The uniform bound on f provides a bound on h. Indeed, for all t ∈ [0, T ], we have

|h(t)| = t 0 f (α), β ds t 0 |f (α)| 2 |β| 2 ds CT .
Thus, all the state and control variables are uniformly bounded, provided that the control and the state satisfy the given constraints. Therefore, we can apply the Filippov-Cesari Theorem (see [START_REF] Hartl | A survey of the maximum principles for optimal control problems with state constraints[END_REF]Theorem 3.1] or [START_REF] Cesari | Optimization-theory and applications[END_REF]Theorem 9.3]) to conclude to the existence of an optimal solution (T * , h * , α * , β * ).

In the next two sections, we consider a simplified model for which we shall explicitly determine the optimal solutions (T * , h * , α * , β * ) of problem (4.8)-(4.13). We first neglect the state constraint (4.13) and we obtain the optimal explicit solutions by directly applying Pontryagin's maximum principle.

We finally study (4.8)-(4.13) by taking in consideration the state constraint on α. To this purpose, we apply an extended maximum principle which takes into account the state constraint and that provides jump conditions for the multipliers.

To obtain the simplified model we note that if α is small enough, according to Proposition 3.5, equation (4.8) can be approximated by

ḣ = M SW α, β ,
The above equation is of independent interest. In particular, it can be seen as a generalization of the system describing a non holonomic integrator, see, for instance Coron ... In the remaining part of this work we study a slight generalization of the above control system, obtained by replacing the matrix M SW by an arbitrary non symmetric matrix M ∈ M L (R).

A simplified optimal control problem without state constraints

In this section we consider the simplified control system described at the end of the last section. More precisely, given an integer L 2, M ∈ M L (R) with M ⊤ = M and h 1 ∈ R * , our aim consists in determining the minimal time T * for which there exists β ∈ L ∞ (0, T * ), R L such that

|β(t)| 2 1 (for a.e. t ∈ (0, T * )) , (5.1)
and the solution (α,

h) ∈ W 1,∞ (0, T * ), R L × W 1,∞ (0, T * ), R of ḣ = M α, β , (5.2
) α = β , (5.3)
h(0) = 0 and α(0) = 0 .

(5.4) satisfies h(T * ) = h 1 and α(T * ) = 0 .

(

We assume that h 1 = 0, since otherwise the time optimal control problem is trivial. Before integrating the Pontryagin maximum principle, we give some elementary facts about this control problem.

Proposition 5.1. The time T * > 0 and the control β * ∈ L ∞ (0, T * ), R L are solutions of the time optimal control system (5.1)-(5.5) if and only if T * and β * are solution of the time optimal control system obtained from (5.1)-(5.5) by replacing (5.2) with the equation

ḣ = 1 2 (M -M ⊤ )α, β . (5.6)
In other words, the optimal time and controls of the system (5.1),(5.2)-(5.5) coincide with the ones of the system (5.1),(5.6),(5.3)-(5.5).

Proof of Proposition 5.1. Note that (5.2) can be written in the form

ḣ = 1 2 (M + M ⊤ )α, α + 1 2 (M -M ⊤ )α, α .
Since M + M ⊤ is symmetric, there exist a diagonal matrix D and an orthogonal matrix

Q such that M + M ⊤ = Q ⊤ DQ. Consequently, (M + M ⊤ )α, α = DQα, Q α = 1 2 d dt DQα, Qα . Integrating (M + M ⊤ )α, α over t ∈ [0, T * ] yields T * 0 (M + M ⊤
)α, α dt = 0 due to the initial and final conditions α(0) = α(T * ) = 0. In other words, the symmetric part of M does not play any role in the control system. This fact clearly implies that the optimal times and the optimal time controls of the two considered systems coincide.

The remaining part of this section is devoted to the application of Pontryagin's maximum principle which yields, at least for the simplified model with no state constraints, an explicit solution of the optimal control problem. We first recall, following [START_REF] Trélat | Contrôle optimal, Mathématiques Concrètes[END_REF], some basic facts on Pontryagin's maximum principle and we adapt the general procedure to our case. To this end we introduce the Hamiltonian H for the minimal time problem of the control system (5.1)-(5.5). To accomplish this goal we gather the state variables in a single vector by setting

q = h α ∈ R L+1 ,
and we denote

F (q, β) = M α, β β ∈ R L+1 .
Then the differential system (5.2)-(5.5) reads q = F (q, β) , (

q(0) = 0, q(T ) = h 1 0 L , 5.7)
where 0 L denotes the null vector in R L . The Hamiltonian of the system (5.7),(5.8) for the optimal time problem is the function

H : R L+1 × R L × R L+1 × (-∞, 0] defined by H(q, β, r, s 0 ) = r, F (q, β) + s 0
with q, r ∈ R L+1 , β ∈ R L and s 0 0. We recall that r is generally designed as the adjoint state. In the present case it is convenient to write r into the form

r = p 0 p ∈ R L+1 ,
with p 0 ∈ R and p ∈ R L . With the above notation, the Hamiltonian can be also written:

H(α, β, p 0 , p, s 0 ) = p 0 M α + p, β + s 0 , (5.9)
with p ∈ R L , p 0 ∈ R and s 0 ∈ R, s 0 0. The Pontryagin maximum principle ensures that if (T, q, β) is an optimal solution to (5.7),(5.8), then there exist a function r : [0, T ] → R L+1 and a scalar s 0 0 such that the pair (r, s 0 ) is nontrivial and such that q = ∂H ∂r = F (q, β) , (5.10)

ṙ = - ∂H ∂q = - ∂F ∂q ⊤ r , (5.11)
q(0) = 0, q(T ) = h 1 0 L , (5.12)
H(α, β, p 0 , p, s 0 ) = max

γ∈R L |γ| 2 1
H(α, γ, p 0 , p, s 0 ) (5.13)

Moreover, since the system (5.7),(5.8) is autonomous, if (T, h, α, β, p 0 , p) is an optimal solution to (5.15)-(5.20) then H α(t), β(t), p 0 (t), p(t), s 0 = 0 (for a.e. t ∈ [0, T ]) .

• Assume that p(0) = 0. Using the relation (5.14) at t = 0 we obtain that s 0 = 0, which is a contradiction. Thus, p(0) = 0.

• Finally, suppose that p 0 = 0. Then, according to (5.18) we have that p(t) = p(0) for all t ∈ [0, T ].

The maximum principle property (5.13) provides the optimal control β(t) = p(0) |p(0)| 2 = β(0) for all t ∈ [0, T ]. Since α(0) = α(T ) = 0, we get β(0) = 0 by integrating (5.16) so that β(t) = 0 for all t ∈ [0, T ]. We deduce from (5.14) that s 0 = 0 which is a contradiction. Thus, p 0 = 0.

We are now in position to determine the optimal control variables β. Using the fact that s 0 < 0, it follows from (5.14) that p 0 M α(t) + p(t) = 0 for every t ∈ [0, T ]. Using the expression (5.9) of the Hamiltonian, together with (5.13) we deduce that the optimal control β is indeed given by (5.21).

The result below shows that, for the simplified control problem (5.1)-(5.5), without state constraints, an optimal solution can be explicitly determined. For the remaining part of the paper, we denote by σ(A) the set of the complex eigenvalues of a matrix A.

Proposition 5.3. The minimal time T * such that (5.1)-(5.5) admits a solution is given by

T * = 2π|h 1 | λ * , (5.26
)

where λ * = max |λ|, λ ∈ σ 1 2 (M -M ⊤ ) > 0.
Moreover, an optimal time control is

β * (t) = exp sign(h 1 ) 2π λ * |h 1 | t (M -M ⊤ ) 2 β 0 (t ∈ [0, T * ]) , (5.27)
where the initial control

β 0 ∈ R L is chosen such that |β 0 | 2 = 1 with β 0 ∈ Ker M -M ⊤ 2 2 + |λ * | 2
Proof. Let us denote by M A the skew-symmetric part of the non-symmetric matrix M i.e. M A = 1 2 (M -M ⊤ ) = 0. According to Proposition 5.1, it is enough to consider the control system (5.1)-(5.5) with M replaced by M A . From Lemma 5.2 with the relation (5.14), we deduce that the optimal control variables β is given by

β(t) = - (p 0 M A α(t) + p(t)) s 0 (t ∈ [0, T ]) .
Hence, differentiating the above relation over t and using (5.16) and (5.18), we obtain

β = δ -1 M A β ,
with δ = -s 0 2p 0 . Integrating the above ordinary differential equation in β yields for all t ∈ [0, T ],

β(t) = exp δ -1 t M A β(0).
(5.28) Now, we turn to the computation of α and h. Using (5.16), we obtain

d dt δ -1 M A α = δ -1 M A β = β
and then we get

M A α(t) = δ (β(t) -β(0)) (t ∈ [0, T ]) .
(5.29) Using (5.29) in (5.15) we obtain that ḣ = M A α, β = δ β(t)β(0), β(t) .

