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

A dispersive estimate for the linearized Water-Waves equations in finite depth

 in the case of infinite depth, we prove a decay with respect to time t of order |t| ´1{3 for solutions with initial data ϕ such that |ϕ| H 1 , |xBxϕ| H 1 are bounded. We also give variants to this result with different decays for a more convenient use of the dispersive estimate. We then give an existence result for the full Water-Waves equations in weighted spaces for practical uses of the proven dispersive estimates.

Introduction

We recall here classical formulations of the Water-Waves problem. We then shortly introduce the meaningful dimensionless parameters of this problem, and then present the main results 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 fixed 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 . There exists a function b : R d Ñ R such that the domain of the fluid at the time t is given by Ω t " tpX, zq P R d`1 , ´H0 `bpXq 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 `bpXqu P " P atm on tz " ζpt, Xqu,

In these equations, V and w are the horizontal and vertical components of the velocity evaluated at the surface. The vector n in the second equation stands for the normal upward vector at the bottom pX, z " ´H0 `bpXqq. 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 second 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 top 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 " Grζ, bsψ " a 1 `|∇ζ| 2 B n Φ |z"ζ ,
where Φ stands for the potential, and solves a 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`b " 0 with the notation, for the normal derivative

B n Φ |z"´H0`bpXq " ∇ X,z ΦpX, ´H0 `bpXqq ¨n

where n stands for the normal upward vector at the bottom pX, ´H0 `bpXqq. 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 ;

(5) The size of bottom topography a bott .

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 , b 1 " b a bott ,
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 re scaling, several dimensionless parameters appear in the equation. They are

a surf H 0 " ε, H 2 0 L 2 x " µ, a bott H 0 " β, L x L y " γ,
where ε, µ, β, γ are commonly referred to respectively as "nonlinearity", "shallowness", "topography" 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 rεζ, βbsψ " 0 B t ψ `ζ `ε 2 |∇ γ ψ| 2 ´ε µ pG µ,γ rεζ, βbsψ `εµ∇ γ ζ ¨∇γ ψq 2 2p1 `ε2 µ | ∇ γ ζ | 2 q " 0, (1.4)
where G µ,γ rεζ, βbsψ stands for the dimensionless Dirichlet-Neumann operator,

G µ,γ rεζ, βbsψ " 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 `βbpXq ă z ă εζpXqu φ |z"εζ " ψ, B n Φ |z"´1`βb " 0. (1.5)
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`βb " 1 a 1 `β2 |∇ γ b| 2 pB z Φ ´µ∇ γ pβbq ¨∇γ Φq |z"´1`βb .

Main result

The linearized Water-Waves equations (1.4) in one dimension around a rest state of a flat surface and a zero velocity, in presence of a flat bottom can be read as

$ ' ' ' & ' ' ' % B t ζ ´1 µ G 0 ψ " 0 B t ψ `ζ " 0 pζ, ψqp0q " pζ 0 , ψ 0 q (1.6)
where pt, xq P R ˆR. We denote

1 µ G 0 " 1 µ Gr0, 0s
the Dirichlet-Neumann operator in ζ " 0 with a flat bottom, which explicit formulation is given by its Fourier transform

1 µ y G 0 f pξq " |ξ| tanhp ? µ|ξ|q ? µ p f pξq, (1.7)
for all f P S 1 pRq where µ is the shallowness parameter (see for instance [START_REF] Lannes | The water waves problem[END_REF] for more details). The equation (1.6) leads to the following equation for ζ:

B 2 t ζ `1 µ G 0 ζ " 0
which is similar to the wave equation for low frequencies, and to the Water-Wave equation in infinite depth

B 2 t ζ `p´∆q 1{2 ζ " 0 where ∆ " B 2
x , for high frequencies. In order to study the solutions of the linearized system (1.6), we are therefore led to study the decay in time of the operator e itωpDq where ω :

