Mésognon-Gireau Benoît
email: benoit.mesognon-gireau@ens.fr

The singular limit of the Water-Waves equations in the rigid lid regime

The re-scaled Water-Waves equations depend strongly on the ratio ε between the amplitude of the wave and the depth of the water. We investigate in this paper the convergence as ε goes to zero of the free surface Euler equations to the so called rigid lid model. We first prove that the only solutions of this model are zero. Due to the conservation of the Hamiltonian, the solutions of the free surface Euler equations converge weakly to zero, but not strongly in the general case, as ε goes to zero. We then study this default of convergence. More precisely, we show a strong convergence result of the solutions of the water waves equations in the Zakharov-Craig-Sulem formulation to the solutions of the linear water-waves equations. It is then easy to observe these latter converge weakly to zero. The simple structure of this system also allows us to explain the mechanisms of the weak convergence to zero. Finally, we show that this convergence to the rigid lid model also holds for the solutions of the Euler equations. To this end we give a new proof of the equivalence of the free surface Euler equations and of the Zakharov-Craig-Sulem equation by building an extension of the velocity and pressure fields.

Introduction

We recall here classical formulations of the Water Waves problem. We then shortly introduce the meaningful dimensionless parameters of this problem, and introduce the so called rigid lid equations. We then present the main result of this paper.

Formulations of the Water Waves problem

The Water Waves problem puts the motion of a fluid with a free surface into equations. We recall here two equivalent formulations of the Water Waves equations for an incompressible and irrotationnal fluid.

Free surface d-dimensional Euler equations

The motion, for an incompressible, inviscid and irrotationnal fluid occupying a domain Ω t delimited below by a flat bottom and above by a free surface is described by the following quantities:

-the velocity of the fluid U " pV, wq, where V and w are respectively the horizontal and vertical components;

-the free top surface profile ζ;

-the pressure P.

All these functions depend on the time and space variables t and pX, zq P Ω t . The domain of the fluid at the time t is given by Ω t " tpX, zq P R d`1 , ´H0 ă z ă ζpt, Xqu, where H 0 is the typical depth of the water. The unknowns pU, ζ, P q are governed by the Euler equations: $ ' & ' % B t U `U ¨∇X,z U " ´1 ρ ∇P ´ge z in Ω t divpU q " 0 in Ω t curlpU q " 0 in Ω t .

(1.1)

We denote here ´ge z the acceleration of gravity, where e z is the unit vector in the vertical direction, and ρ the density of the fluid. Here, ∇ X,z denotes the d `1 dimensional gradient with respect to both variables X and z.

These equations are completed by boundary conditions:

$ ' & ' %
B t ζ `V ¨∇ζ ´w " 0 U ¨n " 0 on tz " ´H0 u P " P atm on tz " ζpt, Xqu.

(1.2)

In these equations, V and w are the horizontal and vertical components of the velocity evaluated at the surface. The vector n in the last equation stands for the normal upward vector at the bottom pX, z " ´H0 q. We denote P atm the constant pressure of the atmosphere at the surface of the fluid. The first equation of (1.2) states the assumption that the fluid particles do not cross the surface, while the last equation of (1.2) states the assumption that they do not cross the bottom. The equations (1.1) with boundary conditions (1.2) are commonly referred to as the free surface Euler equations.

Craig-Sulem-Zakharov formulation

Since the fluid is by hypothesis irrotational, it derives from a scalar potential:

U " ∇ X,z Φ.

Zakharov remarked in [START_REF] Vladimir | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF] that the free surface profile ζ and the potential at the surface ψ " Φ |z"ζ fully determine the motion of the fluid, and gave an Hamiltonian formulation of the problem. Later, Craig-Sulem, and Sulem ( [START_REF] Craig | Numerical simulation of gravity waves[END_REF] and [START_REF] Craig | Nonlinear modulation of gravity waves: a rigorous approach[END_REF]) gave a formulation of the Water Waves equation involving the Dirichlet-Neumann operator. The following Hamiltonian system is equivalent (see [START_REF] Lannes | The water waves problem[END_REF] and [START_REF] Alazard | The water-wave equations: from Zakharov to Euler[END_REF] for more details) to the free surface Euler equations (1.1) and (1.2):

$ & % B t ζ ´Gψ " 0 B t ψ `gζ `1 2 |∇ψ| 2 ´pGψ `∇ζ ¨∇ψq 2 2p1`| ∇ζ | 2 q " 0, (1.3)
where the unknowns are ζ (free surface profile) and ψ (velocity potential at the surface) with t as time variable and X P R d as space variable. The fixed bottom profile is b, and G stands for the Dirichlet-Neumann operator, that is

Gψ " Grζsψ " a 1 `|∇ζ| 2 B n Φ |z"ζ ,
where Φ stands for the potential, and solves Laplace equation with Neumann (at the bottom) and Dirichlet (at the surface) boundary conditions: # ∆ X,z Φ " 0 in tpX, zq P R d ˆR, ´H0 `bpXq ă z ă ζpXqu φ |z"ζ " ψ, B n Φ |z"´H0 " 0, (1.4) with the notation, for the normal derivative

B n Φ |z"´H0 " ∇ X,z ΦpX, ´H0 q ¨n

where n stands for the normal upward vector at the bottom pX, ´H0 q. See also [START_REF] Lannes | The water waves problem[END_REF] for more details.

Dimensionless equations

Since the properties of the solutions depend strongly on the characteristics of the fluid, it is more convenient to non-dimensionalize the equations by introducing some characteristic lengths of the wave motion:

(1) The characteristic water depth H 0 ;

(2) The characteristic horizontal scale L x in the longitudinal direction;

(3) The characteristic horizontal scale L y in the transverse direction (when d " 2);

(4) The size of the free surface amplitude a surf ;

Let us then introduce the dimensionless variables:

x 1 " x L x , y 1 " y L y , ζ 1 " ζ a surf , z 1 " z H 0 ,
and the dimensionless variables:

t 1 " t t 0 , Φ 1 " Φ Φ 0 ,
where

t 0 " L x ? gH 0 , Φ 0 " a surf H 0 L x a gH 0 .
After rescaling, several dimensionless parameters appear in the equation. They are

a surf H 0 " ε, H 2 0 L 2 x " µ, L x L y " γ,
where ε, µ, γ are commonly referred to respectively as "nonlinearity", "shallowness" and "transversality" parameters.

For instance, the Zakharov-Craig-Sulem system (1.3) becomes (see [START_REF] Lannes | The water waves problem[END_REF] for more details) in dimensionless variables (we omit the "primes" for the sake of clarity):

$ ' & ' % B t ζ ´1 µ G µ,γ rεζsψ " 0 B t ψ `ζ `ε 2 |∇ γ ψ| 2 ´ε µ pG µ,γ rεζsψ `εµ∇ γ ζ ¨∇γ ψq 2 2p1 `ε2 µ | ∇ γ ζ | 2 q " 0, (1.5)
where G µ,γ rεζsψ stands for the dimensionless Dirichlet-Neumann operator, G µ,γ rεζsψ " a 1 `ε2 |∇ γ ζ| 2 B n Φ |z"εζ " pB z Φ ´µ∇ γ pεζq ¨∇γ Φq |z"εζ , where Φ solves the Laplace equation with Neumann (at the bottom) and Dirichlet (at the surface) boundary conditions # ∆ µ,γ Φ " 0 in tpX, zq P R d ˆR ´1 ă z ă εζpXqu φ |z"εζ " ψ, B n Φ |z"´1 " 0.

(1.6)

We used the following notations:

∇ γ " t pB x , γB y q if d " 2 and

∇ γ " B x if d " 1 ∆ µ,γ " µB 2 x `γ2 µB 2 y `B2 z if d " 2 and ∆ µ,γ " µB 2 x `B2 z if d " 1
and B n Φ |z"´1 " pB z Φ ´µ∇ γ pβbq ¨∇γ Φq |z"´1 .

The rigid lid model

The Euler system (1.1),(1.2), can be written in the dimensionless variables: $ ' ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' ' % B t V `εpV ¨∇γ `1 µ wB z qV " ´∇γ P in Ω t B t w `εpV ¨∇γ `1 µ wB z qw " ´pB z P q in Ω t B t ζ `εV ¨∇γ ζ ´1 µ w " 0 ∇ µ,γ ¨U " 0 in Ω t curl µ,γ pU q " 0 in Ω t U ¨n " 0 for z " ´1 P " εζ for z " εζ.

(1.7)

We changed the pressure into hydrodynamic pressure P " P ´pz ´εζq and set P atm " 0. The rigid lid model models the motion of the fluid as if the top surface was fixed at z " 0 (see for instance [START_REF] Camassa | Long-time shallow-water equations with a varying bottom[END_REF]). But to derive this model, we do not brutally take ε " 0 in the Euler system (1.7) but rather start by a change of time scale, and a change of scale for the pressure:

t 1 " εt and P 1 " P ε .
Formally, if one takes the limit ε goes to zero in the newly scaled Euler equations, one finds the following so called rigid lid equations (we omit the "primes" for the sake of clarity): $ ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' %

B t V `pV ¨∇γ `1 µ wB z qV " ´∇γ P in Ω B t w `pV ¨∇γ `1 µ wB z qw " ´pB z P q in Ω w " 0 ∇ µ,γ ¨U " 0 in Ω curl µ,γ pU q " 0 in Ω U ¨n " 0 for z " ´1 P " ζ for z " 0 (1.8)
where Ω is now the fixed domain Ω " tpX, zq P R d`1 , ´1 ď z ď 0u.

Reminder on the local existence for the Water-Waves equations with flat bottom

We briefly give some reminders about the local well-posedness theory for the Water-Waves equations and their local existence (see [START_REF] Lannes | The water waves problem[END_REF] Chapter 4 for a complete study, and also [START_REF] Iguchi | A shallow water approximation for water waves[END_REF]). After scaling t 1 " εt as for the rigid lid equation, one gets from (1.5) the equations:

$ ' & ' % εB t ζ ´1 µ Grεζsψ " 0 εB t ψ `ζ `ε 2 |∇ γ ψ| 2 ´ε µ pGrεζsψ `εµ∇ γ ζ ¨∇γ ψq 2 2p1 `ε2 µ|∇ γ ζ| 2 " 0.
(1.9)

As explained above, we prove later that the solutions of the full Water-Waves equations (1.9) converge strongly to the solutions of the linearized equations (2.23). The strategy is to treat the non-linear terms of the Water-Waves equations as a perturbation of the linearized equation.

Let t 0 ą d{2 and N ě t 0 `t0 _ 2 `3{2 (where a _ b " suppa, bq). The energy for the Water-Waves equations is the following (see Section 1.5 for the notations):

E N pU q " |Pψ| H t 0 `3{2 `ÿ |α|ďN |ζ pαq | 2 `|ψ pαq | 2
where ζ pαq , ψ pαq are the so called Alinhac's good unknowns:

@α P N d , ζ pαq " B α ζ, ψ pαq " B α ψ ´εwB α ζ,
and where P is the differential operator defined by

P " |D γ | p1 `?µ|D γ |q 1{2 .
The operator P is of order 1{2 and acts as the square roots of the Dirichlet-Neumann operator, since there exists M 0 " Cp 1 hmin , |ζ| H t 0 `1 q where C is a non decreasing function of its arguments, such that

1 M 0 |P| 2 2 ď pψ, 1 µ Gψq 2 ď M 0 |Pψ| 2 2 .
(1.10)

We consider solutions U " pζ, ψq of the Water-Waves equations in the following space:

E N T " tU P Cpr0, T s ; H t0`2 ˆ. H 2 pR d qq, E N pU p.qq P L 8 pr0, T squ.
The following quantity, called the Rayleigh-Taylor coefficient plays an important role in the Water-Waves problem:

apζ, ψq " 1 `εpεB t `εV ¨∇γ qw " ´ε P 0 ρag pB z P q |z"εζ .

We can now state the local existence result by Alvarez-Samaniego Lannes (see [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF] and [START_REF] Lannes | The water waves problem[END_REF] Chapter 3):

Theorem 1.1 Let t 0 ą d{2,N ě t 0 `t0 _ 2 `3{2. Let U 0 " pζ 0 , ψ 0 q P E N 0 . Let ε, γ be such that 0 ď ε, γ ď 1,
and moreover assume that:

Dh min ą 0, Da 0 ą 0, 1 `εζ 0 ě h min and apU 0 q ě a 0 .

Then, there exists T ą 0 and a unique solution U ε P E N T to (1.9) with initial data U 0 . Moreover, 1 T " C 1 , and sup tPr0;T s

E N pU ε ptqq " C 2 with C i " CpE N pU 0 q, 1 h min , 1 a 0 q for i " 1, 2.
Remark 1.2 Note that the time existence provided by Theorem 1.1 does not depend on ε. Actually, the full Theorem in non flat bottom (see [START_REF] Lannes | The water waves problem[END_REF]) states that the time of existence for the equations in the original scaling (1.5) is of size 1 ε_β where β is the size of the topography. In our case, β " 0 and therefore one gets a time of existence of size 1 ε for (1.5), and of size 1 for the rescaled equations (1.9). In the case of a non flat bottom, this long time result stands true in presence of surface tension (see [START_REF] Mésognon-Gireau | The Cauchy problem on large time for the Water Waves equations with large topography variations[END_REF]).

Main result

We investigate in this paper the rigorous limit ε goes to zero in the rescaled Euler equation (1.7) in view of the mathematical justification of the derivation of the rigid lid model (1.8). However, as one shall see in Section 2.1, the only solutions of the rigid lid equations are trivially null. Though the rigid lid model does not then seem of much interest, it implies that the solutions of the rescaled Euler equations (1.7) converge weakly to zero as ε goes to zero. In [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF], the same limit in the rigid lid regime is investigated by Bresch and Métivier for the Shallow-Water equations with large bathymetry, which consist in an asymptotic model for the Water-Waves problem in the shallow water regime (µ small). They prove that the rescaled equations are well-posed on a time interval independent on ε (which is equivalent to a large time of existence of size 1 ε for the system written in the original variables) and they rigorously pass to the limit as ε goes to zero, and prove the weak convergence of the solutions.

In [START_REF] Mésognon-Gireau | The Cauchy problem on large time for the Water Waves equations with large topography variations[END_REF], a similar long time existence result is proved for the Water-Waves equations with large bathymetry and with surface tension. The local existence result (1.1) implies that (we don't give a precise statement) that there exists T ą 0, and a unique solution pζ, ψq P C 1 pr0; T ε s; H N ˆHN`1{2 pR d qq to the Water-Waves equations (1.5). Moreover, one has:

1 T " C 1 pζ 0 , ψ 0 q, |pζ, ψq| C 1 pr0; T ε s;H N ˆHN`1{2 pR d qq ď C 2 pζ 0 , ψ 0 q, (1.11)
where C i are continuous functions of their arguments, and are independent on µ, ε. The bound on the solutions given by (1.11) is uniform with respect to ε, which allows by a compactness argument to extract a convergent sub-sequence of solutions as ε goes to zero. The limit should be zero (at least for ψ) as we discussed above. The question is to understand if the convergence is strong in Cpr0; T s; H N ´1 ĤN´1{2 pR d qq, i.e. "globally in time and space". In this paper, we complete the study of the rigid lid limit for the Water-Waves problem, and give an answer to this question.