Integrating the above relation on [0, T ] we obtain

h(T ) = h(0) + δ T 0 |β(t)| 2 2 dt -δ β(0), α(T ) -α(0) dt = δ T 0 |β(t)| 2 2 dt .
(5.30)

On the other hand, (5.21) and (5.28) imply that

|β(t)| 2 = |β(0)| 2 = 1 (t ∈ [0, T ]) .
This fact, combined with (5.30) and the final condition h(T ) = h 1 , leads to

h 1 = δT .
(5.31)

We obtain from (5.28) that

β(t) = exp T h 1 t M A β(0).
(5.32)

We now turn to the final condition α(T ) = 0. Since M A is the skew-symmetric part of M , M A is a normal matrix. Thus there exist a unitary matrix U ∈ M L (C) with U U * = U * U = I L and a diagonal matrix Λ = i diag(λ 1 , • • • , λ L ) where (iλ j ) j are the eigenvalues of M A , such that M A = U ΛU * . Let us note that the non-zero eigenvalues of M A are all pure imaginary and they must occur in complex conjugate pairs. Therefore, if iλ is an eigenvalue of M A then -iλ is also an eigenvalue. The columns of the matrix U are formed by the eigenvectors of M A . We clearly that can rearrange the matrices U and Λ such that there exists R ∈ N * with 2R L and λ j = 0 for every j > 2R. Using the above decomposition of M A and relation (5.31), we obtain

β(t) = U             exp i T h 1 λ 1 t 0 . . . 0 exp i T h 1 λ 2R t 0 0 1 0 . . . 0 1             U * β(0) (t ∈ [0, T ]) .
Integrating the above formula on [0, t], we obtain for every t ∈ [0, T ],

α(t) = t 0 β(s) ds = U             h 1 T λ 1 exp i T h 1 λ 1 t -1 0 . . . 0 h 1 T λ 2R exp i T h 1 λ 2R t -1 0 0 t 0 . . . 0 t             U * β(0) (5.33)
The condition α(T ) = 0 with T > 0 leads to

            exp iλ 1 h 1 T 2 -1 0 . . . 0 exp iλ 2R h 1 T 2 -1 0 0 1 0 . . . 0 1             U * β(0) = 0 (5.34)
We deduce from condition (5.34) that

U * β(0) k = 0 for all k > 2R .
Moreover, for every k ∈ {1, . . . , 2R}, we either have

U * β(0) k = 0 or |λ k | |h 1 | T 2 ∈ 2πN * .
Hence, the set of times T > 0 such that condition (5.34) holds with β(0) ∈ R L \ {0} is given by

K = λ>0 iλ∈σ(M A ) 2nπ|h 1 | |λ| , n ∈ N * . (5.35)
The minimum element of this set is

T * = 2π|h 1 | λ * . where λ * = max{λ 1 , . . . , λ 2R } > 0. Finally, we choose β(0) = β 0 ∈ R L with |β 0 | 2 = 1 such that β 0 ∈ Ker M 2
A + λ * 2 and it can be easily checked that the condition (5.34) is satisfied with T = T * i.e. α(T * ) = 0. The proof of Proposition 5.3 is then complete.

In the case where the matrix M is equal to the matrix M SW given by (3.12), the minimal time obtained in Proposition 5.3 tends to a limit when the number L of deformation modes goes to +∞. This limit is a lower bound for the minimal time obtained using a finite dimensional input space and it can be seen as the minimal time obtained by allowing an infinite dimensional input space.

Corollary 5.4. Let us consider the optimal control problem (5.1)-(5.5) with the (L × L)-matrix M = M SW defined by (3.12). We denote by T * L the corresponding minimal time given by (5.26). Then, the map L → T * L is non increasing and lim

L→+∞ T * L = 2 π|h 1 | . ( 5

.36)

Proof. Using the expression (3.12) of the matrix M SW , we obtain that its skew-symmetric part

M A = 1 2 M SW -M SW ⊤ is the (L × L)-bidiagonal matrix given by M A =        0 a 2 0 -a 2 0 a 3 . . . . . . . . . -a L-1 0 a L 0 -a L 0        with a k = -2k 2 + 2k + 1 2(2k + 1)(2k + 3) for k = 2, • • • , L .
According to Proposition 5.3, the minimal time is given by T

* L = 2π|h 1 | λ * L
, where λ * L > 0 is given by λ * L = max{|λ|, λ ∈ σ(M A )}. We first prove that L → T * L is non increasing. For that it suffices to check that λ * L is non decreasing function with respect to L. In order to check this assertion we denote, for every integer k with 2 k L, by A k the complex (k × k)-matrix

A k = i        0 a 2 0 -a 2 0 a 3 . . . . . . . . . -a k-1 0 a k 0 -a k 0        . ( 5

.37)

Let us remark that A k is a Hermitian matrix and thus all its eigenvalues µ i = µ i (A k ) for i = 1, • • • , k are real numbers. We may assume that the eigenvalues µ i are arranged in increasing order, so that µ k (A k ) = max 1 i k µ i (A k ). Moreover, one can easily check that if µ is an eigenvalue of A k then -µ is also an eigenvalue. This implies that µ k (A k ) 0 for every 1 k L, so that µ L (A L ) = λ * L . The inclusion principle (see [START_REF] Horn | Matrix analysis[END_REF]Th.4.3.15]) ensures that µ k (A k ) µ L (A L ) for all 2 k L.

(5.38)

The inclusion property (5.38) leads to λ * L-1 λ * L . Thus, the largest eigenvalue modulus λ * L is an increasing function of L.

Next, we prove that λ * L tends to 1 2 as L goes to +∞. To this end, we first show that: In particular, we have

λ * L 1 2 .
(5.40)

• The lower bound is obtained by using the Rayleigh-Ritz theorem for the Hermitian matrix A L (see [START_REF] Horn | Matrix analysis[END_REF]Th.4.2.2]). The largest eigenvalue modulus |λ * L | is characterized by To conclude, we give a numerical computation of an optimal solution for the simplified control problem (5.1)-(5.5), in the case where the matrix M is the Shapere-Wilczek matrix M = M SW defined by (3.12). The optimal solution corresponding to the final position h 1 = 1 is computed according to the formulae given in Proposition 5.3. Radial and axi-symmetric deformations are considered with a number L = 10 of deformation modes. The position and the shape of the swimmer at several instants t are depicted in Figure 5. The corresponding minimal time is T * ≃ 4.483.

λ * L = µ L (A L ) = max u∈C L u =0 u * A L u u * u (5.41) Let us choose u = (u k ) 1 k L ∈ C L such that u k =        1 if k ≡ 1 (mod4) i if k ≡ 2 (mod4) -1 if k ≡ 3 (mod4) -i if k ≡ 4 (mod4)

Figure 1 :

 1 Figure 1: The fluid domain and the swimmer.

Figure 2 :

 2 Figure 2: Deformation and transport of the swimmer.

Figure 3 :

 3 Figure 3: Spherical coordinates.

  we obtain that b = 0. Since lim r→+∞ |x|=r |u z (x)| dΓ = 0, we deduce that a = 0. This is in contradiction with the boundary condition u z = e z on ∂S. Consequently F(S)(e z ) = 0. The remaining part of this work is devoted to the study of the system (3.7)-(3.8). We first note that the following simple result holds. Proposition 3.3. Assume that the smooth function r ⋆ satisfies (3.3). Then there exists a unique solution h ∈ C ∞ ([0, T ], R) solution of (3.7)-(3.8).