$ ' & ' % R ÝÑ R ξ Þ ÝÑ d |ξ| tanhp ? µ|ξ|q ? µ .
The dispersive nature of the Water-Waves equations in infinite depth plays a key role in the proof of long time or global time results: see for instance [START_REF] Wu | Almost global wellposedness of the 2-d full water wave problem[END_REF] for almost global well-posedness in 2d, [START_REF] Wu | Global wellposedness of the 3-d full water wave problem[END_REF] for 3d global well-posedness, Ionescu-Pusateri [START_REF] Alexandru | Global solutions for the gravity water waves system in 2d[END_REF], Alazard-Delort [START_REF] Alazard | Global solutions and asymptotic behavior for two dimensional gravity water waves[END_REF] and [START_REF] Alazard | Sobolev estimates for two dimensional gravity water waves[END_REF] for Global well-posedness in 2d, [START_REF] Germain | Global solutions for the gravity water waves equation in dimension 3[END_REF] for the global well-posedness in 3d. However, there are to our knowledge only few results on decay estimates for the Water-Waves equations in finite depth (see for instance [START_REF] Mélinand | A mathematical study of meteo and landslides tsunamis : The Proudman resonance[END_REF]). Recently, Aynur Bulut proved in [START_REF] Bulut | An optimal decay estimate for the linearized water wave equation in 2d[END_REF] an L 2 based norm-L 8 decay estimate for the linear Water-Waves equation in infinite depth:

|e itp´∆q 1{4 ϕ| 8 ď Cp1 `|t|q ´1{2 p|ϕ| H 1 `|xB x ϕ| L 2 q. (1.8)
As for all oscillatory integrals estimates, the proof of this result relies only on the behaviour of the operator p´∆q 1{2 , which is the same as the behaviour of 1 µ G 0 (recall the definition (1.7)) for high frequencies. We therefore adapt this proof to get a similar result in the case of a finite depth, with a very special attention given to the dependence in the shallowness parameter µ. As one shall see later, this result gives Bulut's estimate in the limit µ goes to `8. We prove in Section 2 of this paper the following result:

Theorem 1.1 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 p1 `t{ ? µq 1{8 `1 p1 `t{ ? µq 1{2 qp|ϕ| H 1 `|xB x ϕ| 2 q.
Though 1 µ ? G 0 and the square root of the wave operator p´∆q 1{2 have the same behaviour for low frequencies, it is not the case for the second order derivatives of these operators. For this reason, one should not be surprised to have a dispersion result for the Water-Waves equations, while the wave equation in dimension 1 is not dispersive.

The decay in 1 t 1{8 given by Theorem 1.1 is however very bad. As one might be interested to have a better decay result, we also prove the following result, with different spaces: Theorem 1.2 With the notations of Theorem 1.1, the following estimates hold:

1. 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 3{4 1 p1 `t{ ? µq 1{3 |ϕ| L 1 `1 p1 `t{ ? µq 1{2 p|ϕ| H 1 `|xB x ϕ| 2 qq.
2. 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 3{4 1 p1 `t{ ? µq 1{3 |xϕ| L 2 `1 p1 `t{ ? µq 1{2 p|ϕ| H 1 `|xB x ϕ| 2 qq.
As one should remark, the decay given by Theorem 1.2 is better than one of Theorem 1.1. However, for a practical use of such decay, one should prove that the solutions are bounded in L 1 or in |x ¨|2 norm, which is more difficult than proving a local existence result in |xB x ¨|2 -norm. In view of practical use of Theorem 1.1, we therefore prove in Section 3 and in dimensions d " 1, 2 a local existence result for the full Water-Waves equations (1.4) in weighted Sobolev spaces. The proof consists in an adaptation of the local existence result by [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF], and a technical proof of the commutator rG, xs.

Remark 1.3 -All the dispersive effects proved in this paper are in dimension d " 1. A similar result in dimension 2 may however not be difficult to obtain, as the phase of the oscillatory integral studied has a radial symmetry.

-As mentioned before, in all this paper, a very special attention is given to the dependence of the estimates with respect to µ. It allows in the use of the dispersive estimates to identify different regimes, considering the size of the ratio ε µ 3{2 , in which the non-linear effects may overcome or not the linear (and thus dispersive) effects. Such study has been done for example in [START_REF] Mésognon-Gireau | The rigid lid limit for the water waves equations[END_REF].

The plan of the article is the following:

-In Section 2, we prove a dispersive estimate for the linearized Water-Waves equation around a flat bottom and a flat surface in dimension d " 1.

-In Section 3, we give a local existence result for the full Water-Waves equation (1.4) with non flat bottom, in weighted Sobolev spaces and in dimensions d " 1, 2.

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:

∇ 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,γ " t p ? µB x , γ ? µB y , B z q if d " 2 and t p ? µB x , B z q if d " 1.
Remark 1.4 All the results proved in this paper do not need the assumption that the typical wave lengths are the same in both directions, i.e. 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]).