Since the equations (1.5) have the structure of a Hamiltonian equation, the following quantity is conserved:

1 2µ pGψ, ψq 2 `pζ, ζq 2 . (1.12)
We recall that G is symmetric and positive for the L 2 scalar product. Therefore, even after the time change of scale t 1 " εt and the limit ε goes to zero, the Hamiltonian is conserved and the strong convergence in L 2 of the unknowns cannot be (except in particular cases). In Section 2.2, we precisely try to highlight the default of compactness that prevents the strong convergence in dimension 1, in presence of a flat bottom. As one shall see, the default comes from the linear operator of the Water-Waves equations which has quite a similar behavior to the wave equation. In Section 3, we prove the equivalence between the Water-Waves equations and the Euler equations, which completes the study of the weak but not strong convergence to zero in Sobolev spaces for the solutions of the Euler equation (1.7) in the rigid lid regime. To sum up, we give here the plan of this article:

-In Section 2.2, we prove that the solutions of the linearized equation does not converge strongly in L 2 pR d q at a fixed time, in the rigid lid scaling pt 1 " εtq for d " 1, 2.

-In Section 2.3, we prove the strong convergence in L 8 pr0; T r; L 2 ˆH1{2 pR d qq of the solutions of the full Water-Waves equation in rigid lid scaling to the solutions of the linearized equations, for d " 1. It proves that the solutions of the full Water-Waves equations do not converge strongly in L 8 pr0; T r; L 2 ˆH1{2 pR d qq to zero, for d " 1.

-In Section 3, we prove the equivalence between the free-surface Euler equations (1.7) and the Water-Waves equations (1.5) for d " 1, 2. We show that the solutions of Euler equations converge weakly as ε goes to zero, to the solutions of the rigid lid equation (1.8). The convergence is therefore not strong, at least for d " 1, according to the preceding points.

Notations

We introduce here all the notations used in this paper.

Operators and quantities

Because of the use of dimensionless variables (see before the "dimensionless equations" paragraph), we use the following twisted partial operators: Remark 1.3 All the results proved in this paper do not need the assumption that the typical wave lengths are the same in both directions, ie γ " 1. However, if one is not interested in the dependence of γ, it is possible to take γ " 1 in all the following proofs. A typical situation where γ ‰ 1 is for weakly transverse waves for which γ " ? µ; this leads to weakly transverse Boussinesq systems and the Kadomtsev-Petviashvili equation (see [START_REF] Lannes | Weakly transverse Boussinesq systems and the KP approximation[END_REF]).

∇ γ " t pB x , γB y q if d " 2 and ∇ γ " B x if d " 1 ∆ µ,γ " µB 2 x `γ2 µB 2 y `B2 z if d " 2 and ∆ µ,γ " µB 2 x `B2 z if d " 1 ∇ µ,
For all α " pα 1 , .., α d q P N d , we write

B α " B α1 x1 ...B α d x d
and |α| " α 1 `... `αd .

We use the classical Fourier multiplier Λ s " p1 ´∆q s{2 on R d defined by its Fourier transform as F pΛ s uqpξq " p1 `|ξ| 2 q s{2 pF uqpξq for all u P S 1 pR d q. The operator P is defined as

P " |D γ | p1 `?µ|D γ |q 1{2 (1.13)
where F pf pDquqpξq " f pξqF puqpξq is defined for any smooth function f of polynomial growth and u P S 1 pR d q. The pseudo-differential operator P acts as the square root of the Dirichlet-Neumann operator (see (1.10)). We denote as before by G µ,γ the Dirichlet-Neumann operator, which is defined as followed in the scaled variables:

G µ,γ ψ " G µ,γ rεζsψ " a 1 `ε2 |∇ γ ζ| 2 B n Φ |z"εζ " pB z Φ ´µ∇ γ pεζq ¨∇γ Φq |z"εζ ,
where Φ solves the Laplace equation

# ∆ γ,µ Φ " 0 Φ |z"εζ " ψ, B n Φ |z"´1 " 0.
For the sake of simplicity, we use the notation Grεζsψ or even Gψ when no ambiguity is possible.

The Dirichlet-Neumann problem

In order to study the Dirichlet-Neumann problem (1.4), we need to map Ω t into a fixed domain (and not on a moving subset). For this purpose, we introduce the following fixed strip:

S " R d ˆp´1; 0q
and the diffeomorphism

Σ ε t : S Ñ Ω t pX, zq Þ Ñ p1 `εζpXqqz `εζpXq.
(1.14)

It is quite easy to check that Φ is the variational solution of (1.4) if and only if φ " Φ ˝Σε t is the variational solution of the following problem:

# ∇ µ,γ ¨P pΣ ε t q∇ µ,γ φ " 0 φ z"0 " ψ, B n φ z"´1 " 0, (1.15)
and where

P pΣ ε t q " | det J Σ ε t |J ´1 Σ ε t t pJ ´1 Σ ε
t q, where J Σ ε t is the jacobian matrix of the diffeomorphism Σ ε t . For a complete statement of the result, and a proof of existence and uniqueness of solutions to these problems, see [START_REF] Lannes | The water waves problem[END_REF] Chapter 2.

We introduce here the notations for the shape derivatives of the Dirichlet-Neumann operator. More precisely, we define the open set Γ Ă H t0`1 pR d q as: Γ " tΓ " ζ P H t0`1 pR d q, Dh 0 ą 0, @X P R d , εζpXq `1´ě h 0 u and, given a ψ P . H s`1{2 pR d q, the mapping:

Grε¨s : Γ ÝÑ H s´1{2 pR d q Γ " pζ, bq Þ ÝÑ Grεζsψ.
We can prove the differentiability of this mapping. See Appendix A for more details. We denote d j Gph, kqψ the j-th derivative of the mapping at pζ, bq in the direction ph, kq. When we only differentiate in one direction, and no ambiguity is possible, we simply denote d j Gphqψ or d j Gpkqψ.

Functional spaces

The standard scalar product on L 2 pR d q is denoted by p , q 2 and the associated norm | ¨|2 . We will denote the norm of the Sobolev spaces H s pR d q by | ¨|H s .

We introduce the following functional Sobolev-type spaces, or Beppo-Levi spaces: Definition We denote 9

H s`1 pR d q the topological vector space 9 H s`1 pR d q " tu P L 2 loc pR d q, ∇u P H s pR d qu endowed with the (semi) norm |u| 9 H s`1 pR d q " |∇u| H s pR d q .

Just remark that 9 H s`1 pR d q{R d is a Banach space (see for instance [START_REF] Deny | Les espaces du type de beppo levi[END_REF]).

The space variables z P R and X P R d play different roles in the equations since the Euler formulation (1.1) is posed for pX, zq P Ω t . Therefore, X lives in the whole space R d (which allows to take fractional Sobolev type norms in space), while z is actually bounded. For this reason, we need to introduce the following Banach spaces:

Definition The Banach space pH s,k pp´1, 0q ˆRd q, |.| H s,k q is defined by

H s,k pp´1, 0q ˆRd q " k č j"0 H j pp´1, 0q; H s´j pR d qq, |u| H s,k " k ÿ j"0 |Λ s´j B j z u| 2 .
2 The rigid lid limit for the Water-Waves equations

We prove in this section that the solutions of the linearized Water-Waves equations in flat bottom converge weakly but non strongly in L 2 as ε goes to zero, in the rigid lid regime. We then prove the strong convergence in L 8 t L 2

x of the solutions of the full Water-Waves equations to the solutions of the linearized equations, in the same regime, in dimension 1.

Remark 2.1 The hypothesis "d " 1" can be removed, if one can prove a dispersive estimate for the linear operator of the Water-Waves equation of the form of Theorem 2.9, in dimension d " 2. The flat bottom hypothesis seems however less easy to remove, due to technical reasons, such as the complexity of the asymptotic expansion of G with respect to the surface when the bottom is non flat.

Solutions of the rigid lid equations

We recall the formulation of the rigid lid equations with a flat bottom (see also [START_REF] Camassa | Long-time shallow-water equations with a varying bottom[END_REF] for reference): $ ' ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' ' %

B t V `pV ¨∇γ `1 µ wB z qV " ´∇γ P in Ω B t w `pV ¨∇γ `1 µ wB z qw " ´pB z P q in Ω w " 0 ∇ µ,γ ¨U " 0 in Ω curl µ,γ pU q " 0 in Ω U ¨ez " 0 for z " ´1 P " ζ for z " 0 (2.16)
where Ω is the fixed domain

Ω " tpX, zq P R d`1 , ´1 ď z ď 0u.
We prove here that in fact, there is only one trivial solution to the system (2.16): Lemma 2.2 Let us consider the following div-curl problem:

$ ' ' ' & ' ' ' % ∇ µ,γ ¨U " 0 in Ω curl µ,γ pU q " 0 in Ω U ¨ez " 0 for z " ´1 w " 0 (2.17)
with the above notations. Then, the unique solution in 9

H 1 pΩq to the system (2.17) is U " 0.

Proof The vector field U " 0 is of course a solution to the system (2.17). Let us now consider a solution U to this problem. The curl free condition over U states that there exists a potential Φ such that U " ∇ µ,γ Φ in Ω. The divergence free condition provides ∆ µ,γ Φ " 0 in Ω. With the boundary conditions (recall that w is the vertical component of the velocity at the surface), the potential Φ satisfies the following Laplace equation:

$ ' & ' % ∆ µ,γ Φ " 0 in Ω ∇ µ,γ Φ |z"0 ¨n " 0 ∇ µ,γ Φ |z"´1 ¨n " 0,
where n denotes the upward normal vector in the vertical direction. It is well known (see for instance [START_REF] Lannes | The water waves problem[END_REF] Chapter 2 and Appendix A) that the unique solution to this system in 9

H 1 pΩq satisfies ∇ µ,γ Φ " 0. 1 l
Let us explain this result on a physical point of view. The rigid-lid equations are obtained after scaling the time t 1 " εt in the free surface Euler equations, and passing to the limit as ε goes to zero. Therefore, in the original time variable, t " t 1 ε , the rigid lid limit consists in letting the amplitude of the waves goes to zero AND moving forward in time at a rate 1 ε . At the limit, after an infinite time, all the interesting components of the waves moved to ˘8 and there only remains a static fluid (U " 0) with a flat surface. It is possible that some vortices remain, but they are not seen by the model (curlpUq " 0). One could generalize the study lead in this paper to the Water-Waves equation with vorticity (see [START_REF] Castro | Well-posedness and shallow-water stability for a new hamiltonian formulation of the water waves equations with vorticity[END_REF]).

The linearized equation around ζ " 0 in the rigid lid regime

We prove here that the solutions to the Water-Waves equations do not converge strongly in L 2 as ε goes to zero. To this purpose, we study the lack of compactness induced by the linear operator of the Water-Waves equations. We recall that in the rigid lid regime, we perform a time scaling t 1 " εt.

The Water-Waves equations can be written in dimensionless form, and in the rigid lid scaling:

$ ' & ' % εB t ζ ´1 µ Grεζsψ " 0 εB t ψ `ζ `ε 2 |∇ γ ψ| 2 ´ε µ pGrεζsψ `εµ∇ γ ζ ¨∇γ ψq 2 2p1 `ε2 µ|∇ γ ζ| 2 " 0.
(2.18) 1 In fact the result stands even if the bottom is non flat.

Since we are interested in the convergence as ε goes to zero, we write this system under the form:

$ & % εB t ζ ´1 µ G 0 ψ " εf εB t ψ `ζ " εg.
(2.19)

We will treat the non-linear terms f, g present in (2.19) as a perturbation of the linearized equations.

To this purpose, we start to study the solutions of the linearized system:

# εB t ζ ´1 µ G 0 ψ " 0 εB t ψ `ζ " 0, (2.20)
where G 0 " Gr0s is the Dirichlet-Neumann operator in flat bottom and flat surface, and is given by (see for instance [START_REF] Lannes | The water waves problem[END_REF] Chapter 1):

G 0 ψ " ? µ|D γ | tanhp ? µ|D γ |qψ (2.21)
for all ψ P SpR d q. It is easy to check that the solution ζ of (2.20) satisfies the following equation:

ε 2 B t ζ `1 µ G 0 ζ " 0. (2.22)
The equation (2.22) looks like a wave equation, except that G 0 is not exactly an order two operator (since it acts like an order one operator for high frequencies). In the case of the wave equation on R

$ ' & ' % ε 2 B t u ´B2 x u " 0 upx, 0q " u 0 pεB t uqpx, 0q " u 1 ,
with u 0 , u 1 smooth, the solutions have components of the form

u 0 px ´1 ε tq `u0 px `1 ε tq
(plus other terms we do not detail). Each of these two components converges pointwise to 0 as ε goes to zero (since u 0 pt, xq ÝÑ |x|Ñ`8 0) and does not converge strongly in L 2 pRq, since for instance its L 2 pRq norm does not depend on ε. The same behavior, combined with some dispersive effects stands for (2.22). We start to give an existence result for the linearized equations (2.20). Note that it is not difficult to prove formally that if pζ, ψq is the solution of (2.20), then the following quantity, called "Hamiltonian" is conserved through time:

1 2µ pG 0 ψ, ψq 2 `1 2 |ζ| 2 2 .
Therefore, we introduce the following operator:

P " |D γ | p1 `?µ|D γ |q 1{2 .
Note that P has the same behavior as the square root of G 0 defined by (2.21).