•

  The upper bound is obtained by applying the Gershgorin theorem (see[START_REF] Horn | Matrix analysis[END_REF] Th.6.1.1]) for the localization of the eigenvalues of M A . Indeed, the eigenvalues λ of M A are such that|λ| max ([a 2 |, |a 2 | + |a 3 |, • • • , |a L-1 | + |a L |, |a L |) = |a L-1 | + |a L | 1 2 .

Figure 4 :

 4 Figure 4: Minimal time with respect to the number L of deformation modes for the simplified control problem without state constraints. Convergence to the limit value 2 π|h 1 | with h 1 = 1.

Figure 5 :

 5 Figure 5: Simplified control problem without state constraints. Time optimal control for radial and axi-symmetric deformations of the unit sphere with L = 10 modes of deformations.

  According to Corollary 4.3, we see that two scalar inputs are sufficient to steer the sphere to any final position lying on e z . Note that, due to the scallop Theorem, see Purcell[START_REF] Purcell | Life at low Reynolds number[END_REF], at least two controls are necessary. It follows that with six scalar inputs the sphere can deform itself periodically in time in order to reach any final position h 1 ∈ R 3 . Proposition 4.5. For any L 2 and h 1 ∈ R, there exists a minimal time T * 0 such that the control problem (4.8)-(4.11) with constraints (4.12) and (4.13) admits a solution for controls

			12)
	and the state constraint
	|α(t)| 2 c	(t ∈ [0, T ]) .	(4.13)
	Remark 4.4.

This property, combined with the expression (5.9) for the Hamiltonian leads to p 0 M α(t) + p(t), β(t) + s 0 = 0 (for a.e. t ∈ [0, T ]) .

(5. [START_REF] Michelin | Efficiency optimization and symmetry-breaking in a model of ciliary locomotion[END_REF] This fact implies that, going back to the variables h and α, that equations (5.10)-(5.12), reduce to determining

We are now in a position to show that the cost multiplier s 0 is negative and to express the optimal control β in function of the state and adjoint state. Lemma 5.2. Let (T, h, α, β, p 0 , p, s 0 ) be a solution of (5.13)-(5.20). Then s 0 < 0, p(0) = 0 and p 0 = 0. In addition, for every t ∈ [0, T ] we have that p 0 M α(t) + p(t) = 0 and the optimal control variables are given by

Proof. We first remark that since h 1 = 0 we necessarily have T > 0. We start by proving that s 0 , p(0) and p 0 do not vanish. We argue by contradiction.

• Suppose that s 0 = 0. From (5.9) and (5.14), we deduce that H(α, β, p 0 , p, s 0 ) = p 0 M α + p, β = 0 .

(5.22)

On the other hand, (5.17) implies that p 0 is a constant, so that (5.22) and (5.15) give p 0 ḣ(t) + p(t), β(t) = 0 (for a.e. t ∈ [0, T ]) .

(5.23) Furthermore, using (5.15), (5.16) and (5.18), we obtain

Combining (5.24) and (5.23), we deduce that

Integrating the above formula on [0, T ] we get 2p 0 h 1 = 0.

(5.25)

On the other hand, since (p 0 , p) ⊤ and s 0 form a nontrivial pair and s 0 = 0, we necessarily have (p 0 , p) ≡ (0, 0). If p 0 = 0 then p = 0 and the Hamiltonian reduces to H(α, β, p 0 , p, s 0 ) = p, β . ¿From the maximum principle (5.13), we obtain that β = p |p| 2 and H(α, β, p 0 , p, s 0 ) = |p| 2 = 0 which is in contradiction with (5.22). Hence, we have p 0 = 0 and we deduce from (5.25) that h 1 = 0 which is a contradiction. We have thus shown that s 0 = 0.


Frédéric Charve
email: frederic.charve@univ-paris12.fr

Boris Haspot
email: haspot@ceremade.dauphine.fr

Existence of global strong solution and vanishing capillarity-viscosity limit in one dimension for the Korteweg system

) can not be used.

Introduction

We are concerned with compressible fluids endowed with internal capillarity. The model we consider originates from the XIXth century work by Van der Waals and Korteweg VW,fK [START_REF] Van Der Waals | Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung[END_REF]22] and was actually derived in its modern form in the 1980s using the second gradient theory, see for instance fDS,fJL,fTN [START_REF] Dunn | On the thermomechanics of interstitial working[END_REF][START_REF] Jamet | The second gradient method for the direct numerical simulation of liquid-vapor flows with phase change[END_REF][START_REF] Bali | The nonlinear field theories of mechanics[END_REF]. The first investigations begin with the Young-Laplace theory which claims that the phases are separated by a hypersurface and that the jump in the pressure across the hypersurface is proportional to the curvature of the hypersurface. The main difficulty consists in describing the location and the movement of the interfaces. Another major problem is to understand whether the interface behaves as a discontinuity in the state space (sharp interface) or whether the phase boundary corresponds to a more regular transition (diffuse interface, DI). The diffuse interface models have the advantage to consider only one set of equations in a single spatial domain (the density takes into account the different phases) which considerably simplifies the mathematical and numerical study (indeed in the case of sharp interfaces, we have to treat a problem with free boundary).

Let us consider a fluid of density ρ ≥ 0, velocity field u ∈ R, we are now interested in the following compressible capillary fluid model, which can be derived from a Cahn-Hilliard like free energy (see the pioneering work by J.-E. Dunn and J. Serrin in fDS [START_REF] Dunn | On the thermomechanics of interstitial working[END_REF] and also in fA,fC,fGP,HM [START_REF] Anderson | Diffuse-interface methods in fluid mech[END_REF][START_REF] Cahn | Free energy of a nonuniform system, I. Interfacial free energy[END_REF][START_REF] Gurtin | Two-phases binary fluids and immiscible fluids described by an order parameter[END_REF][START_REF] Heida | On compressible Korteweg fluid-like materials[END_REF]). The conservation of mass and of momentum write:

     ∂ ∂t ρ + ∂ x (ρ u ) = 0, ∂ ∂t (ρ u ) + ∂ x (ρ (u ) 2 ) -∂ x (ρ ∂ x u ) + ∂ x (a(ρ ) γ ) = 2 ∂ x K, (1.1) 3systeme
where the Korteweg tensor reads as following:

divK = ∂ x ρ κ(ρ )∂ xx ρ + 1 2 (κ(ρ ) + ρ κ (ρ ))|∂ x ρ | 2 -∂ x κ(ρ )(∂ x ρ ) 2 . (1.2) divK
κ is the coefficient of capillarity and is a regular function of the form κ(ρ) = 2 ρ α with α ∈ R. In the sequel we shall assume that κ(ρ) = 2 ρ . The term ∂ x K allows to describe the variation of density at the interfaces between two phases, generally a mixture liquidvapor. P = aρ γ with γ ≥ 1 is a general γ law pressure term.

corresponds to the controlling parameter on the amplitude of the viscosity and of the capillarity. When we set v = u + ∂ x (ln ρ ), we can write ( 3systeme 1.1) on the following form (we refer to Hprepa [START_REF] Haspot | Blow-up criterion, ill-posedness and existence of strong solution for Korteweg system with infinite energy[END_REF] for the computations):

     ∂ ∂t ρ + ∂ x (ρ v ) -∂ xx ρ = 0, ∂ ∂t (ρ v ) + ∂ x (ρ (u )(v )) -∂ x (ρ ∂ x v ) + ∂ x (a(ρ ) γ ) = 0, (1.3) 1.1
We now consider the Cauchy problem of ( 1.3) with the following initial data: ρ (0, x) = ρ 0 (x) > 0, u (0, x) = u 0 (x),

(1.4) 1.2 such that: lim x→+,-∞ (ρ 0 (x), u 0 (x)) = (ρ +,-, u +,-), with ρ +,-> 0.

We would like to study in the sequel the limit process of system ( 1.1

1.3) when goes to 0 and to prove in particular that we obtain entropic solutions of the Euler system:

     ∂ ∂t ρ + ∂ x (ρv) = 0, ∂ ∂t (ρv) + ∂ x (ρv 2 ) + ∂ x (aρ γ ) = 0, (1.5) 1.3
Let us now explain the interest of the capillary solutions for the hyperbolic systems of conservation laws.