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 denote for all a, b P R: a _ b " maxpa, bq.

We denote, for all ϕ P S 1 pR d q, the Fourier transform of ϕ by F pϕq of more simply p ϕ.

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 (1.9)
where F pf pDquqpξq " f pξqF puqpξq is defined for any smooth function of polynomial growth f and u P S 1 pR d q. The pseudo-differential operator P acts as the square root of the Dirichlet Neumann operator, since P " 1 µ ? G 0 (recall the definition of G 0 given by (1.7)) where the implicit constant does not depend on µ. 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 rεζ, βbsψ " 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`βb " 0.
For the sake of simplicity, we use the notation Grεζ, βbsψ or even Gψ when no ambiguity is possible.

The Dirichlet-Neumann problem

In order to study the Dirichlet-Neumann problem (1.5), we need to map the domain occupied by the water Ω t into a fixed domain (and not on a moving subset). For this purpose, we define:

ζ δ p., zq " χpδz|D γ |qζ, b δ p., zq " χpδpz `1q|D γ |qb
where χ : R ÝÑ R is a compactly supported smooth function equals to one in the neighbourhood of the origin, and δ ą 0. We now introduce the following fixed strip:

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

Σ : S Ñ Ω t pX, zq Þ Ñ pX, p1 `εζ δ pXq ´βδ bpXqqz `εζ δ pXqq . (1.10)
It is quite easy to check that Φ is the variational solution of (1.5) if and only if φ " Φ ˝Σ is the variational solution of the following problem: # ∇ µ,γ ¨P pΣq∇ µ,γ φ " 0 φ z"0 " ψ, B n φ z"´1 " 0, (1.11) and where P pΣq " | det J Σ |J ´1 Σ t pJ ´1 Σ q, where J Σ is the Jacobian matrix of the diffeomorphism Σ.

Remark 1.5 By smoothing the functions ζ and b in the choice of the diffeomorphism Σ as in [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF], we ensure a better estimate for the solutions of (1.11).

For a complete statement of the result, and a proof of existence and uniqueness of solutions to these problems, see later Section 3.2 and also [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 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:

Grε¨, β¨s : Γ ÝÑ H s´1{2 pR d q Γ " pζ, bq Þ ÝÑ Grεζ, βbsψ.
We can prove the differentiability of this mapping. See Appendix B 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 when 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 denote the norms of W k,8 pR d q by | ¨|W k,8 or simply | ¨|8 " | ¨|W 0,8 when no ambiguity is possible.

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 denote the L 2 norm on S by }¨}, and we 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 .
We will denote }¨} H s " }¨} H s,0 when no ambiguity is possible. To sum up, | ¨| will denote a norm on R d while } ¨} will denote a norm on the flat strip S.

A dispersive estimate for the linear Water-Waves equations in dimension 1

We prove in this section the dispersive estimate of Theorem 1.1 and Theorem 1.2. We first introduce some classical results on the oscillatory integrals, and some technical results on the Littlewood-Paley decomposition.

Technical tools

Littlewood-Palay decomposition

We briefly recall the Littlewood-Paley decomposition. Let ψ P C 8 0 pRq be such that suppψ Ă p´1; 1q, ψpξq " 1 for |ξ| ď 1{2.

We now define for all k P Z, a function ψ k by:

@ξ P R, ψ k pξq " ψpξ{2 k q ´ψpξ{2 k´1 q (2.12)
which is compactly supported in 2 k´1 ď |ξ| ď 2 k`1 . We then define the operators P k for all k P Z by:

@ξ P R, y P k f pξq " ψ k pξq p f pξq (2.13)
for all f P S 1 pRq. We recall here Bernstein's Lemma:

Lemma 2.1 Let k P Z and P k defined by (2.13). For every 1 ď p ď q ď 8 and all s ě 0, one has:

|P k g| L q ď C2 kp1{p´1{qq |P k g| L p and |P k g| L p ď C2 ´sk |p´∆q s{2 P k g| L p
for all g P SpRq, where C does not depend on k, s, p, q.

We also give the two following technical results (see for instance [START_REF] Bulut | An optimal decay estimate for the linearized water wave equation in 2d[END_REF] for a complete proof):

Lemma 2.2 Let k P Z and P k defined by (2.13). One has:

|B ξ y P k ϕ| 2 ď C2 ´kp|ϕ| 2 `|xB x ϕ| 2 q,
for all ϕ P SpRq, where C does not depend on k.