Proposition 2.3 Let s ě 1. Let also ζ 0 P H s pR d q and ψ 0 be such that Pψ 0 P H s pR d q. Then, there exists a unique solution pζ, ψq to the equation:

$ ' & ' % εB t ζ ´1 µ G 0 ψ " 0 εB t ψ `ζ " 0 pζp0, Xq, ψp0, Xqq " pζ 0 pXq, ψ 0 pXqq (2.23)
such that pζ, Pψq P CpR, H s pR d qq X C 1 pR, H s´1 pR d qq. Moreover, one has:

@t P R, ˆζ ψ ˙ptq " e ´t ε L ˆζ0 ψ 0 where e tL " 1 2
ˆ1 ´iωpDq ´1 iωpDq 1 ˙e´iωpDqt `1 2 ˆ1 iωpDq 1 iωpDq 1 ˙eiωpDqt (2.24) and ωpDq " d |D γ | tanhp ? µ|D γ |q ? µ . (2.25) Remark 2.4
The notation e tL refers to the linear operator of the Water-Waves, defined by

L " ˆ0 ´1 µ G 0 1 0 ˙.
Proof Let us fix s ě 1, and pζ 0 , ψ 0 q P H s pR d q 2 . If pζ, ψq is a solution to (2.23), then ζ satisfies the following equation:

$ ' ' ' & ' ' ' % ε 2 B 2 t ζ `1 µ G 0 ζ " 0 ζpx, 0q " ζ 0 pεB t ζqpx, 0q " ζ 1 , (2.26) in the space CpR, H s pR d qqXC 1 pR, H s´1 pR d qq, with ζ 1 " 1 µ G 0 ψ 0 .
As usual, we take the Fourier transform in space of the equation (2.26) and we denote p ζpt, ξq the Fourier transform of ζ with respect to the variable X for a fixed t. The distribution x B t ζof S 1 pR ˆRd q is equal to the distribution B t p ζ using the Banach-Steinhaus theorem. We therefore get the following equation in the distributional sense of S 1 pR ˆRd q: with Pψ 0 P H s pR d q and that there exists C 1 , C 2 ą 0 independent of µ, ε such that:

$ ' & ' % ε 2 B 2 t p ζ `1 ? µ |ξ γ | tanhp ? µ|ξ γ |q p ζ " 0 p ζpξ, 0q " p ζ 0 pξq pεB t p ζqpξ, 0q " p ζ 1 pξq.
@ξ P R d , C 1 ď d tanhp ? µ|ξ γ |qp1 `?µ|ξ γ |q ? µ|ξ γ | ď C 2 .
The same method applies for ψ and one gets that if pζ, ψq satisfies (2.23), then necessarily:

p ψpt, ξq " x ψ 0 cospωpξq t ε q ´sinpωpξq t ε q ωpξq p ζ 0 . (2.28)
It is easy to check that pζ, ψq given by (2.27), (2.28) is a solution to the system (2.20). l

The oscillating component of the solution of the linearized Water-Waves equations (2.23) appears in the explicit formula (2.27), but we need to prove an oscillating phase method type result in order to conclude to the weak convergence to zero and strong non-convergence. The phase ω is actually not differentiable at ξ " 0, which prevent the direct use of standards results on stationary phase methods. Proposition 2.5 Let t ą 0 and u P C 1 0 pR d q. Then, one has ż

R d exp i t ε ωpξq upξqdξ ÝÑ εÑ0 0.
Proof One can compute:

@ξ ‰ 0, p∇ωqpξq " tanhp|ξ γ |q `|ξ γ |p1 ´tanh 2 p|ξ γ |qq 2 a |ξ γ | tanhp|ξ γ |q ξ |ξ γ | . (2.29)
It is easy to see that ∇ω does not vanish for ξ ‰ 0. It is also easy to show that ω is not differentiable in ξ " 0. But the derivative ∇ω stay bounded as ξ goes to zero. We treat separately the cases d " 1 and d ą 1.

-Case d " 1 One writes ż R exp i t ε ωpξq upξqdξ " ż 0 ´8 exp i t ε ωpξq upξqdξ `ż `8 0 exp i t ε ωpξq upξqdξ. (2.30)
Since ω is right and left differentiable in ξ " 0, according to the expression (2.29), one can use standard methods of stationary phase. Indeed, one can write:

ż 0 ´8 exp i t ε ωpξq upξqdξ " ε ti ż 0 ´8 1 ω 1 pξq d dξ pexp i t ε ωpξq qupξqdξ " ε ti up0q ω 1´p 0q e i t ε ωp0q ´ε ti ż 0 ´8 e i t ε ωpξq d dξ p upξq ω 1´p ξq qdξ
where ω 1`p 0q and ω 1´p 0q are respectively the right and left derivatives of ω at ξ " 0. We then get the desired result. The same goes for the second integral of (2.30).

-Case d ą 1 We use the standard integrating by part method for the truncated integral:

ż |ξ|ěδ e i t ε ωpξq upξqdξ " ż |ξ|ěδ ε it 1 |ω 1 pξq| 2 d ÿ j"1 B Bξ j ωpξq B Bξ j pe i t ε ωpξq qupξqdξ " ε it ż |ξ|"δ 1 |ω 1 pξq| 2 d ÿ j"1 B Bξ j ωpξqe i t ε ωpξq upξqn j dσpξq ´ε it ż |ξ|ěδ d ÿ j"1 B Bξ j p 1 |ω 1 pξq| 2 B Bξ j ωpξqupξqqe i t ε ωpξq dξ (2.31)
where n j stands for the j-th component of the normal external vector at the surface |ξ| " δ. Now, one can check that:

1 |ω 1 pξq| 2 B Bξ j ωpξq " 2ξ j a |ξ γ | tanhp|ξ γ |q |ξ γ | tanhp|ξ γ |q `p1 ´tanh 2 p|ξ γ |qq|ξ γ | 2
and thus this function is bounded as ξ goes to zero. Moreover, one can check by computation that

| B Bξ j p 1 |ω 1 pξq| 2 B Bξ j ωpξqq| ď C |ξ|
as ξ goes to zero, and consequently this function is integrable at ξ " 0 (remember that d ą 1 here). Therefore, one can pass the limit as δ goes to zero in the formula (2.31) and get

ż R d e i t ε ωpξq upξqdξ " ´ε it ż R d d ÿ j"1 B Bξ j p 1 |ω 1 pξq| 2 B Bξ j ωpξqupξqqe i t ε ωpξq dξ
and we get the desired result.

l Now it is easy to prove the following result:

Theorem 2.6 Let s ě 1. Let pζ 0 , ψ 0 q be such that pζ 0 , Pψ 0 q P H s pR d q. Then, for all t P R, the solution pζ, ψqptq of the system (2.20) converges weakly to zero, and does not converge strongly, in L 2 pR d q.

Proof We prove it for ζ. We first assume that p ζ, p ψ P C 1 0 pR d q. The weak convergence to zero is a consequence of the explicit formulation of the solution given by (2.27) and the Proposition 2.5. To prove the strong non convergence, one computes:

|ζpt, .q| 2 2 " 1 p2πq d | p ζpt, .q| 2 2 " 1 p2πq d ż R d 1 2 `p1 `cosp2ωpξq t ε qq| p ζ 0 pξq| 2 `ωpξq 2 p1 ´cosp2ωpξq t ε qq| x ψ 0 pξq| 2 Ȓep p ζ 0 x ψ 0 qpξq sinp2ωpξq t ε qdξ
where we expanded the square modulus of the explicit formulation of p ζ given by (2.27). We can then conclude, using Proposition 2.5 to the following convergence:

|ζ| 2 2 ÝÑ εÑ0 1 2 p|ζ 0 | 2 2 `|ωpDqψ 0 | 2 2 q
and thus ζpt, .q does not converge strongly to zero.

For the general case, one proceeds by density of C 1 0 pR d q in L 2 pR d q. l Remark 2.7 As explained above, if ζ, ψ is a solution to (2.23), then the Hamiltonian

1 2µ pG 0 ψ, ψq 2 `1 2 |ζ| 2 2
is conserved through time. Since 1 µ pG 0 ψ, ψq 2 " |Pψ| 2 2 , the strong convergence of both ζ and ψ to zero cannot happen (except for zero initial conditions). However, it would have been possible that some transfers of energy occur between ζ and ψ, with one of the unknown strongly converging to zero. The Theorem 2.6 states that such behavior does not happen.

Lack of strong convergence for the full Water-Waves equations in dimension 1

In all this section, we work with a flat bottom and in dimension 1:

d " 1, b " 0.
The study in dimension 2 should however not be hard to do, if one gets a dispersive estimate of the form of Theorem 2.9 for the Water-Waves operator in dimension 2. In the previous section, we proved that the solutions of the linearized Water-Waves equations (2.20) weakly converge as ε goes to zero in L 2 pRq, and do not converge strongly. We now establish a similar result for the solutions of the full nonlinear Water-Waves system (1.9). To this purpose, we write this system under the form:

# εB t ζ ´1 µ G 0 ψ " εf εB t ψ `ζ " εg with f " 1 µε pGrεζsψ ´G0 ψq and g " ´p 1 2 |∇ γ ψ| 2 ´1 µ pGrεζsψ `εµ∇ γ ζ ¨∇γ ψq 2 2p1 `ε2 µ|∇ γ ζ| 2 q.
Remark 2.8 As suggested by the notations, the quantity f is of size Op1q with respect to ε (although it has a 1 ε factor), as given by the asymptotic extension of Grεζ, 0s given later in Proposition 2.13. We are going to treat the non-linear terms f, g as perturbations of the linear equation as ε goes to zero.

For this proof, we need to use a dispersive estimate for the linear Water-Waves equations with flat bottom, proved in [START_REF] Mésognon-Gireau | The cauchy problem on large time for a boussinesq peregrine equation with large topography variations[END_REF]:

Theorem 2.9 Let ω : $ ' & ' % R ÝÑ R ξ Þ ÝÑ d |ξ| tanhp ? µ|ξ|q ? µ .
Then, there exists C ą 0 independent on µ such that, for all µ ą 0:

@t ą 0, @ϕ P SpRq |e itωpDq ϕ| 8 ď Cp 1 µ 1{4 1 pt{ ? µq 1{8 `1 pt{ ? µq 1{2 qp|ϕ| H 1 `|xB x ϕ| 2 q.
Remark 2.10 The dispersion in 1 t 8 stated by this Theorem is absolutely not optimal. Actually, one should expect a decay of order 1 t 1{3 in dimension 1 (see for instance [START_REF] Mésognon-Gireau | The cauchy problem on large time for a boussinesq peregrine equation with large topography variations[END_REF] for variants of Theorem 2.9). However, the 1 t 1{8 decay suffices to prove the main result of this section.

In view of use of Theorem 2.9, we also need a local existence result in weighted Sobolev spaces, proved also in [START_REF] Mésognon-Gireau | The cauchy problem on large time for a boussinesq peregrine equation with large topography variations[END_REF]. For all N P N, we define E N x by

E N x " E N pζ, ψq `ÿ 1ď|α|ďN ´2 |xζ pαq | 2 2 `|Pxψ pαq | 2 2 .
Theorem 2.11 Let us consider the assumption of Theorem 1.1, and then consider pζ, ψq the unique solution provided by the Theorem 1.1, of the Water-Waves equation (1.9). If pζ 0 , ψ 0 q P E N x , then one has pζ, ψq P L 8 pr0; T s, E N x q, with pζ, ψq L 8 pr0;T s,E N x q ď C 2 .

We now state the main result of this section:

Theorem 2.12 Let pζ 0 , ψ 0 q be such that pζ 0 , ψ 0 q P E N x . Let denote T ą 0, pζ W W , ψ W W q the solution of the Water-Waves equations (1.9) in pζ, ψq P L 8 pr0; T s, E N

x q given by Theorem 1.1 on r0; T s and pζ L , ψ L q the solution of the linearized equation (2.20) given by Proposition 2.3, both with initial condition pζ 0 , ψ 0 q. Then, one has:

|pζ L , Pψ L q ´pζ W W , Pψ W W q| L 8 pp0;T q;L 2 pRqq ď p ε 1{8 µ 3{16 `ε1{2 µ 1{4 qC 2 (2.32)
where C 2 is given by Theorem 1.1.

Proof In all this proof, we will denote by C N any constant of the form

C N " CpE N x pζ 0 , ψ 0 q, 1 h min , 1 a 0 q (2.33)
where C is a non decreasing function of its arguments. Let us define pζ, ψq " pζ W W , Pψ W W q´pζ L , Pψ L q which is defined on r0; T s. We use the evolution operator e t ε L defined by (2.24) to write:

B t pe t ε L ˆζ ψ ˙q " e t ε L F p ˆζ ψ ˙qptq
where

F p ˆζ ψ ˙q " ˜1 µε pGpεζqψ ´G0 ψq ´1 2 |∇ γ ψ| 2 2 ´1 µ pGpεζqψ`εµ∇ γ ζ¨∇ γ ψq 2 2p1`ε 2 µ|∇ γ ζ| 2 q ¸(2.34)
and thus one has

@t P r0; T s, ˆζ ψ ˙" ż t 0 e s´t ε L F p ˆζ ψ ˙qpsqds. ( 2

.35)

We set A " ˆI 0 0 P ˙and look for a estimate of the L 2 norm of A t pζ, ψq. The proof consists in using the decay estimate of the linear operator e tL of the Water-Waves equation of Theorem 2.9. More precisely, the operator F is "almost" bilinear (at least up to a Opεq order term), which allows us to control the L 2 norm of the integral (2.35) by writing estimates of the form

| ż t 0 Ae s´t ε L F p ˆζ ψ ˙qpsqds| 2 ď | ż t 0 Ae s´t ε L Bp ˆζ ψ ˙, ˆζ ψ ˙qpsqds| 2 `εC
where C does not depend on ε, and with B a bilinear operator. The proof then consists in using a Strichartz type of estimate, using the dispersive nature of e itL , given by Theorem 2.9 to get a control of the remaining integral of the form

| ż t 0 e s´t ε L Bp ˆζ ψ ˙, ˆζ ψ ˙qpsqds| 2 ď pεq 1{8 t α
with α ą 0.

We start to write F under the form F " B `εR where B is bilinear, and R is a least of size Op1q with respect to ε. To this purpose, recalling that F is given by (2.34), we get inspired by the following Proposition, which gives an asymptotic extension of Grεζ, 0s (see [START_REF] Lannes | The water waves problem[END_REF] Proposition 3.44) with respect to ε: Proposition 2.13 Let t 0 ą d{2, s ě 0 and k " 0, 1. Let ζ P H s`pk`1q{2 X H t0`2 pR d q be such that:

Dh min ą 0, @X P R d , 1 `εζpXq ě h min and ψ P H s`k{2 pR d q. We get:

|Gψ ´G0 ψ ´εG 1 | H s´1{2 ď ε 2 µ 3`k 4 Cp 1 h min , µ max , |ζ| H t 0 `1 , |ζ| H s`pk`1q{2 q|Pψ| H s`k{2 ,
where G 1 " ´G0 pζpG 0 ¨qq ´µ∇ γ ¨pζ∇ γ ¨q.