Viscosity capillarity processes of selection for the Euler system

In addition of modeling a liquid-vapour mixture, the Korteweg also shows purely theoretical interests consisting in the selection of the physically relevant solutions of the Euler model (in particular when the system is not strictly hyperbolic). The typical case corresponds to a Van der Waals pressure: indeed in this case the system is not strictly hyperbolic in the elliptic region (which corresponds to the region where the phase change occurs).

In the adiabatic pressure framework (P (ρ) = ρ γ with γ > 1), the system is strictly hyperbolic and the theory is classical. More precisely we are able to solve the Riemann problem when the initial Heaviside data is small in the BV space. Indeed we are in the context of the well known Lax result as the system is also genuinely nonlinear (we refer to Lax [START_REF] Lax | Hyperbolic systems of conservation laws II[END_REF]). It means we have existence of global C 1 -piecewise solutions which are unique in the class of the entropic solutions. This result as been extent by Glimm in the context of small initial data in the BV-space by using a numerical scheme and approximating the initial BV data by a C 1 -piecewise function (which implies to locally solve the Riemann problem via the Lax result). For the uniqueness of the solution we refer to the work of Bianchini and Bressan ( BB1 [START_REF] Bianchini | Vanishing viscosity solutions of nonlinear hyperbolic systems[END_REF]) who use a viscosity method. In the setting of the Van der Waals pressure, the existence of global solutions and the nature of physical relevant solutions remain completely open. Indeed the system is not strictly hyperbolic anymore.

If we rewrite the compressible Euler system in Lagrangian coordinates by using the specific volume τ = 1/ρ in ( 1 b , ∞) and the velocity u, the system satisfies in (0, +∞) × R the equations:

∂ t τ -∂ x u = 0, ∂ t u -∂ x ( P (τ )) = 0, (1.6) euler
with the function P : ( 1 b , ∞) → (0, ∞) given by:

P (τ ) = P ( 1 τ ), τ ∈ ( 1 b , ∞).
The two eigenvalues of the system are:

λ 1 (τ, v) = --P (τ ), λ 2 (τ, v) = --P (τ ). (1.7) vp
The corresponding eigenvectors r 1 , r 2 are:

w 1 (τ, v) = 1 -P (τ ) , w 2 (τ, v) = 1 --P (τ ) (1.8)
Furthermore by calculus we obtain:

∇λ 1 (τ, v) • w 1 (τ, v) = P (τ ) 2 -P (τ ) , ∇λ 2 (τ, v) • w 2 (τ, v) = -P (τ ) 2 -P (τ ) (1.9)
We now recall the definition of a standard conservation law in the sense of Lax (it means entropy solutions):

• The system is strictly entropic if the eigenvalues are distinct and real.

• The characteristics fields are genuinely nonlinear if we have for all (τ, v),

∇λ 1 (τ, v) • w 1 (τ, v) = 0 and ∇λ 2 (τ, v) • w 2 (τ, v) = 0,
for more details we refer to

Serre [START_REF] Serre | Systems of conservation laws, I-Hyperbolicity, Entropies[END_REF]. The definition of genuine nonlinearity is some kind of extension of the notion of convexity to vector-valued functions (in particular when we consider the specific case of the traveling waves). The previous assumptions aim at ensuring the existence and the uniqueness of the Riemann problem ( see Evans [START_REF] Evans | Partial differential equations[END_REF] and Serre [START_REF] Serre | Systems of conservation laws, I-Hyperbolicity, Entropies[END_REF]). When P is a Van der Waals pressure, we observe that the first conservation law ( Serre,Evans [START_REF] Serre | Systems of conservation laws, I-Hyperbolicity, Entropies[END_REF][START_REF] Evans | Partial differential equations[END_REF]) is far from being a standard hyperbolic system, indeed:

• It is not hyperbolic (but elliptic) in ( 1 α 1 , 1 α 2 ) × R,
• the characteristic fields are not genuinely nonlinear in the hyperbolic part of the state space.

Here the classical Lax-Glimm theory cannot be applied. In particular there doesn't exist any entropy-flux pair, which suggests that the entropy framework is not adapted for selecting the physically relevant solutions. In order to deal with this problem, Van der Waals and Korteweg began by considering the stationary problem with null velocity, and solving ∇P (ρ) = 0. For more details we refer to Rohdehdr [START_REF] Rohde | Approximation of Solutions of Conservation Laws by Non-Local Regularization and Discretization[END_REF]. It consists in minimizing in the following admissible set

A 0 = {ρ ∈ L 1 (Ω)/W (ρ) ∈ L 1 (Ω), Ω ρ(x)dx = m},
the following functionnal

F [ρ] = Ω W (ρ(x))dx.
Unfortunately this minimization problem has an infinity of solutions, and many of them are physically irrelevant. In order to overcome this difficulty, Van der Waals in the XIXth century was the first to regularize the previous functional by adding a quadratic term in the density gradient. More precisely he considered the following functional:

F local = Ω W (ρ (x)) + γ 2 2
|∇ρ | 2 dx, with:

A local = H 1 (Ω) ∩ A 0 .
This variational problem has a unique solution and its limit (as goes to zero) converge to a physical solution of the equilibrium problem for the Euler system with Van der Waals pressure, that was proved by Modica in REF [START_REF] Modica | The gradient theory of phase transitions and the minimal interface criterion[END_REF] with the use of gamma-convergence. By the Euler-Lagrange principle, the minimization of the Van der Waals functional consists in solving the following stationary problem:

∇P (ρ ) = γ 2 ρ ∇∆ρ ,
where the right-hand side can be expressed as the divergence of the capillarity tensor.

Heuristically, we also hope that the process of vanishing capillarity-viscosity limit selects the physical relevant solutions as it does for the stationary system. This problem actually remains open.

Existence of global entropic solutions for Euler system

Before presenting the results of this paper let us recall the results on this topic in these last decades. We shall focus on the case of a γ pressure law P (ρ) = aρ γ with γ > 1 and a positive. Let us mention that these cases are the only ones well-known (essentially because the system is strictly hyperbolic in this case and that we can exhibit many entropy-flux pairs). Here the Lax-Glimm theory can be applied, however at the end of the 70's, one was interested in relaxing the conditions on the initial data by only assuming ρ 0 and u 0 in L ∞ .

In the beginning of the 80's Di Perna initiated this program, consisting in obtaining global entropic solutions for L ∞ initial data.

Indeed in

Di1, Di2

[7, 8], Di Perna prove the existence of global weak entropy solution of ( In

36

[25], Lions et al generalize this result to the general case γ ∈ (1, 3), and finally the case γ = 1 is treated by Hu1 [START_REF] Huang | Convergence of viscosity solutions for isothermal gas dynamics[END_REF]. We would like to mention that these results are obtained through a vanishing artificial viscosity on both density and velocity. The problem of vanishing physical viscosity limit of compressible Navier-Stokes equations to compressible Euler equations was until recently an open problem. However Chen and Perepelista in 10

[5] proved that the solutions of the compressible Navier-Stokes system with constant viscosity coefficients converge to a entropic solution of the Euler system with finite energy. This result was extended in Hu2 [START_REF] Huang | Vanishing viscosity limit for isentropic Navier-Stokes equations with density-dependent viscosity[END_REF] to the case of viscosity coefficients depending on the density. Inspired by 10

[5] and Hu2 [START_REF] Huang | Vanishing viscosity limit for isentropic Navier-Stokes equations with density-dependent viscosity[END_REF], we would like to show that the solution of the Korteweg system ( 1.1 1.3) converges to a entropic solution of the Euler system with finite energy when the pressure is a γ law. To do this, we will prove for the first time up our knowledge the existence of global strong solution for the Korteweg system in one dimension in the case of Saint-Venant viscosity coefficients. By contrast, the problem of global strong solutions for compressible Navier-Stokes equations remains open (indeed one of the main difficulties consists in controlling the vacuum). This result justifies that the Korteweg system allows us to select the relevant physical solutions of the compressible Euler system at least when the pressure is adiabatic (P (ρ) = aρ γ with γ > 1). The problem remains open in the case of a Van der Waals pressure.

Results

Let us now describe our main result. In the first theorem we prove the existence of global strong solution for the Korteweg system ( 1.1

1.3).