Lemma 2.3 Let k P Z and P k defined by (2.13). For all s ą 1{2 one has

| y P k ϕ| 8 ď C2 ´sk p|ϕ| H s `|xB x ϕ| 2 q,
for all ϕ P SpRq, where C does not depend on k, s.

Some results on oscillatory integrals

We invoke later in this paper the following Van der Corput Lemma, which is a refinement of the stationary phase lemma:

Lemma 2.4 Let Φ P C k pRq, and a ă b be such that, either:

( 1) @x P ra; bs,

|Φ pkq pxq| ě 1 if k ą 1;
( 2) @x P ra; bs, |Φ 1 pxq| ě 1 and Φ 1 is monotonic.

Then, there exists C ą 0 which only depends on k such that

@t ą 0, | ż b a e itΦpξq dξ| ď C t 1{k .
Note that in the above Lemma, C does not depend on a nor b.

Proof of the main result

We prove in this section the following dispersive estimate for the linearized Water-Waves equations in dimension 1:

Theorem 2.5 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 p1 `t{ ? µq 1{8 `1 p1 `t{ ? µq 1{2 qp|ϕ| H 1 `|xB x ϕ| 2 q.
Note that the dependence of µ in the dispersive estimate has been precisely mentioned. This is crucial in view of using a decay estimate for the Water-Waves equations, since the properties of the solutions, and even the dispersive properties of the problem may completely vary with respect to the size of the shallowness parameter µ, as one should see by studying for example the asymptotic regimes when µ goes to zero. See for instance the Chapter 5 on shallow water models in [START_REF] Lannes | The water waves problem[END_REF] any assumption on the size of µ (while µ ď µ 0 is a common assumption in the Water-Waves results).

In [START_REF] Mélinand | A mathematical study of meteo and landslides tsunamis : The Proudman resonance[END_REF], a similar decay as one given by Theorem 2.5 is proved but only for functions ϕ such that p ϕp0q " 0 and with L 1 and H 2 weighted spaces, which are less convenient for practical use than H 1 space and L 2 -weighted space. However, a short adaptation of the proof of [START_REF] Mélinand | A mathematical study of meteo and landslides tsunamis : The Proudman resonance[END_REF] shows that a decay of order 1 t 1{3 can be obtained if p ϕp0q ‰ 0, which is a better decay than one of Theorem 2.5. We can however adapt the proof of Theorem 2.5, and still get some better estimates than [START_REF] Mélinand | A mathematical study of meteo and landslides tsunamis : The Proudman resonance[END_REF] (without any assumption on p ϕp0q) that we also prove in this paper: Theorem 2.6 With the notations of Theorem 2.5, the following estimates hold:

1. 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 3{4 1 p1 `t{ ? µq 1{3 |ϕ| L 1 `1 p1 `t{ ? µq 1{2 p|ϕ| H 1 `|xB x ϕ| 2 qq.
2. There exists C ą 0 independent on µ such that, for all µ ą 0: Proof The result is easy to get for |t{ ? µ| ď 1 by using the continuous injection H 1 pRq Ă L 8 pRq, and therefore we assume that t ą ? µ (the case t ă ´?µ is similar). Let fix x P R with x ‰ 0. In all this proof, we will denote by C any constant which does not depend on µ, x, t, k. As explained above, the derivative of the phase, Φ 1 , may vanish and therefore one needs a close study of the second derivative Φ 2 " ω 2 . Note that ωpξq " Taking the last remark into account, we first assume that p ϕ is compactly supported in some r0; y0 ? µ r for some y 0 ą 0. In this case, we only need to control the first term of (2.16). One sets δ ą 0 and splits the integral into two parts (recall that the phase Φ is defined by (2. (2.17)

@t ą 0, @ϕ P SpRq |e itωpDq ϕ| 8 ď Cp 1 µ 3{4 1 p1 `t{ ? µq 1{3 |xϕ| L 2 `1 p1 `t{ ? µq 1{2 p|ϕ| H 1 `
We now use Cauchy-Schwarz inequality to control the first integral of the right hand side of (2.17):

| ż |ξ|ďδ e itΦpξq p ϕpξqdξ| ď ? 2δ|ϕ| 2 . (2.18)
For the second integral of the right hand side of (2.17), we only consider the integral over rδ; y 0 { ? µs where Φ is smooth (the integral over r´y 0 { ? µ; ´δr is controlled by the exact same technique using the symmetry of Φ 2 ). We integrate by parts in the second integral of the right hand side of (2.17) (remember that p ϕ is compactly supported in r0; We now assume that p ϕ has its support in r y0 ? µ ; `8r. In this case, we only need to focus on the second term of the right hand side of (2.16). We are led to control in L 8 |P k e itωpDq ϕpxq|.