We therefore write

1 µε pGpεζq ´G0 qψ " 1 µ G 1 ´1 µε pG 1 `G0 ´Gpεζqqψ (2.36)
where the second term of the right hand side satisfies the following estimate, using Proposition 2.13 with k " 1 and s " 1{2:

| 1 µε pG 1 `G0 ´Gpεζqq| 2 ď Cp 1 h min , µ max , |ζ| H t 0 `1 , |ζ| H 1{2`1 q|Pψ| H 1 .
(2.37)

The second component of F (recall that it is given by (2.34)) is easier to decompose, and using Proposition A.1 one gets

´1 2 |∇ γ ψ| 2 ´1 µ pGpεζqψ `εµ∇ γ ζ ¨∇γ ψq 2 2p1 `ε2 µ|∇ γ ζ| 2 q " ´1 2
| ż t 0 Ae s´t ε L F p ˆζ ψ ˙qpsqds| 2 ď | ż t 0 e s´t ε L ABpsqds| 2 `εC N .
We denote in the following lines p¨, ¨qL 2

x the L 2 scalar product with respect to the space variable. One writes:

| ż t 0 e s´t ε L ABpsqds| 2 2 " | ż t 0 ż t 0 pe s´t ε L ABpsq, e u´t ε L ABpuqq L 2 x duds|.
Now, note that Ae itL is sum of terms of the form ˆ1 iωpDq `P iωpDq P ˙eitωpDq and therefore, we are led to estimate terms of the form ż t

0 ż t 0 pe s´t ε L f 2 psq, e u´t ε L f 1 puqq L 2 x duds, (2.43)
where f 1 , f 2 are terms of the form

1 µ G 1 , ωpDq 1 µ Gpεζq 2 , ωpDq|∇ γ ψ| 2 , P ωpDq 1 µ G 1 , P 1 µ Gpεζq 2 , P|∇ γ ψ| 2 2 .
(2.44)

One has to notice that the control related to the latter three terms can be deduced from the control of the first three terms since P acts like the square root ωpDq of G 0 . Indeed, one has

ωpDq " |D γ | p1 `?µ|D γ |q 1{2 d tanhp ? µ|D γ |qp1 `?µ|D γ |q ? µ|D γ | " P d tanhp ? µ|D γ |qp1 `?µ|D γ |q ? µ|D γ |
and there exists C 1 , C 2 ą 0 independent of µ, ε such that:

@ξ P R d , C 1 ď d tanhp ? µ|ξ γ |qp1 `?µ|ξ γ |q ? µ|ξ γ | ď C 2 .
(2.45)

We now estimate terms of the form (2.43) using a similar technique as for Strichartz estimates for dispersives PDE's. One computes, using the symmetry of e itL :

ż t 0 ż t 0 pe s´t ε L f 2 psq, e u´t ε L f 1 puqq L 2 x duds " ż t 0 ż t 0 pf 2 psq, e u`s´2t ε L f 1 puqq L 2 x duds.
Now recall that f i are of the form (2.44). We are not treating all the possible cases, but only the most difficult one (the others are treated by the same technique), which is

f 1 " f 2 " 1 µ G 1 ψ.
Using the definition of G 1 given in Proposition 2.13, one computes, integrating by parts:

ż t 0 ż t 0 pf 2 psq, e u`s´2t ε L f 1 puqq L 2 x duds " ż t 0 ż t 0 p´1 µ G 0 pζG 0 pψqqpsq ´∇γ ¨pζ∇ γ ψqpsq, e u`s´2t ε L 1 µ G 1 ψpuqq L 2 x duds " ż t 0 ż t 0 p´1 µ ζG 0 pψqpsq, e u`s´2t ε L G 0 1 µ G 1 puqq L 2 x `pζ∇ γ ψpsq, e u`s´2t ε L ∇ γ 1 µ G 1 ψpuqq L 2 x duds. (2.46)
We only control the first integral of the right hande side of (2.46) (the other one is estimated by a similar technique). We set f2 " ´1 µ ζG 0 ψpsq,

f1 " G 0 1 µ G 1 ψpuq (2.47)
and we now use the dispersive estimate of Theorem 2.9:

| ż t 0 ż t 0 p f2 psq, e u`s´2t ε L p f1 q L 2 x duds| ď ż t 0 ż t 0 | f2 | L 1 x |e u`s´2t ε L p f1 q| L 8 x ď C N ż t 0 ż t 0 ε 1{8 µ 3{16 1 p2t ´u ´sq 1{8 `ε1{2 µ 1{4 p2t ´u ´sq 1{2 p| f1 puq| H 1 x `|xB x f1 puq| L 2 x qduds ď p ε 1{8 µ 3{16 t 7{8 `ε1{2 µ 1{4 t 3{2 qC N
if one can prove the following controls:

sup uPr0;T s | f2 puq| L 1 x `| f1 | H 1 x `|xB x f1 puq| L 2 x ď C N , i " 1, 2.
(2.48) One has, using Cauchy-Schwarz inequality:

| f2 puq| L 1 x ď |ζ| 2 | 1 µ G 0 ψ| 2 ď |ζ| 2 |Pψ| H 1 ď C N
where we used Proposition A.1 with ζ " b " 0 to control 1 µ G 0 ψ. We now focus on the most difficult term of (2.48) which is

|xB x G 0 1 µ G 1 ψ| L 2
x . We use again the definition of G 1 given by Proposition 2.13, and we control 1 µ |xB x G 0 G 0 pζG 0 pψqq| 2 (the other one is similar). One computes:

1 µ |xB x G 0 G 0 pζG 0 pψqq| 2 ď 1 µ |xG 2 0 pB x ζqG 0 ψ| 2 `1 µ |xG 2 0 pζG 0 pB x ψqq| 2 .
(2.49)

We only control the first term of the right hand side of (2.49) (the other one is similar):

|xG 2 0 pB x ζqG 0 ψ| 2 ď |G 2 0 xpB x ζqG 0 ψ| 2 `|rx, G 2 0 spB x ζqG 0 ψ| 2 .
Now, note that for all f P SpRq, one has:

|rx, G 2 0 sf | 2 " | d dξ p ? µ|ξ| tanhp ? µ|ξ|qq 2 p f | 2 ď µ 2 |f | H 3
using the definition of G 0 given by (2.21). Therefore, one has:

1 µ |xG 2 0 pB x ζqG 0 ψ| 2 ď µ|xpB x ζqG 0 ψ| H 4 `µ|pB x ζqG 0 ψ| H 3 ď µ 2 |xpB x ζq| H 4 |G 0 ψ| H 4 ,
where we used the gross estimate |G 2 0 u| ď |u| H 4 . One can conclude using Proposition A.1 with ζ " b " 0, and Theorem 2.11.

This achieves the proof of the controls (2.48).

Conclusion

: We proved |pζ, Pψqptq| 2 ď p ε 1{8 µ 3{16 t 7{8 `ε1{2 µ 1{4 t 3{2 qC N .
One can then take the supremum over t P r0; T s and get the desired result. l Remark 2.14 It is very important to note that according to Theorem 2.12 the linear effects are a good approximation in the rigid lid regime if the ratio ε µ 3{2 is not too large. For instance, if ε " µ 3{2 , the strong convergence of the solutions of the fully nonlinear system to the solution of the linear system is not true anymore. In this case, the dispersive effects would be at the same order as the non-linear effects, and one should expect a behavior of solutions similar to one of the Korteweg de Vries's equation (such as solitary waves type behavior, see for instance [START_REF] Brooke | The stability of solitary waves[END_REF]). In particular, it is no longer possible to treat the non-linearities f, g as perturbations of the linearized system. A more precise study should be lead, and one should consider as approximation of the full nonlinear Water-Waves equation an equation where the nonlinear terms of size ε are taken into account (see for instance [START_REF] Saut | Well-posedness on large time for a modified full dispersion system of surface waves[END_REF]). If ε ! µ 3{2 , then the non-linear effects are dominants on the dispersive effects, and one should expect breaking waves. This is the main interest of having a dispersive result of the form of Theorem 2.9 which depends on the small parameters.

It is now easy to prove the following Corollary:

Corollary 2.15 Let pζ 0 , ψ 0 q be such that pζ 0 , ψ 0 q P E N

x . Let denote T ą 0, pζ, ψq the solution of the Water-Waves equations defined on r0; T s of (1.9). Then, pζ, Pψq converge weakly to zero in L 8 t L 2 x and does not converge strongly in this space.

Proof Let us denote pζ L , ψ L q the solution of the linearized equation (2.20) with initial condition pζ 0 , ψ 0 q given by Theorem 2.11 on r0; T s (just take ε " 0 in the equation (1.9) to get the solutions). Using Proposition 2.3, one has pζ L , ψ L qptq " e ´t ε L pζ 0 , ψ 0 q.

Using the definition of e tL given by (2.24), ζ, Pψ, are sum of terms of the form e itωpDq ε pζ 0 , ψ 0 q and therefore one has, using the dispersive estimate of Theorem 2.9:

@t P r0; T s, |pζ L ptq, Pψ L ptqq| ď p ε 1{8 µ 3{16 t 7{8 `ε1{2 µ 1{4 t 3{2 qp|pζ 0 , Pψ 0 q| H 1 `|xB x ζ 0 , PxB x ψ 0 q| L 2
and therefore one gets the weak convergence of pζ, Pψq in L 8 pp0; T q; L 2 x pRqq as ε goes to zero. The convergence is not strong in L 8 pp0; T q; L 2

x pRqq since the quantity

1 2µ pG 0 ψ, ψq 2 `1 2 |ζ| 2 2 " 1 2 p|Pψ| 2 2 `|ζ| 2 2 q
is conserved through time. Now, according to Theorem 2.12, pζ, Pψq ´pζ L , ψ L q converges strongly to zero as ε goes to zero in L 8 pp0; T q; L 2

x pRqq. Therefore, pζ L , ψ L q converges weakly, but not strongly as ε goes to zero, in this space. l

Equivalence between the free surface Euler and Water-Wave equation

In all this Section, we do not make any assumption on the dimension, and we set the bottom to zero:

d " 1, 2, b " 0.
We consider the standard dimensionless version of the Water-Waves equations with flat bottom:

$ ' & ' % B t ζ ´1 µ Grεζ, 0sψ " 0 B t ψ `ζ `ε 2 |∇ γ ψ| 2 ´ε µ pGrεζ, 0sψ `εµ∇ γ ζ ¨∇γ ψq 2 2p1 `ε2 µ|∇ γ ζ| 2 " 0. (3.50)
In Section 2, we studied the rigid lid limit for the Water-Waves equations (2.18). In order to complete the study of the rigid lid equations (2.16), we study in this Section the rigid lid limit for the Euler equations (1.7). To this purpose, we prove that one can build rigorously solutions to the free surface Euler equations from the solutions of the Water-Waves equations. Such problem has been studied for instance in [START_REF] Alazard | The water-wave equations: from Zakharov to Euler[END_REF]. The first problem is to define rigorously a solution to the free surface Euler equations. Because the free surface Euler equations (1.7) are posed on a domain Ω ε t which depends on ε and t, we have to be careful with the functional spaces used. The idea is that, by hypothesis, the height 1`εζpt, Xq of the water is bounded with respect to X P R d at a fixed time t. By continuity in time, we can include Ω ε t in a fixed strip S ˚, which does not depends either on t nor ε.

We can then define rigorously what is a solution to the Euler equation (1.7): it is pU, P q living in a functional space like "continuous in time with value to a Sobolev-type space in X on the fixed strip S ˚" such that there exists T ą 0 such that, for all 0 ď t ď T , the derivative (in the sense of the distributions of D 1 X,z,t pr0, T s ˆS˚q ) satisfies the equalities of (1.7).

Main result

In order to extract a convergent sub-sequence of solutions to the Water-Waves equations (3.50) as ε goes to zero, we need some compactness results. Indeed, a space such as Cpr0; T s; H N pR d qq is not compactly embedded into L 8 pr0; T s; H N pR d qq. We need a control for time derivatives of the unknowns. For this reason, the stated local existence Theorem 1.1 is not sufficient. We recall here a slightly different version, including time derivatives in the energy space. The following framework is used for instance to prove the local existence for the Water-Waves equations with surface tension in [START_REF] Lannes | The water waves problem[END_REF] Chapter 9, and in [START_REF] Mésognon-Gireau | The Cauchy problem on large time for the Water Waves equations with large topography variations[END_REF] for a long time existence result. We first introduce the following energy:

Definition Let N ě 1 and t 0 ą d{2. We define

E N 1 " |Pψ| 2 H t 0 `3{2 `ÿ αPN d`1 ,|α|ďN |ζ pαq | 2 2 `|Pψ pαq | 2 2 (3.51) with ζ pαq " B α ζ, ψ pαq " B α ψ ´εwB α ζ and w " Gψ `εµ∇ γ ζ ¨∇γ ψ 1 `ε2 µ|∇ γ ζ| 2
with the notation: @α " pα 1 , ..., α d , kq

P N d`1 , B α " B α1 X1 ...B α d X2 B k t .
Remark 3.1 -It is very important to notice that we consider here also time derivatives in the energy.

-We recall that w coincides as suggested with the horizontal component of the velocity evaluated at the surface.

We now give a slightly different version of the local existence Theorem for the Water-Waves equations (3.50) proved by Alvarez-Samaniego and Lannes (see also [START_REF] Lannes | The water waves problem[END_REF] Chapter 4 and Chapter 9): Theorem 3.2 (Alvarez Samaniego,Lannes) Let t 0 ą d{2, N ą t 0 `t0 _ 2 `3{2, and pε, µ, γq be such that 0 ď ε, µ, γ ď 1.

Let pζ 0 , ψ 0 q with E N 1 pζ 0 , ψ 0 q ă 8, where E N 1 is defined by (3.51). We assume that:

Dh min ą 0, Da 0 ą 0, 1 `εζ 0 ě h min and apζ 0 , ψ 0 q ě a 0 .

Then, there exists T ą 0 and a unique solution pζ, ψq P Cpr0; T s; H t0`2 ˆ9 H 2 pR d qq to the Water-Waves equation (3.50) such that E N 1 pζ, ψq P L 8 pr0; T ε sq. Moreover, one has

1 T " C 1 and sup tPr0; T ε s E N 1 ptq " C 2
with C i a constant of the form C i " CpE N pζ 0 , ψ 0 q, 1 hmin , 1 a0 q for i " 1, 2.

We use from now on the notations H s,k for the Beppo-Levy spaces, and Σ ε t for the diffeomorphism from the flat strip to the water domain (see Section 1.5 for notations). We also recall that Ω ε t denote the domain occupied by the water at time t. We now state the main result of this Section: Theorem 3.3 Under the hypothesis of Theorem 3.2, let T ą 0 be given by Theorem 3.2. Then, for all ε ą 0, there exists a flat strip S ˚" R d ˆp´pk `1q; kq with k ě 1 and pU ε , ζ ε , P ε q where U ε " pV ε , w ε q in the following spaces:

$ ' & ' % @k ď N, B k t U ε P L 8 pp0; T ε q; H N ´k´1{2,N ´k´1 pS ˚qd`1 q @k ď N, B k t ζ ε P L 8 pr0; T ε s; H N ´kpR d qq @k ď N, B k t P ε P L 8 pp0; T ε q; H N ´k´1{2,N ´k´1 pS ˚qq (3.52)
with bounds uniform in ε in these spaces, such that for all t P r0; T ε s, for all ε ą 0, the domain Ω ε t is included in S ˚. Moreover, pV ε , w ε , ζ ε , P ε q satisfy in the sense of the distributions of D 1 X,z,t pr0, T ε s ˆS˚q the Euler equations (1.7).

The proof follows these steps:

-We claim some regularity for the velocity potential φ solving (1.15) on the fixed strip S " R d p´1;

0q.

-We extend φ to a larger strip than S.

-We recover regularity for the velocity potential Φ defined originally on the water domain by φ " Φ ˝Σε t , after defining it on a large strip containing all the fluid domain Ω ε t for all t, ε.