Theorem 1.1 Let ρ > 0. Assume that the initial data ρ 0 and u 0 satisfy:

0 < m 0 ≤ ρ 0 ≤ M 0 < +∞, ρ 0 -ρ ∈ H 1 (R), v 0 ∈ H 1 (R) ∩ L ∞ (R).
(1.10) 2.5

Then there exists a global strong solution (ρ, v) of (

1.1 1.3) on R + × R such that for every T > 0: ρ -ρ ∈ L ∞ (0, T, H 1 (R)), ρ ∈ L ∞ (0, T, L ∞ (R)), v ∈ L ∞ (0, T, H 1 (R)) ∩ L 2 (0, T, H 2 (R)) and v ∈ L ∞ (0, T, L ∞ (R)).
Finally this solution is unique in the class of weak solutions satisfying the usual energy inequality.

theo Remark 1 We would like to point out that the problem remains open in the case of the Saint-Venant system, which corresponds to system ( 1.1

1.3) without capillarity.

In the following theorem, we are interested in proving the convergence of the global solutions of system ( 1.3) with initial data (ρ 0 , v 0 ) as in theorem ( theo 1.1).Then, when → 0, there exists a subsequence of (ρ , m ) that converge almost everywhere to a finite entropy solution (ρ, ρv) to the Cauchy problem ( [5]) Let ψ ∈ C 2 0 (R), (η ψ , q ψ ) be a weak entropy pair generated by ψ. Assume that the sequences (ρ (x, t), v (x, t)) defined on R × R + with m = ρ v , satisfies the following conditions:

1. For any -∞ < a < b < +∞ and all t > 0, it holds that: where C(t) > 0 is independent of .

For any compact set

K ⊂ R, it holds that t 0 K (ρ ) γ+θ + ρ |v | 3 dxdτ ≤ C(t, K), (1.12
) 1.9

where C(t, K) > 0 is independent of .

The sequence of entropy dissipation measures

η ψ (ρ , m ) t + q ψ (ρ , m ) x are compact in H -1 loc (R 2 + ). (1.13) 1.10
Then there is a subsequence of (ρ , m ) (still denoted (ρ , m )) and a pair of measurable functions (ρ, m) such that:

(ρ , m ) → (ρ, m), a.e as → 0. (1.14) 1.11 theo2
Remark 2 We would like to recall that the estimate ( 1.9

1.12) was first derived by Lions et al in

35

[26] by relying the moment lemma introduced by Perthame in Per [START_REF] Perthame | Higher moments lemma: application to Vlasov-Poisson and Fokker-Planck equations[END_REF].

The paper is arranged as follows. In section 2 Mathematical tools section2 Definition 2.1 A pair of functions (η(ρ, v), H(ρ, v)) or (η(ρ, m), q(ρ, m)) for m = ρv, is called an entropy-entropy flux pair of system ( 1.1 1.3), if the following holds:

[η(ρ, v)] t + [H(ρ, v)] x = 0,
for any smooth solution of ( Such η satisfy the wave equation:

∂ tt η = θ 2 ρ γ-3 ∂ xx η.
From

35

[26], we obtain an explicit representation of any weak entropy (η, q) under the following form:

η ψ (ρ, m) = R χ(ρ, s -v)ψ(s)ds, H ψ (ρ, m) = R (θs + (1 -θ)u)χ(ρ, s -v)ψ(s)ds,
(2.15) 1.4 where the kernel χ is defined as follows:

χ(ρ, v) = [ρ 2θ -v 2 ] λ + , λ = 3 -γ 2(γ -1) > - 1 2
, and θ = γ -1 2 , and here:

t λ + =t λ for t > 0, =0 for t ≤ 0, Proposition 2.1 (see 35 [26])
For instance, when ψ(s) = 1 2 s 2 , the entropy pair is the mechanical energy and the associated flux:

η * (ρ, m) = m 2 2ρ + e(ρ), q * (ρ, m) = m 3 2ρ 2 + e (ρ), (2.16) 1.5
where e(ρ) = κ γ-1 ρ γ represents the gas internal energy in physics.

In the sequel we will work far away of the vacuum that it why we shall introduce equilibrium states such that we avoid the vacuum. Let (ρ(x), v(x)) be a pair of smooth monotone functions satisfying (ρ(x), v(x)) = (ρ -,+ , v -,+ ) when -+ x ≥ L 0 for some large L 0 > 0. The total mechanical energy for (

1.1 1.
3) in R with respect to the pair of reference function (ρ(x), v(x)) is:

E[ρ, v](t) = R 1 2 ρ(t, x)|v(t, x) -v(x)| 2 + e * (ρ(t, x), ρ(x)) dx (2.17) 1.7
where e * (ρ, ρ) = e(ρ) e (ρ) -e (ρ(ρ -ρ) ≥ 0. The total mechanical energy for system ( 3system ??) with κ(ρ) = κ ρ is:

E 1 [ρ, u](t) = R 1 2 ρ(t, x)|u(t, x) -ū(x)| 2 + e * (ρ(t, x), ρ(x)) + 2 (∂ x ρ 1 2 ) 2 dx (2.18) 1.7
and the total mechanical energy for system (

1.1 1.3) is: E 2 [ρ, v](t) = R 1 2 ρ(t, x)|v(t, x) -v(x)| 2 + e * (ρ(t, x), ρ(x)) dx (2.19) 1.7
Definition 2.3 Let (ρ 0 , v 0 ) be given initial data with finite-energy with respect to the end states: (ρ ± , v ± ) at infinity, and

E[ρ 0 , v 0 ] ≤ E 0 < +∞. A
E[ρ, v](t) ≤ C(E 0 , t).
2. The entropy inequality:

η ψ (ρ, v) t + q ψ (ρ, v) x ≤ 0,
is satisfied in the sense of distributions for all test functions ψ(s) ∈ {±1, ±s, s 2 }.

3. The initial data (ρ 0 , v 0 ) are obtained in the sense of distributions.

We now give our main conditions on the initial data ( Definition 2.4 Let (ρ(x), v(x)) be some pair of smooth monotone functions satisfying (ρ(x), v(x)) = (ρ -,+ , v -,+ ) when -+ x ≥ L 0 for some large L 0 > 0. For positive constant C 0 , C 1 and C 2 independent of , we say that the initial data (ρ 0 , v 0 ) satisfy the condition H if they verify the following properties:

• ρ 0 > 0, R ρ 0 (x)|u 0 (x) -ū(x)| ≤ C 0 < +∞, • The energy is finite: R 1 2 ρ 0 (x)|v 0 (x) -v(x)| 2 + e * (ρ 0 (x), ρ(x)) dx ≤ C 1 < +∞, • 2 R |∂ x ρ 0 (x)| 2 ρ 0 (x) 3-2α dx ≤ C 2 < +∞.
In this section, we would like to recall some properties on the pair of entropy for the system (

∂ t η(ρ, u) + ∂ x H(ρ, u) = 0,
if and only if:

η ρρ = P (ρ) ρ 2 η uu . (2.

20) ondes

We supplement the equation ondes 2.20 by giving initial conditions:

η(0, u) = 0, η ρ (0, u) = ψ(u). (2.

21) initial

We are now going to give a sequel of proposition on the properties of η, we refer to

35

[26] for more details.

proputile Proposition 2.2 For ρ ≥ 0, u, ω ∈ R,

• The fundamental solution of ( ondes 2.20)-( initial 2.21) is the solution corresponding to η ρ (0, u) = δ(u) is given by:

χ(ρ, ω) = (ρ γ-1 -ω 2 ) λ + with λ = 3 -γ 2(γ -1)
.

(2.22)

• The solution of (

ondes 2.20)-( initial 2.21
) is given by:

η(ρ, u) = R ψ(ξ)χ(ρ, ξ -u)dξ, (2.23) entropie
• η is convex in (ρ, ρu) for all ρ, u if and only if g is convex.