The term S 1 is controlled by using Bernstein's Lemma 2.1:

S 1 ď ÿ 2 k ďλptq |P k e itωpDq ϕpxq| 8 ď C ÿ 2 k ďλptq 2 k{2 |P k e itωpDq ϕpxq| 2 ď Cλptq 1{2 |ϕpxq| 2 ď Cp1 `|t{ ? µ|q ´1{2 |ϕ| 2 .
(2.23)

Using again Bernstein's Lemma 2.1, one gets the control of S 3 :

S 3 ď ÿ 2 k ěΛptq |P k e itωpDq ϕpxq| 8 ď ÿ 2 k ěΛptq 2 k{2 |P k e itωpDq ϕpxq| 2 ď C ÿ 2 k ěΛptq 2 ´k{2 |P k e itωpDq ϕpxq| H 1 ď Cp1 `|t{ ? µ|q ´1{2 |ϕ| H 1 . (2.24)
For the control of S 2 , we need a close study of oscillatory integrals of the form ş R e itpx{tξ`ωpξqq y P k ϕpξqdξ. Therefore, we need precise bounds for the oscillatory phase ξ Þ Ñ px{tqξ `ωpξq and its derivatives. We are led to split the summation set of S 2 into three parts:

I 1 " tk P Z, λptq ď 2 k ď Λptq, 2 k{2 ď |t{x|{C 2 u, I 2 " tk P Z, λptq ď 2 k ď Λptq, |t{x|{C 2 ď 2 k{2 ď C 2 |t{x|u, I 3 " tk P Z, λptq ď 2 k ď Λptq, 2 k{2 ě C 2 |t{x|u,
where C 2 has to be set. We therefore set

S 2j " ÿ Ij | ż R e ipxξ`tωpξqq y P k ϕpξqdξ|, j " 1, 2, 3.
The contributions of I 1 and I 3 are the most easy to get. One writes, for all k P I 1 :

ż R e itΦpξq y P k ϕpξqdξ " ż R d dξ p ż ξ 2
|x{t `ω1 psq| ě C 3 µ 1{4 1 ? 2 2 ´k{2 ´2´k{2 C 2 ě C 3 µ 1{4 2 ´k{2 provided C 3 µ 1{4 ? 2 ´C2 ě Cµ 1{4
with C independent on k, µ, x. We therefore set We now sum over k P I 1 . Since the set has a Oplogp|t|q number of elements (recall that it is included in tλptq ď 2 k ď Λptqu), we get:

C 2 " C 3 2 ? 2µ 1{4 . ( 2
S 21 ď µ 1{4 C |t| ÿ kPI1 p|ϕ| 2 `|xB x ϕ| 2 q ď Cµ 1{4 |t| logp|t|qp|ϕ| 2 `|xB x ϕ| 2 q ď Cµ 1{4 |t| ´1{2 p|ϕ| 2 `|xB x ϕ| 2 q.
(2.29)

The control for S 23 is similar and therefore we omit it and focus on the most difficult term which is S 22 . One starts to notice that there is a finite number of terms which is of the form C logpµq with C independent on t, x, µ in the set I 2 . Indeed, if k P I 2 then one has ´logpC 2 q `logp|t{x|q ď k ď logpC 2 q `logp|t{x|q, and C 2 has been set in (2.26). Therefore, it suffices to control the integrals ş R e itΦpξq y P k ϕpξqdξ for k P I 2 by a term of the form C t 1{2 with C independent on x, t, k. For k P I 2 , the derivative of the phase Φ may vanishes, and one needs to control the second derivative and use a Van der Corput type result. Let c be the minimum of |Φ 1 | on r2 k´1 ; 2 k`1 s.