Regularity of the solutions of the Water Waves problem

According to Theorem 3.2, for all ε, there exists a unique solution pζ ε , ψ ε q to (3.50) on a time interval r0, T ε s, with T only depending on the initial energy (and not on ε), with the following control:

@t P r0; T ε s, E N 1 pU ε qptq ď C 2 (3.53)
with C 2 a constant of the form CpE N 1 pζ 0 , ψ 0 q, 1 hmin , 1 a0 q, where E N 1 is defined by (3.51). We can then prove the following regularity result: Proposition 3.4 Let pζ ε , ψ ε q be the solution of 3.50 given by Theorem 3.2, and T ą 0 such that (3.53) is satisfied. Then, the following regularity results stand:

1) For all 0 ď k ď N , one has:

B k t ζ ε P L 8 pr0; T ε s; H N ´kpR d qq and |B k t ζ ε | L 8 pr0; T ε s;H N ´k pR d qq ď C 2 ;
2) For all 0 ď k ď N ´1, one has:

B k t Pψ ε P L 8 pr0; T ε s; H N ´k´1{2 pR d qq and |B k t Pψ ε | L 8 pr0; T ε s;H N ´k´1{2 pR d qq ď C 2 ,
with C 2 a constant of the form CpE N pζ 0 , ψ 0 q, 1 hmin , 1 a0 q.

Proof Recall that the energy is:

E N pU q " |Pψ| 2 H t 0 `3{2 `Σ αPN d`1 ,|α|ďN |ζ pαq | 2 2 `|Pψ pαq | 2 2 .
1) Since the time derivatives of the unknowns appear in the energy, the control given by (3.53) implies that for all 0 ď k ď N , B k t ζ ε is in L 8 t pH N ´kpR d qq, with the desired estimate.

2) Let us fix k such that 0 ď k ď N ´1. We have to get a control of PB k t B α ψ by Pψ pα,kq . Adapting a proof from Lemma 4.6 in [START_REF] Lannes | The water waves problem[END_REF], one computes:

|PB k t ψ| H N ´k´1{2 ď ÿ βPN d ,|β|ďN ´1´k |PB k t B β ψ| H 1{2 ď ÿ βPN d ,|β|ďN ´1´k p|Pψ pβ,kq | H 1{2 `ε|PpwB k t B β ζq| H 1{2 q
where we used the definition of ψ pβ,kq given by Definition 3.1. We now use the identity |Pf | H 1{2 ď maxt1, µ ´1{4 u|∇f | 2 to conclude:

|PB k t ψ| H N ´k´1{2 ď ÿ βPN d ,|β|ďN ´1´k p|Pψ pβ,kq | H 1 `ε maxt1, µ ´1{4 u|w| H t 0 `1 |B k t ζ| H N ´k q.
One already has by the first point that

|B k t ζ| H N ´k ď C 2 . Using Proposition A.1, we have |w| H t 0 `1 ď µ 3{4 M |Pψ| H t 0 `3{2 ď C 2 .
We now control ψ pβ,kq in H 1 norm, for all |β| ď N ´1 ´k. One computes:

|Pψ pβ,kq | H 1 ď |Pψ pβ,kq | 2 `ÿ δPN d ,|δ|"1 |B β Pψ pβ,kq | 2 ď |Pψ pβ,kq | 2 `ÿ δPN d ,|δ|"1 p|Pψ pβ`δ,kq | 2 `ε|PpB δ wB k t B β ζq| 2 q.
Since |β `δ| `k ď N , the first component of the right hand side from above is controlled by a constant of the form C 2 . For the second component, we use the identity:

DC ą 0, @f P SpR d q, |Pf | ď Cµ ´1{4 |f | H 1{2
where C does not depend on µ. Therefore,

|PpB δ wB β ζq| 2 ď Cµ ´1{4 |B δ wB β ζ| H 1{2 ď Cµ ´1{4 |w| H t 0 `1 |B k t ζ| H N ´k .
We now use as before Proposition A.1 to control µ ´1{4 |w| H t 0 `1 by C 2 and we remark that |Pψ pβ`δ,kq | 2 ď C 2 for all |β `δ| `k ď N . We obtained the control:

|PB k t ψ| H N ´k´1{2 ď C 2
with C 2 independent on the time. Taking the supremum on time over r0; T ε s, one gets the desired result.

l

In order to define a velocity U " ∇ µ,γ Φ on the fluid domain Ω ε t , where Φ solves the Dirichlet-Neumann problem (1.6) on the fluid domain, we first study the regularity of φ where φ is the solution of the Dirichlet-Neumann problem (1.15) on the flat strip S " R d ˆp´1; 0q. We recall the relation φ " Φ ˝Σε t , where Σ ε t is the diffeomorphism defined by (1.14).

Proposition 3.5 Let φ be the unique variational solution to (1.15). We have, for all 0 ď k ď N ´1:

B k t ∇ µ,γ φ P L 8 pp0, T ε q; H N ´k´1{2,N ´k´1 pSqq with |B k t ∇ µ,γ φ| L 8 pp0, T ε q;H N ´k´1{2,N ´k´1 pSqq ď C 2 where C 2 is a constant of the form CpE N pζ 0 , ψ 0 q, 1 hmin , 1 a0 q.
This notation is a bit heavy, but we have to deal with the fact that we only use integer derivatives in z.

Proof Using the notations of the Proposition A.7, we recall that:

φ " A ψ pζ, 0q.

By taking the derivative in time, B k t φ is sum of terms of the form:

d j A B k´j t ψ pζ, 0qpB l1 t ζ, ..., B lj t ζq
with 0 ď j ď k and l 1 `... `lj " j. Let l max " max 1ďmďj l m . We assume l max " l 1 by symmetry. We then have two cases:

1) Case l 1 ď N ´t0 ´1. By using Proposition A.7, with t 1 0 " N ´l1 ´1 (note that t 1 0 ě t 0 ą d{2) and s ď N ´k ´1{2 we have:

|Λ s ∇ µ,γ d j A B k´j t ψ pζ, 0qpB l1 t ζ, ..., B lj t ζq| 2 ď ? µM 0 j ź m"1 |εB lm t ζ| H N ´l1 |PB k´j t ψ| H s ď C 2
where we used the H N ´k´1{2 regularity of PB k´j t ψ given by Proposition 3.4.

2) Case l 1 ě N ´t0 ´1. Let us use Proposition A.8, with t 1 0 " N ´1 ´pk ´l1 q (just note that t 1 0 ě 3{2 ą d{2 for d " 1, 2). We have, for all s ď N ´k ´1{2 (we have N ´k ´1{2 ě t 1 0 ):

|Λ s ∇ µ,γ d j A B k´j t ψ pζ, 0qpB l1 ζ, ..., B lj ζq| 2 ď ? µM 0 |εB l1 t ζ| H s`1{2 ź mą1 |εB lm t ζ| H t 1 0 `1 |PB k´j t ψ| H t 1 0 `1{2 ď C 2 because t 1 0 `1{2 `k ´j ď N ´1{2
, and where we used again the Proposition 3.4 for the H N ´k´1{2 regularity of PB k´j t ψ.

We proved that B k t ∇ µ,γ φ P L 8 pp0, T ε ; H N ´k´1{2 q. Using the last part of Proposition A.7 and Proposition A.8, we can adapt the previous proof to get

B k t ∇ µ,γ φ P L 8 pp0, T ε q; H N ´k´1{2,N ´k´1 pSqq
with the desired control for the norm. l

Extension of the solution φ

We want to extend the distribution φ defined on the strip S " R d ˆp´1; 0q to a larger strip. This is the point of the following result:

Theorem 3.6 Let s P R `and k P N. Let us denote S j " p´pj `1q, jq ˆRd for all j P N. Then, there exists an extension P :

H s,k pS 0 q Ñ H s,k pS j q u Þ Ñ P u such that: @u P H s,k pS 0 q, |u| H s,k pSj q ď Cpk, jq|u| H s,k pS0q (3.54)
where Cpk, jq only depends on k and j.

The proof is postponed to Appendix B. We now have the full artillery to extend the solution Φ defined in the moving domain Ω ε t into a function (with the same regularity) defined on a fixed strip S ˚.

Let Σ ε t be the following diffeomorphism, mapping the flat strip S " R d ˆp´1, 0q into Ω ε t :

Σ ε t : R d`1 Ñ R d`1 pX, zq Þ Ñ pX, p1 `εζpt, Xqqz `εζpt, Xqq.
It is easy to check that Σ ε t is a homeomorphism that maps S exactly into Ω ε t . We now include Ω ε t into a fixed strip S ˚:

Proposition 3.7 With the notations of Theorem 3.6, there exists a strip S k , which does not depend either in ε or t such that:

@0 ď t ď T ε , Ω ε t Ă S k .
Proof Proposition 3.4 gives that:

r0; T ε s ˆRd Ñ R pt, Xq Þ Ñ 1 `εζpt, Xq
is (in particular) in Cpr0; T ε s, H 2 pR d qq with a bound uniform in ε. The continuous embedding H 2 pR d q Ă L 8 pR d q for d ď 2 gives the desired result. l

Now, with the uniform (with respect to t, X, ε) bound of ζpt, Xq, it is easy to check that all strip S j is mapped by Σ ε t into a strip S k : @j P N ˚Dk P N ˚, @t P r0, T ε s, @ε ď 1, Σ ε t pS j q Ă S k .

Definition Let k, l, j be such that:

S l Ă S j Ω ε t Ă S k for all t, ε S l Ă Σ ε t pS j q for all t, ε S k Ă Σ ε t pS l q for all t, ε
We denote S k " S ˚, φ the extension of φ provided by theorem 3.6 into S l , Φ " φ ˝pΣ ε t ´1q and Φ " φ ˝pΣ ε t ´1q.

Φ is the potential on the domain Ω ε t , while Φ is an extension into the fixed strip S ˚.

Remark 3.8 The purpose of this definition is to have a distribution Φ extending Φ and defined on a flat strip S k . We therefore extend φ to S l , and Φ " φ ˝pΣ ε t ´1q is then defined on S k (since it contains Σ ε t pS l q) which also contains Ω ε t . The fact that we need to use three strips instead of two is purely technical (see later the proof of Proposition 3.9).

Regularity after diffeomorphism

We now need to recover regularity for Φ. Proposition 3.9 Using the notations of Definition 3.3, let φ be a distribution on r0; T ε s ˆS with the following regularity:

@0 ď k ď N ´1, B k t ∇ µ,γ φ P L 8 pp0; T ε q; H N ´k´1{2,N ´k´1 pSqq
with a C 2 bound. Let φ be an extension of φ on the strip S l , and Φ " φ ˝Σε t ´1. Then, we have, for all k ď N ´1:

B k t ∇ µ,γ Φ P L 8 pp0; T ε q; H N ´k´1{2,N ´k´1 pS ˚q with a C 2 bound.
The proof is postponed in Section B for the sake of clarity.

Craig-Sulem-Zakharov formulation to Euler formulation

We now build solutions to the free surface Euler equation (1.7) from the potential Φ introduced before. We first prove that Φ satisfies the Bernoulli equations on the domain Ω ε t . One considers the following distribution of D 1 pp0; T ε q; S ˚q:

B t Φ `ε 2 |∇ µ,γ Φ| 2 `1 ε z,
which exists in L 8 pp0; T ε q; H N ´1{2,N ´1pS ˚qq.

Let us consider a measurable function F of the variable t, such that for almost every t P p0; T ε q, F is equal to this quantity. For almost every t, F is an element of H N ´1{2,N ´1pS ˚qq which trace on the boundary z " εζ of Ω ε t is equal to zero (this statement implies the second equation of the Craig-Sulem-Zakharov formulation (3.50)). For a complete and rigorous proof of this statement, see [START_REF] Alazard | The water-wave equations: from Zakharov to Euler[END_REF]. In particular, for almost every t, F is equal to an element of

H N ´1{2,N ´1 0,surf pS ˚q " S ˚ztz " εζu ||.|| H N ´1{2,N ´1pS ˚q .
We denote this element ´1 ε P . Of course, P corresponds physically to the pressure. We can rewrite this as the Bernoulli equation:

B t Φ `ε 2 |∇ µ,γ Φ| 2 `1 ε z " ´1 ε P (3.55)
in D 1 pp0; T ε q; Ω ε t q, where pp0; T ε q; Ω ε t q " tpt, X, zq P p0; T ε q ˆS˚, pX, zq P Ω ε t u (note that this is an open set of p0; T ε q ˆS˚) .

Remark 3.10 It is easy to deduce from the Bernoulli equation (3.55) that for all k ď N 0 :

B k t P P L 8 pp0; T ε q; H N ´k´1{2,N ´k´1 pS ˚q with a bound uniform in ε.

Now, defining U " ∇ µ,γ Φ, we have immediately $ ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' %

B t V `εpV ¨∇γ `1 µ wB z qV " ´1 ε ∇ γ P in D 1 pp0; T ε q; Ω ε t q B t w `εpV ¨∇γ `1 µ wB z qw " ´1 ε pB z P `1q in D 1 pp0; T ε q; Ω ε t q B t ζ `εV ¨∇γ ζ ´1 µ w " 0 in D 1 pp0; T ε q; R d q
∇ µ,γ ¨pU q " 0 in D 1 pp0; T ε q; Ω ε t q curl µ,γ pU q " 0 in S Ů ¨n " 0 on z " ´1 -Note that the regularity of U " ∇ µ,γ Φ is given by Proposition 3.9

4 The lack of strong convergence from Euler to rigid-lid

We now have a proper set of solutions pU ε , ζ ε , P ε q to the free surface Euler equations (1.7), defined in the following spaces:

$ ' & ' % @k ď N ´1, B k t U ε P L 8 pp0; T ε q; H N ´k´1{2,N ´k´1 pS ˚qq @k ď N ´1, B k t ζ ε P L 8 pr0; T ε s; H N ´kpR d qq @k ď N ´1, B k t P ε P L 8 pr0; T ε s; H N ´k´1{2,N ´k´1 pS ˚qq (4.57)
with bounds uniform in ε of the form CpE N pζ 0 , ψ 0 q, 1 hmin , 1 a0 q. In order to prove the weak convergence to the solutions of the rigid lid model, we need to prove some compactness results in the Beppo-Levy spaces in order to extract a convergent sub-sequence of solutions.

Compactness result

We recall that A ĂĂ B means that there exists a compact K such that A Ă K Ă B. Lemma 4.1 Let k ě 1. We have the following compact embedding:

H s,k loc pS ˚q Ă H s´1,k´1 loc pS ˚q @1 ď k ď s
Proof The proof remains the same as for the Rellich-Kondrachov theorem, in the case of Sobolev spaces (see [START_REF] Brezis | Analyse fonctionnelle[END_REF]). Let ω ĂĂ S ˚.

1) We first show the compact embedding of H s,1 pωq into H s´1,0 pωq. We write, for h " ph 1 , h 2 q P R ˆRd , |h| ă dpω, pS ˚qA q and u P C 8 pS ˚q: ż ω |pΛ s´1 uqpX ´h1 , z ´h2 q ´pΛ s´1 uqpX, zq| and the result stands true for all u P H s,1 pS ˚q by density of C 8 pS ˚q in this space.