• The entropy flux H associated with η is given by:

H(ρ, u) = R ψ(ξ)[θξ + (1 -θ)ξ]χ(ρ, ξ -u)dξ where θ = γ -1 2 . ( 2

.24) flux

We now give a important result on the entropy pair (see

35

[26], lemma 4) .

pair35 Proposition 2.3 Taking ψ(s) = 1 2 s|s|, then there exists a positive constant C > 0, depending only on γ > 1, such that the entropy pair (η ψ , H ψ ) satisfies:

|η ψ (ρ, u)| ≤ (ρ|u| 2 + ρ γ ), H ψ (ρ, u) ≥ C -1 (ρ|u| 3 + ρ γ+θ ), for all ρ ≥ 0 and u ∈ R, |η ψ m (ρ, u)| ≤ (ρ|u| + ρ θ ), |η ψ mm (ρ, u)| ≤ Cρ -1 .
(2.25) 2.37

We are now going to give recent results on the entropy pair (η ψ , q ψ ) generated by ψ ∈ C 2 0 (R) (we refer to 10

[5] for more details).

propChen Proposition 2.4 For a C 2 function ψ : R → R, compactly supported on the interval [a, b], we have:

supp(η ψ ), supp(q ψ ) ⊂ {(ρ, m) = (ρ, ρu) : u + ρ θ ≥ a, u -ρ θ ≤ b} : (2.26) 3.2
Furthermore, there exists a constant C ψ such that, for any ρ ≥ 0 and u ∈ R, we have:

• For γ ∈ (1, 3], |η ψ (ρ, m)| + |q ψ (ρ, m)| ≤ C ψ ρ.
(2.27) 3.3

• For γ ∈ (3, +∞),

|η ψ (ρ, m)| ≤ C ψ ρ, |q ψ (ρ, m)| ≤ C ψ (ρ + ρ θ+1 ).
(2.28) 3.4

• If η ψ is considered as a function of (ρ, m), m = ρu then

|η ψ m (ρ, m)| + |ρη ψ mm (ρ, m)| ≤ C ψ ,
(2.29) 3.5 and, if η ψ m is considered as a function of (ρ, u), then

|η ψ m (m, u)| + |ρ 1-θ η ψ mρ (ρ, ρu)| ≤ C ψ .
(2.30) 3.6

We now would like to express the kinetic formulation of (

1.3 1.5) introduced in ( 35 [26]). Theorem 2.4 Let (ρ, ρv) ∈ L ∞ (R + , L 1 (R))
have finite energy and ρ ≥ 0, then it is an entropy solution of ( 1.3

1.5) if and only if there exists a non-positive bounded measure m on R + × R 2 such that the function χ(ρ, ξ -u) satisfies:

∂ t χ + ∂ x [(θξ + (1 -θ)u)χ] = ∂ ξξ m(t, x, ξ).
(2.31) We would like to start with recalling an important result due to Solonnikov (see

Sol

[34]). Let ρ 0 the initial density such that:

0 < m 0 ≤ ρ 0 ≤ M 0 < +∞. ( 3

.32) initiald

When the viscosity coefficient µ(ρ) satisfies:

µ(ρ) ≥ c > 0 for allρ ≥ 0, (3.33) visco
we have the existence of strong solution for small time. More exactly, we have: [START_REF] Rowlinson | Translation of J.D van der Waals. The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density[END_REF] and assume that µ satisfies ( visco 3.33), then there exists T 0 > 0 depending on m 0 , M 0 , ρ 0 -ρ H 1 and v 0 H 1 such that (

Proposition 3.5 Let (ρ 0 , v 0 ) satisfy ( initiald 3 .
1.1 1.3) has a unique solution (ρ, v) on (0, T 0 ) satisfying: ρ -ρ ∈ L ∞ (H 1 (R), ∂ t ρ ∈ L 2 ((0, T 1 ) × R), v ∈ L 2 (0, T 1 , H 2 (R)), ∂ t v ∈ L 2 ((0, T 1 ) × R) for all T 1 < T 0 .
Remark 3 The main point in this theorem is that the time of existence T 0 depends only of the norms of ρ 0 which gives us a low bounds on T 0 of the system ( 1.1

1.3).

In view of this proposition, we see that if we introduce a truncated viscosity coefficient µ n (ρ):

µ n (ρ) = max(ρ, 1 n ),
then there exists approximated solutions (ρ n , v n ) defined for small time (0, T 0 ) of the system (

1.1

1.3). In order to prove theorem theo 1.1 , we only have to show that (ρ n , v n ) satisfies the following bounds uniformly with respect to n and T large:

0 < m 0 ≤ ρ n ≤ M 0 < +∞, ∀t ∈ [0, T ], ρ n -ρ ∈ L ∞ T (H 1 (R)), v n ∈ L ∞ T (H 1 (R)).
(3.34)

We are going to follow the method of Lions et al in

36

[25], indeed the main point is to prove that we can extend the notion of Riemann invariant or more precisely the kinetic formulation of proposition cinetique 2.4 to the system ( 1.1 1.3). We recall that system (

     ∂ ∂t ρ n + ∂ x (ρ n v n ) -∂ xx ρ n = 0, ∂ ∂t (ρ n v n ) + ∂ x (ρ n v n v n ) -∂ x (∂ x ρ n v n ) -∂ x (ρ n ∂ x v n ) + ∂ x (a(ρ n ) γ = 0, (3.35) 1.1a
and we have finally:

     ∂ ∂t ρ n + ∂ x (ρ n v n ) -∂ xx ρ n = 0, ∂ ∂t (ρ n v n ) + ∂ x (ρ n v n v n ) -∂ x ∂ x (ρ n v n ) + ∂ x (a(ρ n ) γ = 0, (3.36) 1.1b
Following 36

[25] and setting m n = ρ n v n we have for any pair of entropy flux (η(ρ, u), H(ρ, u)) defined by ( entropie 2.23) and ( flux 2.24) where η is a convex function of (ρ n , m n ). We write η = η(ρ n , m n ):

∂ t η + ∂ x H = ηρ ∂ xx ρ n + ηm ∂ xx m n , = ∂ xx η -(η ρρ (∂ x ρ n ) 2 + 2η ρm (∂ x ρ n )(∂ x m n ) + ηmm (∂ x m n ) 2 ).
Here we define µ n such that:

µ n = ηρρ (∂ x ρ n ) 2 + 2η ρm (∂ x ρ n )(∂ x m n ) + ηmm (∂ x m n ) 2
By proposition proputile 2.2, we can check that µ n ≥ 0. We obtain then that:

∂ t η(ρ n , v n ) + ∂ x H((ρ n , v n ) -ηρ ∂ xx ρ n ≤ 0 in R × (0, +∞).
By applying the same method than for proving the theorem cinetique 2.4, we obtain the following kinetic formulation:

∂ t χ + ∂ x ([θξ + (1 -θ)v n ]χ) -∂ xx χ = ∂ ξξ mn on R 2 × (0, +∞), (3.37) riemann
where mn is a nonpositive bounded measure on R 2 × (0, +∞). Finally we recover the classical maximum principle by multiplying ( riemann 3.37) by the convex functions g(ξ) = (ξξ 0 ) + and g(ξ) = (ξ -ξ 0 ) -and integrating over R 2 × (0, +∞). Indeed as we have that:

-C ≤ min x (v 0 -ρ θ 0 ) ≤ max x (v 0 + ρ θ 0 ) ≤ C,
and that: suppξ = [v -ρ θ , v + ρ[θ].
For ξ 0 large enough, we can show that:

suppξ 0 ∩ suppχ = ∅.
We have obtain then that:

-C ≤ min x (v 0 -ρ θ 0 ) ≤ v n -ρ θ n ≤ v n + ρ θ n ≤ max x (v 0 + ρ θ 0 ) ≤ C.
In particular we obtained that ρ n and v n are uniformly bounded in

L ∞ (0, T n , L ∞ (R)) or: sup x∈R,t∈(0,Tn) |ρ n (t, x)| + |v n (t, x)|) ≤ C 0 , (3.38) imp2 4 Proof of theorem theo1 1.2 4.
1 Uniform estimates for the solutions of (

First we assume that (ρ , v ) is the global solutions of Korteweg's equations (

(ρ , v )(x, t) = (ρ ± , u ± ). (4.40) 2.2
Here we are working around a non constant state (ρ, v) with:

lim x→±∞ (ρ, v)(x, t) = (ρ ± , u ± ).
It is a simple extension of theorem For simplicity, throughout this section, we denote (ρ, v) = (ρ , v ) and C > 0 denote the constant independent of . We start with recalling the inequality energy for system ( 1.1 1.3), indeed by the introduction of the effective velocity we obtain new entropies (see Hprepa [START_REF] Haspot | Blow-up criterion, ill-posedness and existence of strong solution for Korteweg system with infinite energy[END_REF]).