-1st case : Φ 1 pcq " 0 We split the integral into three terms:

ż where we used the fact that |c ´s| ě δ for s P r2 k´1 ; ξs and ξ P r2 k´1 ; c ´δs. Therefore, using Van (2.36)

The above right hand side is minimal with respect to δ if δ " 2 sk{2 µ 1{8 2 k{2 t 1{2 . We therefore set s " 1 and get the control:

| ż R e itΦpξq y P k ϕpξqdξ| ď µ 1{8 C t 1{2 .
At last, one gets by summation on k P I 2 (recall that there is a Oplogpµqq number of terms in I 2 ):

S 22 ď µ 1{4 t 1{2 p|ϕ| H 1 `|xB x ϕ| 2 q.
(2.37) -2nd case : Φ 1 pcq ‰ 0 It is the same technique as above, noticing that in this case Φ 1 is monotonic and does not vanish, and therefore c is one of the bounds of the integral.

Conclusion :

Putting together (2.22),(2.23),(2.24),(2.29) and (2.37), and taking the supremum over all x P R ˚(note that all these estimates are independent on x), one gets: and one gets the second estimate of Theorem 2.6. l

|e itωpDq ϕ| 8 ď Cp 1 µ 1{4
Remark 2.7 It should be possible to prove a dispersive decay in dimension 2, using the previous results, since the phase Φ introduced in the proof of Theorem 2.5 has a radial symmetry.

3 Local existence for the Water-Waves equations in weighted spaces in dimension d " 1, 2

In view of practical use of Theorem 2.5 to prove some long time (or even global time) results in the case of the full Water-Waves equations (1.4), one may need to control xB x ϕ in L 2 norm, for ϕ a solution of the equations. We prove in this section a local existence result for the Water-Waves equations in weighted spaces. To this purpose, we briefly give some reminders about the Water-Waves equations, and we state the local existence result proved by [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF]. We then give a commutator estimate which is the key point for local existence in weighted spaces. Note that in this Section we do not make any assumption on the dimension unlike in the previous one, and that all the results proved in this section stand in dimensions d " 1, 2. The local existence result in weighted spaces is used for instance in [START_REF] Mésognon-Gireau | The rigid lid limit for the water waves equations[END_REF], where the default of compactness in the rigid lid limit for the Water-Waves equations is investigated.

The Water-Waves equations

We briefly give some reminders about the Water-Waves equations and its local existence (see [START_REF] Lannes | The water waves problem[END_REF] Chapter 4 for a complete study). 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 1.3 for the notations):

E N pU q " |Pψ| H t 0 `3{2 `ÿ |α|ďN |ζ pαq | 2 `|ψ pαq | 2 (3.38)
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 with w " Gψ `εµ∇ γ ζ ¨∇γ ψ 1 `ε2 µ|∇ γ ζ| 2 .
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 `εpB t `εV ¨∇γ qw " ´ε P 0 ρag pB z P q |z"εζ where V " ∇ γ ψ ´εw∇ γ ζ.

As suggested by the notations, V and w are respectively the horizontal and vertical components of the velocity evaluated at the surface. We recall that the notation a _ b stands for maxpa, bq. 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 for reference):

Theorem 3.1 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 , b P H N `1_t0`1 pR d q. Let ε, β, γ, µ be such that 0 ď ε, β, γ ď 1, 0 ď µ ď µ max
with µ max ą 0 and moreover assume that:

Dh min ą 0, Da 0 ą 0, 1 `εζ 0 ´βb 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 ε_β
to (1.4) with initial data U 0 . Moreover,

1 T " C 1 , and
sup tPr0; T ε_β 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.

A commutator estimate

The key point of the local existence result we prove in this Section is the commutator result of Proposition 3.5 below. We first need to introduce some technical results about the resolution of the Dirichlet-Neumann problem (1.5). We use here the notations of Section 3.1. We recall the introduction of the diffeomorphism Σ defined by (1.10) which maps Ω into S " R d ˆp´1, 0q. We also recall that Φ is a solution of (1.5) if and only if φ " Φ ˝Σ is a solution of the following problem:

# ∇ µ,γ ¨P pΣq∇ µ,γ φ " 0 φ z"0 " ψ, B n φ z"´1 " 0, (3.39)
where

P pΣq " | det J Σ |J ´1 Σ t pJ ´1 Σ q, (3.40)
where J Σ is the Jacobian matrix of the diffeomorphism Σ. For the sake of clarity in the proof of the main result of this section, we introduce the following notations, for all ζ, b, t 0 satisfying the hypothesis of Theorem 3.1:

M 0 " Cp 1 h min , µ max , |ζ| H t 0 `1 , |b| H t 0 `1 q, M " Cp 1 h min , µ max , |ζ| H t 0 `2 , |b| H t 0 `2 q, M psq " M 0 " CpM 0 , |ζ| H s , |b| H s q, (3.41)
where C denotes a non decreasing function of its arguments. One can prove (see [START_REF] Lannes | The water waves problem[END_REF] Chapter 2 and equation (2.26)) that P pΣq "

I d `QpΣq (3.42) with }QpΣq} H s,1 ď M 0 |pεζ, βbq| H s`1{2 . (3.43)
One can also prove the coercivity of P pΣq:

@Θ P R d , P pΣqΘ ¨Θ ě kpΣqΘ 2 (3.44)
where 1 kpΣq ď M 0 .