It follows from the theorem of Riesz-Frechet-Kolmogorov that if pu n q n is bounded in H s,1 pS ˚q, then pΛ s´1 u n|ω q n is precompact in L 2 pωq. Such result is true for all ω ĂĂ S ˚and therefore, by a diagonal sub-sequence argument, one proves the compact embedding H s,1 loc pS ˚q into H s´1,0 loc pS ˚q.

2) For the general case, let k P N ˚and s ě k, let pu n q n be bounded in H s,k`1 pS ˚q with s ě k `1. By the definition of Beppo-Levy spaces (recall Section 1.5), to prove the convergence of a sub-sequence in H s´1,k loc pS ˚q, one must prove the existence of a sub-sequence pu ϕpnq q n such that for all 0 ď l ď k, the sequence pB l z u ϕpnq q n converges in L 2 loc pp´pk `1q, kq, H s´1´l loc pR d qq (recall that S ˚" p´pk `1q; kq). From the case k " 1, there exists u P L 2 pS ˚q and a sub-sequence pu ϕpnq q n convergent to u in L 2 loc pS ˚q. In particular, pB l z u ϕpnq q n is convergent to B l z u in D 1 pS ˚q. But one has also pB l z u ϕpnq q n P H s´l,k`1´l pS ˚q and therefore this sequence converges, up to a sub-sequence, in L 2 loc pp´pk `1q, kq, H s´1´l loc pR d qq, since k `1 ´l ě 1. By uniqueness of the limit in D 1 pS ˚q, the limit is B l z u. l

Extraction of a convergent sub-sequence for U ε

We now perform the rigid lid time scaling: t 1 " εt and the change of unknown P 1 " 1 ε 2 pP `zq. We now have U ε P L 8 pp0; T q; H N ´1{2,N ´1pS ˚qd`1 q, ζ ε P L 8 pp0; T q; H N pR d qq and the following equations are satisfied: $ ' ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' ' %

B t V ε `pV ε ¨∇γ `1 µ w ε B z qV ε " ´1 ε ∇ γ P ε in D 1 pp0; T q; Ω ε t q B t w ε `pV ε ¨∇γ `1 µ w ε B z qw ε " ´1 ε pB z P ε q in D 1 pp0; T q; Ω ε t q B t ζ ε `V ε ¨∇γ ζ ε ´1 µε 2 w ε " 0 in D 1 pp0; T q; R d q ∇ µ,γ ¨pU ε q " 0 in D 1 pp0; T q; Ω ε t q curl µ,γ pU ε q " 0 in S Ů ε ¨n " 0 on z " ´1 (4.58)
We now prove the following Theorem:

Theorem 4.2 There exists U, P, ζ distributions of D 1 pp0; T q; Ωq such that U P H N ´3{2,N ´2pS ˚q, U P H N ´5{2,N ´3pS ˚q, ζ P H N ´1pR d q are solutions in the distributional sense of D 1 pp0; T q; Ωq of the rigid lid equations (1.8). Moreover, the solutions of (4.58) converge in the distributional sense of D 1 pp0; T q; Ω ε t q to these solutions of the rigid-lid equation 1.8. The convergence of pU ε q ε is not strong in L 8 pp0; T q; L 2 pΩqq. By Lemma 4.1, there exists U ε P H N ´3{2,N ´2pS ˚q such that pU ε q ε converges up to a sub-sequence to U in H N ´3{2,N ´2 loc pS ˚q. Moreover, since pU ε q ε P L 8 pp0; T q; H N ´1{2,N ´1pS ˚qd`1 q, one has also U P L 8 pp0; T q; H N ´3{2,N ´2pS ˚qd`1 q. Therefore, pU ε q ε converges up to a sub-sequence to U in D 1 pp0; T q; S ˚q. Therefore, pB t U ε q ε converges (up to a sub-sequence) to B t U in D 1 pp0; T q; S ˚q. In view of taking the limit as ε goes to zero in the first equation of (4.58), one must prove the convergence of the non-linear terms. We do it for the term V ε ¨∇γ V ε . One uses the following estimate, which proof can be found in [START_REF] Lannes | The water waves problem[END_REF] Corollary B.5.

|f g| H N ´3{2,N ´2 ď C|f | H N ´1{2,N ´1 |g| H N ´3{2,N ´2
for all N ´2 ě d{2 (which is true for our choice of N , since N ´2 ě t 0 `3{2 ą d{2). Such estimate proves that V ε ¨∇γ V ε P L 8 H N ´3{2,N ´2, and therefore one has the convergence of this term (up to a sub-sequence) as ε goes to zero, in D 1 pp0; T q; S ˚q. It proves, using the equation (4.58) that p∇ γ P ε q ε converges in D 1 pp0; T q; S ˚q to an element G. By De Rham's Theorem, there exists P P D 1 pp0; T q; S ˚q such that G " ∇ γ P . The same study can be done for the second equation of 4.58 and one obtains, passing to the limit as ε goes to zero: B t V `pV ¨∇γ `1 µ wB z qV " ´∇γ P in D 1 pp0; T q; Ωq B t w `pV ¨∇γ `1 µ wB z qw " ´1 ε pB z P q in D 1 pp0; T q; Ω with U " pV, wq P L 8 pp0; T q; H N ´3{2,N ´2pS ˚qd`1 , ∇ µ,γ P P L 8 pp0; T q; H N ´5{2,N ´3pS ˚qd`1 .

We now pass to the limit in the third equation (on ζ ε ) of (4.58). The same reasoning as before proves that pζ ε q ε converges up to a sub-sequence to a ζ P H N ´1pR d q in H N ´1 loc pR d q. Therefore, the same convergence occurs in D 1 pp0; T q; S ˚q, and pB t ζq ε converges up to a sub-sequence to B t ζ in D 1 pp0; T q; S ˚q. We prove as before the convergence of the bi-linear term pV ε ¨∇γ ζ ε q ε in D 1 pp0; T q; S ˚q. Multiplying the third equation of 4.58 by ε 2 , one finds at the limit the following equation:

w " 0 in D 1 pp0; T q; Ω ε t q. Finally, one obtains at the limit ε goes to zero, the following equations: $ ' ' ' ' ' ' ' ' ' ' ' & ' ' ' ' ' ' ' ' ' ' ' % B t V `pV ¨∇γ `1 µ wB z qV " ´∇γ P in Ω B t w `pV ¨∇γ `1 µ wB z qw " ´pB z P q in Ω w " 0 ∇ µ,γ ¨U " 0 in Ω curl µ,γ pU q " 0 in Ω U ¨n " 0 for z " ´1 and the study above prove the following regularity for the unknowns: U P H N ´3{2,N ´2pS ˚q, ∇ µ,γ P P U P H N ´5{2,N ´3pS ˚q, ζ P H N ´1pR d q. The sequence pU ε q ε does not converge in L 8 pp0; T q; L 2 pΩq. Indeed, it would imply the strong convergence of Pψ in L 2 pR d q which is not satisfied, according to Theorem 2.6. l

A The Dirichlet Neumann Operator

Here are for the sake of convenience some technical results about the Dirichlet Neumann operator, and its estimates in Sobolev norms. See [START_REF] Lannes | The water waves problem[END_REF] Chapter 3 for complete proofs. The first two propositions give a control of the Dirichlet-Neumann operator.

Then, for all ψ 1 , ψ 2 P .

H s`1{2 pR d q, for all ph, kq P H t0`1 pR d q one has

|pΛ s d j Gph, kqψ 1 , Λ s ψ 2 q| ď µM 0 j ź m"1 |pεh m , βk m q| H t 0 `1 |Pψ 1 | s |Pψ 2 | s ,
where M 0 is a constant of the form Cp 1 h0 , |ζ| H t 0 `1 , |b| H t 0 `1 q.

The following Proposition gives the same type of estimate that the previous one:

Proposition A.5 Let t 0 ą d{2 and pζ, bq P H t0`1 be such that Dh 0 ą 0, @X P R d , εζpXq ´βbpXq `1 ě h 0 .

Then, for all 0 ď s ď t 0 `1{2,

|d j Gph, kqψ| H s´1{2 ď M 0 µ 3{4 j ź m"1 |pεh m , βk m q| H t 0 `1 |Pψ| H s
We need the following commutator estimate:

Proposition A.6 Let t 0 ą d{2 and ζ, b P H t0`2 pR d q such that:

Dh 0 ą 0, @X P R d , εζpXq ´βbpXq `1 ě h 0
For all V P H t0`1 pR d q 2 and u P H 1{2 pR d q, one has

ppV ¨∇γ uq, 1 µ Guq ď M |V | W 1,8 |Pu| 2 2 ,
where M is a constant of the form Cp 1 h0 , |ζ| H t 0 `2 , |b| H t 0 `2 q.

In order to have regularity estimates of the potential Φ (recall that it solves the Dirichlet-Neumann problem (1.4)), we need to introduce the following mapping:

Definition Let t 0 ą d{2.

(1) We denote by Γ Ă H t0`1 pR d q 2 the open set:

Γ " tΓ " pζ, bq P H t0`1 pR d q 2 , Dh 0 ą 0, @X P R d , εζpXq `1 ´βbpXq ě h 0 u

(2) One can define, for all 0 ď s ď t 0 `1{2 and ψ P 9 H s`1{2 pR d q a mapping A ψ as

A ψ : Γ Ñ 9 H s`1 S Γ Þ Ñ Φ Γ
where Φ Γ is the unique variational solution to (1.15).

One can prove the analicity of A ψ , which means that the solution φ Γ of (1.15) is analytic with respect to the boundaries. The following result then gives some estimates about the derivatives of the mapping A ψ Proposition A.7 Let t 0 ą d{2 and 0 ď s ď t 0 `1{2, ψ P H s`1{2 pR d q, and Γ " pζ, bq P Γ. For all j P N and ph, kq " ph 1 , ..., h j , k 1 , ..., k j q P H t0`1 pR d q 2 , one has

|Λ s ∇ µ,γ d j A ψ pΓqph, kq| 2 ď ? µM 0 j ź m"1 |pεh m , βk m q| H t 0 `1 |Pψ| H s ,
where M 0 is a constant of the form Cp 1 h0 , |ζ| H t 0 `1 , |b| H t 0 `1 q. If s " t 0 `1{2, then the same estimate holds on ||∇ µ,γ d j A ψ pΓqph, kq|| H t 0 `1{2,k pSq for k ď t 0 `1{2.

In Proposition A.7 below, we need at least the H t0`1{2 -norm of the component ph, kq, even if s " 0. The following proposition allows to relax this constraint, by using only the H s`1{2 norm of the first component of ph, kq.

Proposition A.8 Let t 0 ą d{2 and 0 ď s ď t 0 , ψ P H t0`1{2 pR d q, and Γ " pζ, bq P Γ. For all j P N and ph, kq " ph 1 , ..., h j , k 1 , ..., k j q P H t0`1 pR d q 2 , one has

|Λ s ∇ µ,γ d j A ψ pΓqph, kq| 2 ď ? µM 0 |pεh 1 , βk 1 q| H s`1{2 j ź mą1 |pεh m , βk m q| H t 0 `1 |Pψ| H t 0 `1{2 ,
where M 0 is a constant of the form Cp 1 h0 , |ζ| H t 0 `1 , |b| H t 0 `1 q.

B Extensions on Beppo-Levy Spaces

This section contains proofs of the existence and regularity of extensions used in Section 3.

Theorem B.1 Let s P R `and k P N. Let us denote S j " p´pj `1q, jq ˆRd for all j P N. Then, there exists an extension P : H s,k pS 0 q Ñ H s,k pS j q u Þ Ñ P u such that: @u P H s,k pS 0 q, |u| H s,k pSj q ď Cpk, jq|u| H s,k pS0q (B.60)

where Cpk, jq only depends on k and j.

Proof We first construct an extension to H s,k pp´1, 1q ˆRd q.

The proof requires a small adaptation from the case k " 1 which is proved in [START_REF] Brezis | Analyse fonctionnelle[END_REF]. When k " 1, u is extended by reflection: ũpX, zq " " upX, zq @z P p´1, 0q upX, ´zq @z P p0, 1q.

But such extension ũ is indeed in H s,1 pS 1 q but is not in general in H s,2 pS 1 q since derivatives in z of ũ differs for z " 0: B z ũpX, 0 `q ‰ ´Bz ũpX, 0 ´q

We should therefore define an extension which derivatives of order i ď k ´1 have the same value in 0 ànd in 0 ´. We are looking for an extension under the form: P upX, zq " " upX, zq @z P p´1, 0q ř k´1 i"0 c i upX, ´αi zq @z P p0, 1q.

The condition over the α i and c i , 0 ď i ď k ´1 is: @0 ď j ď k ´1, k´1 ÿ i"0 c i p´α i q j B j z upX, 0 `q " B j z upX, 0 ´q i.e. ř k i"0 c i p´α i q j " 1. We should therefore find pc 0 , ..., c k´1 q and pα 0 , ..., α k´1 q such that: ¨1 ¨¨¨1 ´α1 ¨¨¨´α k´1 p´α 1 q 2 ¨¨¨p´α k´1 q 2 . . . . . . . . . which is a Vandermonde system. It suffices to take the α i distincts, non zero, and taken in p0, 1q in order for upX, ´αi zq to make sense. Then the c i are defined as solutions of the Vandermonde system (B.61).

Now, let us prove that such defined extension P u maps continuously H s,1 pS 1 q into H s,1 pp´1, 1q ˆRd q.

1) Case u P C 8 pS 0 q It is clear that P u is measurable. Let us check that it is L 2 pp´1, 1q, H s pR d q:

ż 1 ´1 ż R d Λ 2s |P upX, zq| 2 dXdz " ż 0 ´1 ż R d Λ 2s |upX, zq| 2 dXdz `ż 1 0 ż R d Λ 2s | k´1 ÿ i"0 c i upX, ´αi zq| 2 dXdz ď |u| 2 H s,k pp´1,0qˆR d q `k´1 ÿ i"0 c 2 i α i ż 0 ´αi ż R d Λ 2s |upX, zq| 2 dXdz ď C k |u| 2
H s,k pp´1,0qˆR d q . The first inequality is Cauchy-Schwarz's inequality. The constant C k only depends on k. Now, let us check that P u is H k´j pp´1, 1q; H s´k`j pR d qq for all 0 ď j ď k. Let ϕ P C 8 0 pp´1, 1q ˆRd q. One set j such that 0 ď j ď k, and computes B j z pP uq in the distributional sense of D 1 pp´1, 1q ˆRd q: by integrating by parts, (recall that u P C 8 pS 1 q). Note that these calculus does not make sense for u P H s,k pS 0 q since ϕ is not in C 8 0 pS 0 q.