Lemma 1 Suppose that E 1 [ρ 0 , u 0 ] ≤ E 0 < +∞ for some E 0 > 0 independent of . It holds that:

sup 0≤τ ≤t E 1 [ρ, u](τ ) + t 0 R ρu 2 x dxdτ ≤ C(t), (4.41) 2.3a
and:

sup 0≤τ ≤t E 2 [ρ, v](τ ) + t 0 R ρv 2 x dxdτ + t 0 R ρ γ-2 ρ 2 x dxdτ ≤ C(t), (4.42) 2.3
where C(t) depends on E 0 , t, ρ, and ū but not on . Proof: It suffices to writes the energy inequalities for system ( 1.1

1.3) and from system ( 1.3 1.5). More exactly we have:

d dt R (η(ρ, m) -η(ρ, ρū)dx + R ρu 2 x dx = q(ρ -, m -) -q(ρ + , m + ),
with the entropy pair:

η(ρ, m) = m 2 2ρ + e(ρ), q(ρ, m) = m 3 2ρ 2 + me (ρ),
with e(ρ) = a γ-1 ρ γ . Since we have:

(ρ, ρ) ≥ ρ(ρ θ -ρθ ) 2 , θ = γ -1 2 ,
we can classically bootstrap on the left hand-side the term q(ρ -, m -) -q(ρ + , m + ).

Remark 4 Since vacuum could occur in our solution, the inequality 2 will be more tricky to obtain.

t 0 R ρu 2 x dxdτ ≤ C(t), in ( 2.
The following higher order integrability estimate is crucial in compactness argument. where C(t) > 0 depends on E 0 , a, b, γ, t, ρ, ū but not on .

Remark 5 The proof follows the same ideas than in the case of compressible Navier-Stokes equations when we wish to obtain a gain of integrability on the density. We refer to fL2 [START_REF] Lions | Mathematical Topics in Fluid Mechanics[END_REF] for more details. The proof is also inspired from Huang et al in Hu2 [START_REF] Huang | Vanishing viscosity limit for isentropic Navier-Stokes equations with density-dependent viscosity[END_REF].

Proof. Choose ω ∈ C ∞ 0 (R) such that: 0 ≤ ω(x) ≤ 1, ω(x) = 1 for x ∈ [a, b],
and suppω = (a -1, b + 1).

By the momentum equation of (

ρP (ρ)ω 2 = -ρ 2 uvω 2 + ρ 2 v x ω 2 -(ρω x -∞ ρv ωdy) t -(ρu) x ω( x -∞ ρu ωdy) + ρω x -∞ [(ρuv + P (ρ))ω x -ρv x ω x ]dx, = ρ 2 v x ω 2 -(ρω x -∞ ρv ωdy) t -(ρuω x -∞ ρv ωdy) x + ρuω x x -∞ ρv ωdy + ρω x -∞ [(ρuv + P (ρ))ω x -ρv x ω x ]dx, (4.46) 2.24
We now integrate ( 2.24

4.46) over (0, t) × R and we get:

t 0 R aρ γ+1 ω 2 dxdτ = t 0 R ρ 2 v x ω 2 - R (ρω x -∞ ρv ωdy)dx + R (ρ 0 ω x -∞ ρ 0 v 0 ωdy)dx + t 0 R ρuω x x -∞
ρv ωdy dxdτ 4.48), for any (t, x) there exists a point x 0 = x 0 (t, x) such that |x -x 0 | ≤ d(t) and ρ(t, x 0 ) = ρ. Here we choose β = γ+1 2 > 0, 4.50), Young inequalities and Hölder's inequalities, the first term on the right hand side of ( 2.25 4.47) is treated as follows:

+ t 0 R ρω x -∞ [(ρuv + P (ρ))ω x -ρv x ω x ]dx dxdτ. ( 4
supp x∈supp(ω) ρ β (t, x) ≤ ρ β + supp x∈supp(ω)∩A ρ β (t, x), ≤ 2 ρ β + supp x∈supp(ω)∩A | ρ β (t, x) -ρ β (t, x 0 )|, ≤ 2 ρ β + supp x∈supp(ω)∩A x 0 +d(t) x 0 -d(t) |β|| ρ β-1 (t, x)ρ x |dx, ≤ 2 ρ β + b+1+2d(t) a-1-2d(t) |β|ρ 2β-1 dx + R 2 ρ -1 ρ 2 x dx, ≤ C(t) + b+1+2d(t) a-1-2d(t) ρ γ dx, ≤ C(t).
t 0 R ρ 2 v x ω 2 dxdτ ≤ 1 2 t 0 R ρ 3 ω 4 dxdτ + 1 2 t 0 R ρv 2 x dxdτ, ≤ C(t) + t 0 R ρ 3 ω 2 dxdτ, ≤ C(t) + C(t) t 0 R ρ 4-β ω 2 dxdτ, ≤ C(t) + δ t 0 R ρ γ+1 ω 2 dxdτ, (4.51) 2.29
Here we have used the fact that γ > 5 3 . By lemma

ρv 2 dy) Similarly, we have: Lemma 3 Suppose that (ρ 0 (x), v 0 (x) satisfy the conditions in the lemmas lemma1 1. Furthermore there exists M 0 > 0 independent of , such that as a function depending of (ρ, v), we have for all ρ ≥ 0 and v ∈ R:

t 0 R ρω x -∞ (ρuv + P (ρ))ω x dy dxdτ ≤ C(t), (4.54) 2.33 and t 0 R ρω x -∞ ρv x ω x dy dxdτ ≤ t 0 R ρω R ρ|v x | |ω x |dy dxdτ , ≤ t 0 R ρωdx R ρv 2 x dy + R ρω 2 x dy dτ , ≤ C(t). ( 4
R ρ 0 (x)|v 0 (x) -v(x)|dx ≤ M 0 < +∞, ( 4
|η ψ mv (ρ, v)| ≤ C, |η ψ mρ (ρ, v)| ≤ Cρ θ-1 . (4.58) 2.38
For this weak entropy pair (η ψ , H ψ ), we observe that:

η ψ (ρ, 0) = η ψ ρ (ρ, 0) = 0, H ψ (ρ, 0) = θ 2 ρ 3θ+1 R |s| 3 [1 -s 2 ] λ + ,
and:

η ψ m (ρ, 0) = βρ θ with β = R |s|[1 -s 2 ] λ + ds.
By Taylor formula, we have: for some constant C > 0. Now we introduce a new entropy pair ( η, H) such that,

η * (ρ, m) = βρ θ m + r(ρ, m), ( 4
η(ρ, m) = η ψ (ρ, m -ρv -), H(ρ, m) = H ψ (ρ, m -ρv -) + v -η ψ (ρ, m -ρv -),
with m = ρv which satisfies: 4.58), we obtain: We have now: Now we are going to deal with the last term on the right hand side of ( 4.2 H -1 loc (R 2 + ) Compactness

η(ρ, m) = βρ θ+1 (v -v -) + r(ρ, ρ(v -v -)), r(ρ, ρ(v -v -)) ≤ Cρ(v -v -) 2 . (4.61) 2.41 Integrating ( 1.1 1.3) 1 × η ρ + ( 1.1 1.3) 2 × η m over (0, t) × (-∞, x), we have: x -∞ η(ρ, m) -η(ρ 0 , m 0 ) dy + t 0 q * (ρ, ρ(v -v -)) + v -η * (ρ, ρ(v -v -))dτ = tq * (ρ -, 0) + t 0 η m ρv x dτ - t 0 x -∞ ( η mu ρv 2 x + η mρ ρρ x v x )dydτ. ( 4
t 0 x -∞ η mu ρv 2 x dydτ ≤ C t 0 R ρv 2 x dy dτ ≤ C(t), (4.63) 2.43 t 0 x -∞ η mρ ρρ x v x dydτ ≤ C t 0 R ρ θ-1 ρ|ρ x v x |dy dτ ≤ C(t), ≤ C t 0 R ρv 2 x dy dτ + C t 0 R ρ γ-2 ρ 2 x dy dτ ≤ C(t). ( 4
In this section we are going to take profit of the uniform estimates obtained in the previous section in order to prove the following lemma, which gives the H -1 loc (R 2 + )-compactness of the Korteweg solution sequence (ρ , v ) on a entropy-entropy flux pair. Let K ⊂ R be compact, using proposition This shows that:

-η ψ mu (ρ , m )ρ (v x ) 2 -η ψ mu (ρ , m )ρ v x ρ x are bounded in L 1 ([0, T ] × K), (4.75) 3.9

and thus it is compact in W -1,p 1 loc (R 2 + ), for 1 < p 1 < 2. Moreover we observe that

|η ψ mu (ρ , ρ v )| ≤ C ψ ,

1 . 1 1. 3 )

 113 when the fluid is away from vacuum. Namely, we shall study ( 1.1

  for γ = 1 + 2 2d+1 and γ = 2k + 3 2k + 1 (with k ≥ 1), d ≥ 2 by using the so-called "compensated compactness" introduced by Tartar in Ta [35]. This result was extended by Chen in Chen [4] in the case γ ∈ (1, 5 3 ] and by Lions et al in 35 [26] in the case γ ∈ [3, ∞).