Technical results about the boundary problem (3.39)

As usual for an elliptic problem of the form (3.39) with a Dirichlet condition, we are looking for solutions in the space ψ `H1 0,surf pSq where H 1 0,surf pSq is the set of functions of H 1 pSq with a vanishing trace at z " 0 (recall that S is the flat strip R d ˆp´1; 0q). More precisely, we have: Definition We define H 1 0,surf as the completion of DpR d ˆr´1; 0rq endowed with the H 1 norm of S. We now define the variational solutions to the elliptic equation (3.39). To this purpose, we introduce for all ψ P S 1 pR d q the smoothed distribution

ψ : p., zq " χp ? µz|D γ |qψ (3.45)
where χ is a smooth compactly supported function equals to 1 in the neighbourhood of the origin.

Definition For all ψ P 9 H 1{2 a variational solution to (3.39) is φ " φ `ψ: such that ż S ∇ µ,γ φ ¨P pΣq∇ µ,γ ϕ " ´żS ∇ µ,γ ψ : ¨P pΣq∇ µ,γ ϕ for all ϕ P H 1 0,surf pSq.

Remark 3.2 As expected for an elliptic problem of the form (3.39), if ψ P H s pR d q, the solution Φ should be in H s`1{2 pSq (as ψ is the trace of Φ on z " 0). Therefore, if one uses ψ instead of ψ : in Definition 3.2.1, then the formulation provides the same regularity for Φ as for ψ. Instead of brutally considering ψ (which is a function defined on R d ) as a function of S, we introduce ψ : which is indeed 1{2 more regular than ψ and is defined on all S.

We are now able to give the existence result for the problem (3.39) (recall the notations of (3.41) for the constants M and M psq, and see [START_REF] Lannes | The water waves problem[END_REF] for reference):

Proposition 3.3 Let t 0 ą d{2 and s ě 0. Let ζ, b P H s`1{2 X H t0`1 pR d q be such that Dh min ą 0, @X P R d , 1 `εζpXq ´βbpXq ě h min .

Then, for all ψ P 9 H s`1{2 pR d q, there exists a unique variational solution φ to (3.39). Moreover, this solution satisfies:

@0 ď s ď t 0 `3{2, }Λ s ∇ µ,γ φ} 2 ď ? µM ps `1{2q|Pψ| H s , @t 0 `3{2 ď s, }Λ s ∇ µ,γ φ} 2 ď ? µM p|Pψ| H s `|pεζ, βbq| H s`1{2 |Pψ| H t 0 `3{2 q.
Moreover, if s ě ´t0 `1, the same estimates hold on }∇ µ,γ φ} H s,1 .

Main result

We prove in this section an estimate for the commutator r 1 µ G, xsB x in H s norm, where x is one of the variable of R d (we denote x instead of x j for the sake of clarity). In the case of a flat bottom and a flat surface in dimension 1, one has for all ϕ P SpR d q and all ξ ą 0:

{ r 1 µ G, xsB x ϕ " d dξ p tanhp ? µξq ? µ ξqξ p ϕpξq " `tanhp ? µξq ? µ `p1 ´tanhp ? µξq 2 qξ ˘ξ p ϕpξq,
and thus one should expect a control of the form

|r 1 µ G, xsB x ϕ| H s`1{2 ď C|Pϕ| H s`3{2 ,
with C independent on µ, where we recall that P acts like the square root of the Dirichlet-Neumann operator and is defined by

P " |D γ | p1 `?µ|D γ |q 1{2 . Remark 3.4
-The operator 1 µ G has to be seen as a 3{2 order operator instead of an one order operator if one needs a bound which is not singular with respect to µ. Indeed, the brutal bound

tanhp ? µξq ? µ ď 1 µ 1{2 is singular in µ.
We still gain one derivative in the commutator r 1 µ G, xs and have controls uniforms with respect to µ.