"For all N P N and θ P L 8 p0; T 0 ; H N `1{2,N pS j qq if Θ " θ ˝Σε t ´1, then we have: where C does not depend on ε"

B k
1) k = 0
Let us prove that Θ P L 8 p0; T 0 , H N `1{2,N pS ˚qq. We are using the following characterization of H N `1{2 pR d q:

|u| 2

H N `1{2 pR d q " |u| 2

H N pR d q `żR d ż R d
|D N upxq ´DN upyq| 2 |x ´y| 2p1{2`d{2q dxdy Let first prove that Θ P L 8 pp0; T 0 q; L 2 p´pk `1q; kq, H N `1{2 pR d qq with the control (B.65) by induction on N . Let the induction hypothesis be:

"For for all θ P L 8 pp0; T 0 q; L 2 p´pl `1q; lq, H N `1{2 pR d qq,if Θ " θ ˝Σε t ´1 then we have:

Θ P L 8 pp0; T 0 q; L 2 p´pk `1q; kq, H N `1{2 pR d qq with a norm control such as (B.65)"

For N " 0, we need the previous characterization of H 1{2 pR d q: with C uniform in ε. We used the change of variable pX, zq " Σ ε t pX, uq and pY, uq " Σ ε t pY, vq in the integrals. The first inequality comes from S k Ă Σ ε t pS l q. The last comes from S l Ă Σ ε t pS j q.

ż R d ż R d ż k ´pk`1q
Suppose the result true for N ´1 with N ě 1. Let prove it for N . We have: ∇ X,z Θ " ∇ X,z pθpX, z ´εζ h B qq " p∇ X,z θqpX, z ´εζ h B q `∇X,z p z ´εζ h B qpB z θqpX, z ´εζ h B q

We have p∇ X,z θq P L 8 pp0; T 0 q; L 2 p´pk `1q; kq, H N ´1`1{2 pR d qq, so by induction hypothesis, this term is controlled. For the latter one, just remark that ∇ X,z p z´εζ hB q P H t0 pR d q , so with classical product estimates: using the induction hypothesis and S l Ă S j . The constant C does not depends either on t or ε, so the result is proved.

ż k ´pk`1q
2) Now, suppose this result is proved for k ´1, with k ě 1. Let prove it for k. We write:

B t Θ " B t θpt, X, z ´εζ h B q " pB t θqpt, X, z ´εζ h B q `Bt p z ´εζ h B qpB z θqpt, X, z ´εζ h B q Since B t θ is L 8 p0; t0q; H N ´1`1{2,N ´1pR d q, by induction hypothesis the pk ´1q´th time derivative of the first term of the r.h.s. is controlled as in (B.65).

The same argument stands for pB z θqpt, X, z´εζ hB q. The term B t p z´εζ hB q is harmless, since it is in L 8 pp0; T 0 q; H N ´1`1{2,N ´1q with N ´1 ě t 0 , so standard product estimates in Sobolev spaces will finally give the control of the second term of the pk ´1q´th time derivative of the r.h.s. l

The author has been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01.

Remark 3 . 11 -

 311 The third equation of the Euler system (3.56) is just the first equation of the Zakharov-Craig-Sulem equation.

2

 2

c

  i p´α i q j pB j z uqpX, ´αi zqϕpX, zqdXdz

  ´1 2 |∇ γ ψ| 2 ´1 µ Gpεζqψq 2 ˙(2.41)

										|∇ γ ψ| 2 ´1 µ	pGpεζqψq 2 `εRpψq	(2.38)
	with
	|Rpψq| 2 L ď C N , where C N is given by (2.33). Using (2.36) with the control (2.37), and (2.38) with the control (2.39), (2.39)
	one gets:	F p ˆζ ψ	˙q " Bp ˆζ ψ	˙q,	ˆζ ψ	˙qq `εR	ˆζ ψ	˙q	(2.40)
	with	Bp ˆζ ψ	˙q,	ˆζ ψ	˙qq " ˆ1 µ G 1 ψ
	and				|R	ˆζ ψ	˙q| 2 ď C N .	(2.42)
	We therefore get, using (2.40) and (2.42) (recall that C N is a constant of the form (2.33)):

  Λ s uqpX ´th 1 , z ´th 2 q.ph 1 , h 2 qdt| 2 dXdz. |∇ X,z Λ s´1 upX, zq| 2 |h| 2 dtdXdz

	2 dXdz |pΛ s´1 uqpX ´h1 , z ´h2 q ´pΛ s´1 uqpX, zq| 2 dXdz | ż 1 0 ż ω ω ż ż ď S ˚ż 1 0 p∇ X,z We now use Jensen's inequality to write: ď ď |u| 2 H s,1 pS ˚q|h|

  t Θ P L 8 p0; T 0 ; H N ´k`1{2,N ´kpS ˚qq with a control |B k t Θ| L 8 pp0;T0q;H N ´k`1{2,N ´kpS ˚qq ď C|B k t θ| L 8 pp0;T0q;H N ´k`1{2,N ´kpS j qq (B.65)

  |ΘpX, zq ´ΘpY, zq| 2 |X ´Y | 2p1{2`d{2q dXdY dz |θpX, vq ´θpY, vq| 2 |X ´Y | 2p1{2`d{2q |J Σ ε t pXq||J Σ ε t pY q|dXdY dv ď C|θ| L 8 pp0;T0q;H 1{2,0 pS j qq

	" " ď "	ż R d ż Σ ε ż t ´1pkqz R d ż k ´pk`1q Σ ε t ´1p´pk`1qqz ż R d |θ ˝Σε t ´1pX, zq ´θ ˝Σε t ´1pY, zq| 2 |X ´Y | 2p1{2`d{2q ż R d |θpX, uq ´θ ˝Σε t ´1pY, uq| 2 |X ´Y | 2p1{2`d{2q ż l ´pl`1q ż R d ż R d |θpX, uq ´θ ˝Σε t ´1pY, uq| 2 |J Σ ε t pXq|dXdY du dXdY dz |J Σ ε t pXq|dXdY du |X ´Y | 2p1{2`d{2q ż Σ ε t ´1plqz Σ ε t ´1p´pl`1qqz ż R d ż R d |θpX, vq ´θpY, vq| 2 |X ´Y | 2p1{2`d{2q |J Σ ε t pXq||J Σ ε t pY q|dXdY dv ż j ż ż
	ď	´pj`1q	R d	R d

  ´1`1{2 pR d q dz " C|pB z θq ˝Σε t ´1| L 2 pp´pk`1q;kq;H N ´1`1{2 pR d qq ď C|B z θ| L 2 pp´pl`1q;lq;H N ´1`1{2 pR d qq ď C|B z θ| L 2 pp´pj`1q;jq;H N ´1`1{2 pR d qq

	|∇ X,z p ´pl`1q ż l |∇ X,z p z ´εζ h B z ´εζ qpB z θqpX, h B q| 2 H t 0 |pB z θqpX, z ´εζ h B ď C ď ż k ´pk`1q |pB z θqpX, z ´εζ h B q| 2 H N	q| 2 H N ´1`1{2 pR d q dz z ´εζ h B q| 2 H N ´1`1{2 pR d q dz

Proposition A.1 Let t 0 >d/2, 0 ď s ď t 0 `3{2 and pζ, βq P H t0`1 X H s`1{2 pR d q such that Dh 0 ą 0, @X P R d , εζpXq ´βbpXq `1 ě h 0 [START_REF] Alazard | The water-wave equations: from Zakharov to Euler[END_REF] The operator G maps continuously . H s`1{2 pR d q into H s´1{2 pR d q and one has |Gψ| H s´1{2 ď µ 3{4 M ps `1{2q|Pψ| H s , where M ps `1{2q is a constant of the form Cp 1 h0 , |ζ|

(2) The operator G maps continuously .

H s`1 pR d q into H s´1{2 pR d q and one has

where M ps `1q is a constant of the form Cp 1 h0 , |ζ|

Moreover, it is possible to replace G by w in the previous result, where w " Gψ`εµ∇ γ ζ¨∇ γ ψ 1`ε 2 µ|∇ γ ζ| 2 (vertical component of the velocity U " ∇ X,z Φ at the surface).

Proposition A.2 Let t 0 ą d{2, and 0 ď s ď t 0 `1{2. Let also ζ, b P H t0`1 pR d q be such that Dh 0 ą 0, @X P R d , 1 `εζpXq ´βbpXq ě h 0 Then, for all ψ 1 , ψ 2 P . H s`1{2 pR d q, we have

The second result gives a control of the shape derivatives of the Dirichlet-Neumann operator. More precisely, we define the open set Γ Ă H t0`1 pR d q 2 as:

Γ " tΓ " pζ, bq P H t0`1 pR d q 2 , Dh 0 ą 0, @X P R d , εζpXq `1 ´βbpXq ě h 0 u and, given a ψ P . H s`1{2 pR d q, the mapping:

We can prove the differentiability of this mapping. The following Theorem gives a very important explicit formula for the first-order partial derivative of G with respect to ζ:

Then, for all h P H t0`1 pR d q, one has dGphqψ " ´εGphwq ´εµ∇ γ ¨phV q, with w " Gψ `εµ∇ γ ζ ¨∇γ ψ 1 `ε2 µ|∇ γ ζ| 2 , and V " ∇ γ ψ ´εw∇ γ ζ.

The following result gives estimates of the derivatives of the mapping (A.59).

Proposition A.4 Let t 0 >d/2, 0 ď s ď t 0 `1{2 and pζ, βq P H t0`1 pR d q such that:

Dh 0 ą 0, @X P R d , εζpXq ´βbpXq `1 ě h 0 Since we have the identity @0 ď j ď k ´1, k´1 ÿ i"0 c i p´α i q j B j z upX, 0q " B j z upX, 0q

the two integrals over R d cancel one another. Therefore, we have:

i"0 c i pα i q j pB j z uqpX, ´αi zqϕpX, zqdXdz (B.62) which prove that B j z pP uq " p1 ´sgnpzqqB j z upX, zq `p1 `sgnpzqq k´1 ÿ i"0 c i p´α i q j pB j z uqpX, ´αi zq on D 1 pp´1, 1q ˆRd q, with the notation sgnpzq " 1 if z ě 0 and sgnpzq " ´1 if z ă 0. It is then quite easy to check that B j z pP uq P H k´j pp´1, 1q; H s´k`j pR d q (proceed as for case j " 0) with

By density of C 8 pS 0 q in H s,k pS 0 q, it is easy to check that (B.62) stands true in H s,k pS 0 q, and the result is proved, with the control (B.63). We can construct by exactly the same way an extension of u P H s,k pS 0 q into H s,k pp´2, 0q ˆRd q and then combine the two extensions to have an extension to H s,k pS 1 q. By the same way it is easy to construct the extension into H s,k pS k q. Finally, the main of the theorem is proved, with the control (B.60). l Proposition B.2 Using the notations of Definition 3.3, let φ be a distribution on r0; T 0 s ˆS with the following regularity: @0 ď k ď N, B k t ∇ µ,γ φ P L 8 pp0; T 0 q; H N ´k´1{2,N ´k´1 pSqq Let φ be an extension of φ on the strip S l , and Φ " φ ˝Σε t ´1. Then, we have, for all k ď N :

B k t ∇ µ,γ Φ P L 8 pp0; T 0 q; H N ´k´1{2,N ´k´1 pS ˚q with a bound uniform in t and ε.

Proof The proof consists in proving that if Θ is a function defined on r0; T 0 s ˆS˚, with θ " Θ ˝Σε t such that: @k ď N, B k t θ P L 8 pp0; T 0 q; H N ´k`1{2,N ´kpS j qq then we have the same regularity for Θ: @k ď N, B k t Θ P L 8 pp0; T 0 q; H N ´k`1{2,N ´k pS ˚qq with a control of the norm: Let us do it by induction on k. The induction hypothesis is the following:


Athmane Bakhta

Virginie Ehrlacher

Global existence of bounded weak solutions to degenerate cross-diffusion equations in moving domain

Introduction

This note focuses on some issues for the analysis of a system of degenerate cross-diffusion partial differential equations (PDEs). This family of models are encountered in a wide variety of contexts, such as population dynamics [START_REF] Jüngel | Boundedness of weak solutions to crossdiffusion systems from population dynamics[END_REF], biology [START_REF] Hillen | A user's guide to PDE models for chemotaxis[END_REF][START_REF] Painter | Continuous models for cell migration is tissues and applications to cell sorting via differential chemotaxis[END_REF], chemistry or materials science. The application we have in mind here is the modeling of the evolution of the concentration of chemical species composing a crystalline solid. The functions, that are the solutions of the system of PDEs of interest, represent the local densities of the different components of the material, and thus should be nonnegative, bounded and satisfy some volumic constraints which will be made precise later in the note. These systems are useful for instance for the prediction of the chemical composition of thin solid films grown by Chemical Vapor Deposition (CVD) [START_REF] Schropp | Hot wire CVD of heterogeneous and polycrystalline silicon semiconducting thin films for application in thin film transistors and solar cells[END_REF]. In this process, a solid wafer is exposed to gaseous precursors, corresponding to the different species entering the composition of the film, which react or decompose on the substrate surface to produce the desired deposit. This process generally occurs at high temperature and takes several hours, so that the diffusion of the different atomic species within the bulk of the solid has to be taken into account in addition to the evolution of the surface of the film.

More precisely, let d ∈ N * (the space dimension), T > 0 (the duration of the process), and for all t ∈ [0, T ], let Ω t ⊂ R d denote a bounded open domain occupied by the solid at time t. Let us assume that there are n + 1 different atomic species (with n ∈ N * ) composing the material, whose local concentrations are respectively denoted by u 0 (t, x), u 1 (t, x), • • • , u n (t, x) for t ∈ (0, T ) and x ∈ Ω t .

The system of equations we consider here reads:

∂ t u i -div x   0≤j =i≤n K ij (u j ∇ x u i -u i ∇ x u j )   = 0 in Ω t , t ∈ (0, T ), ∀ 0 ≤ i ≤ n, (1)
where for all 0 ≤ i = j ≤ n, the non-negative real number K ij = K ji ≥ 0 denotes the cross-diffusion coefficient of atoms of type i with atoms of type j. This system of equations can be formally derived as the continuous limit of some discrete stochastic 1 lattice hopping model. We do not explain the derivation of these equations here for the sake of brievity and refer the reader to [START_REF] Bakhta | Global existence of bounded weak solutions to degenerate cross-diffusion equations in a moving domain[END_REF][START_REF] Burger | Nonlinear Cross-Diffusion with Size Exclusion[END_REF][START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF] for more details. The initial condition (u 0 0 , u 0

1 , • • • , u 0 n ) ∈ L 1 (Ω 0 ; R n+1
) of this system is assumed from now on to satisfy the constraints

∀ 0 ≤ i ≤ n, for almost all x ∈ Ω 0 , u 0 i (x) ≥ 0, n i=0 u 0 i (x) = 1 and u i (0, x) = u 0 i (x).
(2) The last constraint n i=0 u 0 i (x) = 1 is a natural volumic constraint which encodes the fact that each site of the crystalline lattice of the solid has to be occupied (vacancies being treated as a particular type of atomic species).