1 . 1 1. 3 )Theorem 1 . 2

 11312 to entropic solutions of the Euler system ( Let γ > 5 3 and (ρ , v ) with m = ρ v be the global solution of the Cauchy problem (

1 . 3 1Remark 1

 131 .5) with initial data (ρ 0 , ρ 0 v 0 ).theo1 We would like to point out that Lions et al in36[25] had obtained the existence of global entropic solution for γ > 1 by a viscosity vanishing process, and the considered regularizing system was exactly the Korteweg system modulo the introduction of the effective velocity.One important basis of our problem for theoremtheo1 1.2 is the following compactness theorem established in 10 [5]. Theorem 1.3 (Chen-Perepelitsa

10

 10

  γ+1 dxdτ ≤ C(t, a, b), (1.11) 1.8

section2 2 we

 2 recall some important results on the notion of entropy enrtopy-flux pair for Euler system and on the kinetic formulation of Lions et al in 35 [26]. In section section3 3, we show theorem theo 1.1 and in the last section

section4 4 .

 4 1 we prove theorem theo1 1.2.

1 . 3 1

 13 .5). Furthermore (η(ρ, v) is called a weak entropy if: η(0, u) = 0, for any fixed v. Definition 2.2 An entropy η(ρ, m) is convex if the Hessian ∇ 2 η(ρ, m) is nonnegative definite in the region under consideration.

  satisfy the conservation laws:

  satisfying: ρ (t, x) ≥ c (t), for some c (t) > 0, (4.39) 2.1 and lim x→±∞

theo 1 . 1 .

 11 Our goal is now to check the properties (

  lemma1

3

 3

4 .

 4 42) is much weaker than the corresponding one in 10 [5]. That is why lemma lemme2

lemme2 Lemma 2 ρ

 2 If the conditions of lemma lemma1 1 hold, then for any -∞ < a < b < +∞ and all t > 0, it holds that: γ+1 dxdτ ≤ C(t, a, b), (4.43) 2.21

1 2 ≤

 2 C(t).

( 4 .

 4 53) 2.32

( 4 .

 4 68)2.48

0 (ρv 2 +

 02 v 0 -ρv -)dy -t P (ρ) -ρuu --P (ρ -

  ρ(y, τ ), (ρv)(y, τ ))dy≤ x -∞ ( η(ρρv) -βρ θ+1 (v -v))dy + | θ -(ρ -) θ )ρ(v -v))dy + β(ρ -) θ x -∞ ρ(v -v))dy , ≤ C(t) + β(ρ -) θ x -∞ ρ(v -v))dy .

  ) we conclude the proof of the lemma.

Lemma 4 1 1. 3 ,1. 3 )

 4133 Let ψ ∈ C 2 0 (R), η ψ , H ψ ) be a weak entropy pair generated by ψ. Then for the solutions (ρ , v ) with m = ρ v of Korteweg system ( 1.the following sequence:η ψ (ρ , m ) t + q ψ (ρ , m ) x are compact in H -1loc (Now we are going to prove the lemma. A direct computation on ( 2 × η ψ m (ρ , m ) gives:η ψ ρ (ρ , m ) t + H ψ ρ (ρ , m ) x = η ψ ρ (ρ , m )(ρ )v x -η ψ mu (ρ , m )(ρ )(v x ) 2 -η ψ mu (ρ , m )(ρ )v x ρ x .(4.73) 3.7

C t 0 K

 0 (ρ , m )(ρ )|(v x ) 2 + |η ψ mu (ρ , m )(ρ )v x ρ x |dxdt ≤ (ρ )|(v x ) 2 dxdτ + C t 0 K (ρ ) γ-2 (ρ x ) 2 dxdτ ≤ C(t).

( 4 .

 4 74) 3.8

  pair of measurable functions (ρ, u) : R 2 + → R 2 + is called a finite-energy entropy solution of the Cauchy problem ( The total energy is bounded in time such that there exists a bounded function C(E, t), defined on R + × R + and continuous in t for each E ∈ R + with for a.e t > 0:

	1.3 1.5)
	if the following properties hold:
	1.

  P (ρ))ω x -ρv x ω x . (4.45) 2.23

	Integrating (	2.22 4.44) with respect to spatial variable over (-∞, x), we obtain:
		x	x
	P (ρ)ω = -ρuvω + (ρv x ω) x -( [(ρuv + Multiplying ( -∞ ρv ωdy) t + -∞ 2.23 4.45) by ρω, we have
		1.1 1.3) and by localizing, we have
	(P (ρ)ω) (4.44) 2.22

x = -(ρuvω) x + (P (ρ) + ρuv)ω x -(ρv) t ω + (ρv x ω) x -ρv x ω x .

  .47) 2.25

	Let
		A = {x : ρ(t, x) ≥ ρ}, where ρ = 2 max(ρ+, ρ-),	(4.48) 2.26
	then we have the following estimates by (	2.3 4.42):
		|A| ≤	C(t) e * (2ρ, ρ)	= d(t).	(4.49) 2.27
	By (	2.26

  .55)2.34

	Substituting (	2.29 4.51), (	2.32 4.53)-( 4.55) into ( 2.34 4.47) and noticing the smallness of δ, we proved 2.25
	lemma

  .56) 2.35 then for any compact set K ⊂ R, it holds that:

	t
		(ρ γ+θ + ρ|v| 3 )dxdτ ≤ C(t, K),	(4.57) 2.36
	0	K
	where C(t, K) is independent of .
	Remark 6 In order to prove the inequality ( in 35 [26] where this inequality was obtained for the first time. 2.36 4.57), we will use the same ingredients than
	Proof. We are now working with the function ψ of proposition	pair35 2.3. If we consider η ψ m

  .62) 2.42

	By using (	2.38

  .64) 2.44 (ρ, ρ(v -v -)|dxdτ + C

	Substituting ( 4.63) and ( 2.43	2.44 4.64) into (	2.42 4.62), then integrating over K and using (	2.37 2.25), we
	obtain:
	t
		ρ θ+γ + ρ|v -v -| 3 dxdτ
	0	K
	≤ C(t) + C	t	|η t	ρ|v||v x |dxdτ	(4.65) 2.45
				0	K	0	K
		t				x
	+ C			ρ 1+θ |v x |dxdτ + 2 sup	(	eta(ρ(y, τ ), (ρv)(y, τ ))dy)dx .
		0	K			τ ∈[0,t]	K	-∞
	Applying lemma	lemma1 1, we have:
						t
						|η (4.66) 2.46
						0	K
	By Hölder's inequality and (	2.28 4.50), we get:
						t	t
						ρv 2 x dxdτ + C	ρ 1+2θ dxdτ,
						0	K	0	K
						≤ C(t) + C(t)	t	ρ θ dxdτ,	(4.67) 2.47
						0	K
						≤ C(t).

* * (ρ, ρ(v -v -)|dxdτ ≤ C(t). t 0 K ρ 1+θ |v x |dxdτ ≤ C

  ρv -) t + (ρv 2 + P (ρ) -ρuu -) x = (ρv x ) x . (4.69) 2.49

	2.45 4.65). (	1.1 1.3) implies
	that:
	(ρv -Integrating ( 2.49

4.69) over [0, t] × (-∞, x) for x ∈ K, we get:

, then we obtain: 4.75), we obtain where p 3 = γ + 1 > 2 when γ ∈ (1, 3], and p 3 = γ+θ 1+θ > 2 when γ > 3. By interpolation we conclude the proof of the lemma

References fA