From a physical point of view, it is naturally expected that the solutions (u 0 , • • • , u n ) of such a system satisfy similar constraints as the initial condition, i.e. for all t ∈ (0, T ),

∀ 0 ≤ i ≤ n, ∀t ∈ [0, T ], for almost all x ∈ Ω t , u i (t, x) ≥ 0 and n i=0 u i (t, x) = 1. (3)
Indeed, the last equality can be infered at least formally from (1) since by summing all these equations for 0 ≤ i ≤ n, we obtain that ∂ t ( n i=0 u i ) = 0. Despite their importance in chemistry or biology, it appears that the mathematical analysis of these systems, taking into account constraints (3), is quite recent [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF][START_REF] Griepentrog | Maximal regularity for nonsmooth parabolic problems in Sobolev-Morrey spaces[END_REF][START_REF] Liero | Gradient structures and geodesic convexity for reactiondiffusion systems[END_REF][START_REF] Burger | Nonlinear Cross-Diffusion with Size Exclusion[END_REF]. Let us first consider the case of a fixed domain, i.e. when Ω t = Ω 0 for all t ∈ [0, T ], and of no flux boundary conditions

0≤j =i≤n K ij (u j ∇u i -u i ∇u j ) • n = 0 on (0, T ) × ∂Ω 0 , ∀ 0 ≤ i ≤ n, (4)
where n denotes the outward normal unit vector to ∂Ω 0 . When all the coefficients K ij are identically equal to some constant K > 0, it can be easily seen that ( 1) boils down to a set of decoupled heat equations

∀ 0 ≤ i ≤ n, ∂ t u i -div x [K∇ x u i ] = 0 in (0, T ) × Ω 0 ,
with Neumann boundary conditions K∇ x u i • n = 0 on (0, T ) × ∂Ω 0 whose analysis becomes trivial. However, at least to our knowledge, the first proof of existence of global weak solutions of (1) satisfying constraints (3) with non-identical cross-diffusion coefficients is given in [START_REF] Burger | Nonlinear Cross-Diffusion with Size Exclusion[END_REF] for n = 2 with coefficients K ij such that K i0 > 0 for i = 1, 2 and K 12 = K 21 = 0. These results were later extended in [START_REF] Jüngel | Analysis of degenerate cross-diffusion population models with volume filling[END_REF] to a general number of species n ∈ N * with cross-diffusion coefficients satisfying K i0 > 0 and K ij = 0 for all 1 ≤ i = j ≤ n; the authors proved in addition the uniqueness of such weak solutions.

In [START_REF] Jüngel | Boundedness of weak solutions to crossdiffusion systems from population dynamics[END_REF], the case n = 2 with arbitrary positive coefficients K ij > 0 is covered, though no uniqueness result is provided. The main difficulty of the mathematical analysis of such equations relies in the bounds (3), which are not obvious since no maximum principle can be proved for these systems in general. In all the articles mentioned above, the analysis framework used by the authors is the so-called boundedness by entropy method, which was first used in [START_REF] Burger | Nonlinear Cross-Diffusion with Size Exclusion[END_REF] in the particular case mentioned above and then further developped by Jüngel in [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF] to a more general theoretical setting. The main idea is to write the above system of equations as a formal gradient flow and derive estimates on the solutions (u 0 , • • • , u n ) using the decay of some well-chosen entropy functional. The diffusivity matrix [START_REF] Zinsl | Transport distances and geodesic convexity for systems of degenerate diffusion equations[END_REF] obtained for these systems is not a concave function of the densities though, so that standard gradient flow theory arguments (such as the minimizing movement method) cannot be applied in this context. In the first part of this note, we prove that for any n ∈ N * , in the case when K ij = K ji > 0 for all 0 ≤ i = j ≤ n, the above system can be analyzed using the general theoretical framework introduced in [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF]. The proof heavily relies on ideas of [START_REF] Jüngel | Boundedness of weak solutions to crossdiffusion systems from population dynamics[END_REF]. Uniqueness of the solutions remains an open issue.

Interestingly, it seems that very few works deal with the mathematical analysis of such cross-diffusion systems in the case of moving boundary domains, despite their importance for instance in the modelisation of the CVD processes mentioned above. In the second part of this note, we present preliminary existence results for one-dimensional systems with moving boundaries, extending existence results obtained in the general theoretical framework introduced by Jüngel [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF].

Existence on a fixed domain: formal gradient flow structure

For the sake of completeness, let us recall here Theorem 2 of [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF]. (H2) For all 1 ≤ i ≤ n, there exist α * i > 0 and m i > 0 such that for all z = (z

1 , • • • , z n ) T ∈ R n and u = (u 1 , • • • , u n ) T ∈ D, z T D 2 h(u)A(u)z ≥ n i=1 α i (u i ) 2 z 2 i ,
where either

α i (u i ) = α * i (u i -a) m i -1 or α i (u i ) = α * i (b -u i ) m i -1 ; (H3) There exists a * > 0 such that for all u ∈ D and 1 ≤ i, j ≤ n such that m j > 1, it holds that |A ij (u)| ≤ a * |α j (u j )|.
Let u 0 ∈ L 1 (Ω 0 ; D). Then there exists a weak solution u with initial condition u 0 to

∂ t u = div x (A(u)∇ x u) in (0, T ) × Ω 0 , and A(u)∇u • n = 0 on (0, T ) × ∂Ω 0 , (5)
such that for

(t, x) ∈ (0, T ) × Ω 0 , u(t, x) ∈ D with u ∈ L 2 ((0, T ); H 1 (Ω 0 , R n )) and ∂ t u ∈ L 2 ((0, T ); H -1 (Ω 0 ; R n )).
film during a CVD process. In this case, the evolution of the boundary of the film has to be taken into account as well as the diffusion phenomena occuring in the bulk. We introduce and analyze a simplified one-dimensional model, considered as a preliminary step before tackling more complicated systems. In this section, d = 1 and at time t = 0, the domain occupied by the solid is an interval Ω 0 = (0, e 0 ) for some initial thickness e 0 > 0. The external fluxes of volatile precursors during the deposition process are denoted by (φ 0 , • • • , φ n ) ∈ L ∞ ((0, T ); R n+1 ) and are assumed to satisfy φ i (t) ≥ 0 for all t ∈ (0, T ) and 0 ≤ i ≤ n. At time t ∈ (0, T ), the domain occupied by the solid is equal to Ω t = (0, e(t)) where the depth e(t) of the film is assumed to obey an evolution law characterized only by the values of the external fluxes:

e(t) = e 0 + t 0 n i=0 φ i (s) ds.
The concentrations (u 0 , • • • , u n ) satisfy the evolution law given in (1) with the following boundary conditions: for all 0 ≤ i ≤ n and t ∈ (0, T ),

  0≤j =i≤n K ij (u j ∂ x u i -u i ∂ x u j )   (t, 0) = 0 and   0≤j =i≤n K ij (u j ∂ x u i -u i ∂ x u j )   (t, e(t))+e ′ (t)u i (t, e(t)) =
These boundary conditions ensure that the total mass of each chemical species is conserved in the sense that for all t ∈ (0, T ), ∂ t e(t) 0

u i (t, x) dx = φ i (t).
Actually, we can prove the following existence result in a slightly more general framework, directly inspired from the setting of [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF]. Let e 0 > 0, u 0 ∈ L 1 ((0, e 0 ); D) and (φ 0 , • • • , φ n ) ∈ L ∞ ((0, T ); R n+1 + ). We denote by φ := (φ 1 , • • • , φ n ) T . Then there exists a weak solution u with initial condition

u 0 to        e(t) = e 0 + t 0 n i=0 φ i (s) ds, ∂ t u -∂ x (A(u)∂ x u) = 0 for t ∈ (0, T ) and x ∈ (0, e(t)), (A(u)∂ x u)(t, e(t)) + e ′ (t)u(t, e(t)) = φ(t) for t ∈ (0, T ), (A(u)∂ x u)(t, 0) = 0 for t ∈ (0, T ). (10) such that for (t, x) ∈ (0, T ) × (0, e(t)), u(t, x) ∈ D with u ∈ L 2 ((0, T ); H 1 ((0, e(t)); R n )) and ∂ t u ∈ L 2 ((0, T ); H -1 ((0, e(t)); R n )).
Note that the main difference between the assumptions of Proposition 3.1 and Theorem 2.1 is the fact that the entropy function h is asked in addition to belong to L ∞ (D; R). Indeed, a crucial point in our proof is that this functional h has to be bounded from above and from below on D, which is indeed the case for the function h defined in [START_REF] Schropp | Hot wire CVD of heterogeneous and polycrystalline silicon semiconducting thin films for application in thin film transistors and solar cells[END_REF]. We refer the reader to [START_REF] Bakhta | Global existence of bounded weak solutions to degenerate cross-diffusion equations in a moving domain[END_REF] for a proof of this result. The idea of the proof is to define the problem onto a reference fixed domain and derive entropy estimates which enable to carry on the same analysis as in [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF].

Conclusion

In this note, we have presented preliminary existence results for a cross-diffusion model with a moving boundary domain, modeling the evolution of the concentrations of different chemical species in a thin film solid layer grown by CVD.

In the first section, we have slightly extended results of [START_REF] Jüngel | Boundedness of weak solutions to crossdiffusion systems from population dynamics[END_REF] in the sense that we have proved that the system of PDEs [START_REF] Jüngel | Analysis of degenerate cross-diffusion population models with volume filling[END_REF] with boundary conditions (4) defined on a fixed domain, with coefficients K ij = K ji > 0 for all 0 ≤ i = j ≤ n, fell into the analysis framework developped by Jüngel in [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF]. In the second section, we have proposed and analyzed a simplified one-dimensional model in order to describe the growth of a thin film layer during a CVD process, as well as the diffusion of the different chemical species inside the bulk of the solid. The uniqueness of the solution of such a system of equations remains an open issue in both cases.

These results should be seen as preliminary steps before tackling more challenging systems of PDEs. In particular, the analysis of more realistic two-dimensional and threedimensional models for CVD deposition taking into account surface diffusion phenomena is currently work in progress.

Theorem 2 . 1 (

 21 Theorem 2 of [4]). Let D be an open domain of R n such that D ⊂ (a, b) n for some a, b ∈ R. Let A : u ∈ D → A(u) := (A ij (u)) 1≤i,j≤n ∈ R n×n be a matrix-valued functional defined on D satisfying A ∈ C 0 (D; R n×n ) and the following assumptions: (H1) There exists a bounded from below convex function h ∈ C 2 (D, R) such that its derivative Dh : D → R n is invertible on R n ;

Proposition 3 . 1 .

 31 Let D := {(u 1 , • • • , u n ) T ∈ (R * + ) n , n i=1 u i < 1} ⊂ (0, 1) n and A : u ∈ D → A(u) := (A ij (u)) 1≤i,j≤n ∈ R n×n be a matrix-valued functional defined on D satisfying A ∈ C 0 (D; R n×n ) and assumptions (H1)-(H2)-(H3) of Theorem 2.1 with an entropy functional h satisfying in addition (H4) h ∈ L ∞ (D).

Acknowledgements

We are very grateful to Eric Cancès and Tony Lelièvre for very helpful advice and discussions. We would like to thank Jean-François Guillemoles, Marie Jubeault, Torben Klinkert and Sana Laribi from IRDEP (Institut de Recherche et Développement sur l'Energie Photovoltaïque), who introduced us to the problem of modeling of CVD processes for the fabrication of thin film solar cells. The EDF company is acknowledged for funding. We would also like to thank Martin Burger, Ansgar Jüngel and Daniel Matthes for very helpful discussions on the theoretical part.

In this first section, we prove that the system of cross-diffusion equations introduced above, on a fixed domain Ω 0 ,

and with no flux boundary conditions [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF], can be analyzed using the general theoretical framework introduced in [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF]. Actually, the following existence result is a direct consequence of Theorem 2 of [START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF]: 2), the above system can then be reformulated equivalently, writing

we denote by ρ := n i=1 u i . The system (7) introduced above can be rewritten under the form (5) using the notation u :

Let us prove that A satisfies the assumptions of Theorem 2.1 with the following entropy functional, which is the same as in [START_REF] Liero | Gradient structures and geodesic convexity for reactiondiffusion systems[END_REF][START_REF] Jüngel | The boundedness-by-entropy method for cross-diffusion systems[END_REF][START_REF] Jüngel | Analysis of degenerate cross-diffusion population models with volume filling[END_REF]:

4

The function h ∈ C 0 (D; R) ∩ C 2 (D; R) (thus is bounded on D), is strictly convex on D, and its derivative Dh : D → R n is invertible. For all u ∈ D, Dh(u) = (ln u iln(1ρ)) 1≤i≤n and for all w ∈ R n , Dh -1 (w) = e w i 1+ n j=1 e w j 1≤i≤n

. As a consequence, h satisfies assumption (H1) of Theorem 2.1.

Let us now prove that assumption (H2) of Theorem 2.1 is satisfied with m i = 1 2 for all 1 ≤ i ≤ n. Thus, there will be no need to check assumption (H3) for the existence result to hold and Corollary 2.2 will be a direct consequence of Theorem 2.1. To this aim, we follow the same strategy of proof as the one used in [START_REF] Jüngel | Boundedness of weak solutions to crossdiffusion systems from population dynamics[END_REF]. We are going to prove that if the coefficients K ij are assumed to be strictly positive then, there exists α > 0 such that for all u ∈ D,

(9) This inequality implies (H2) with

Introducing P (u) := (P ij (u)) 1≤i,j≤n the matrix such that for all 1 ≤ i, j ≤ n,

it holds that H(u)P (u) = Λ(u). Thus, we can write H(u)A(u) -αΛ(u) = H(u)(A(u) -αP (u)). We can easily check that A(u) -αP (u) = A(u) + αD(u), where A(u) has the same structure as A(u) but with diffusion coefficients K ijα instead of K ij , and D(u) := (D ij (u)) 1≤i,j≤n where D ij (u) = u i for all 1 ≤ i ≤ n.

On the one hand, H(u)D(u) = 1 1-ρ Z where Z := (Z ij ) 1≤i,j≤n with Z ij = 1 for all 1 ≤ i, j ≤ n. Since the matrix Z is a positive matrix, so is H(u)D(u).

On the other hand, since h is strictly convex on D, H(u) A(u) is positive if and only if M (u) := A(u)H(u) -1 is positive. Indeed, for all z ∈ R n , we have z T H(u) A(u)z = (H(u)z) T A(u)H(u) -1 (H(u)z). Besides, it can be easily checked that M (u) = (M ij (u)) 1≤i,j≤n , where for all

and that this matrix is indeed a positive matrix. Hence we have proved inequality [START_REF] Zinsl | Transport distances and geodesic convexity for systems of degenerate diffusion equations[END_REF], assumption (H2) and Corollary 2.2.

A simple one-dimensional model with moving boundary

As announced in the introduction, the second part of this note deals with a model describing the evolution of the concentration of different chemical species in a solid thin