-In the statement of Proposition 3.3, we distinguish the cases s `1{2 ą t 0 and s `1{2 ă t 0 . This is to have tame estimates with respect to the H s norms of the unknowns, for high values of s.

The following Proposition shows that the result stands true with non flat bottom and surface, and in all dimensions. We denote xay cond " a is cond is satisfied, and else 0.

Proposition 3.5 Let t 0 ą d{2, s ě ´1{2, and ζ, b P H t0`2 pR d q be such that Dh min ą 0, @X P R d , 1 `εζpXq ´βbpXq ě h min .

We denote x one of the variables of R d . Then, one has for all ϕ P 9 H s`2 pR d q:

|r 1 µ G, xsB x ϕ| H s`1{2 ď µM ps `1q|Pϕ| H s`3{2 `x|Pϕ| H t 0 `2 |pεζ, βbq| H s`1 qy s`1{2ąt0 .
Moreover, if ∇f P H t0`1{2 X H s`1{2 , one has for all ϕ P 9 H s`2 pR d q:

|r 1 µ G, xsf B x ϕ| H s`1{2 ď µM ps `1q|∇ γ f | H t 0 `1{2 |Pϕ| H s`3{2 `x|Pϕ| H t 0 `2 |pεζ, βbq| H s`1 q|∇ γ f | H s`1{2 y s`1{2ąt0 .
Remark 3.6 The fact that the commutator is applied to B x ϕ instead of ϕ is crucial in this result. This is due to the fact that one only controls ∇ µ,γ Φ instead of Φ, where Φ solves (3.39), and thus some terms of the form xΦ are not controlled (while there derivatives are controlled). Remark that the second point of the Proposition implies the first one (just take f " 1), but its proof requires to use the first point as one shall see during the proof below.

Proof The proof is an adaptation of the commutator estimate rΛ s , 1 µ Gs which is proved in r11s, using a duality argument. We set v " B x ϕ.

For all v P 9 H 1{2 , we will denote v h the solution of the Dirichlet-Neumann problem (3.39) with boundary condition v h z"0 " v. This notation stands for "harmonic extension of v". We recall the notation ψ : for all ψ P S 1 pR d q given by (3.45). We now write, for all u P SpR d q: pΛ s`1{2 u, rG, xsvq 2 " pΛ s`1{2 u, Gxvq 2 ´pxΛ s`1{2 u, Gvq 2 .

Since pΛ s`1{2 xuq :

z"0 " Λ s`1{2 xu, we get using Green's identity that pu, rG, xsvq "

ż S ∇ µ,γ Λ s`1{2 u : ¨P pΣq∇ µ,γ pxvq h ´żS P pΣq∇ µ,γ v h ¨∇µ,γ pxΛ s`1{2 u : q " ż S ∇ µ,γ Λ s`1{2 u : ¨P pΣq∇ µ,γ `pxvq h ´xv h ˘`ż S ∇ µ,γ Λ s`1{2 u : ¨P pΣqp∇ µ,γ xqv h ´żS P pΣq∇ µ,γ v h ¨p∇ µ,γ xqΛ s`1{2 u : . (3.46)
We start to control the easiest term of (3.46), using Cauchy-Schwarz inequality (recall that | ¨| stands for norms on R d while } ¨} stands for norms on the flat strip S " R d ˆp´1; 0q):

| ż S P pΣq∇ µ,γ v h ¨p∇ µ,γ xqΛ s`1{2 u : | ď ? µ}u : } 2 › › ›Λ s`1{2 P pΣq∇ µ,γ v h › › › 2
where the ? µ factor comes from the definition of ∇ µ,γ x " t p ? µB x , γ ? µB y , B z qx (see Section 1.3) and the fact that B z x " 0. Using the definition of u : , one has easily

|u : | 2 ď }u} 2 .
(3.47)

We now use the product estimate of Proposition A.1 and the decomposition P pΣq " I d `Q of (3.42) to write:

@0 ď s `1{2 ď t 0 , › › ›Λ s`1{2 P pΣq∇ µ,γ v h › › › 2 ď Cp1 `}Q} L 8 H t 0 q › › ∇ µ,γ v h › › H s`1{2,0 @t 0 `3{2 ă s `1{2, › › Λ s`1 P pΣq∇ µ,γ v h › › 2 ď Cp1 `}Q} L 8 H t 0 q › › ∇ µ,γ v h › › H s`1{2,0 `}Λ s`1{2 Q} 2 }∇ µ,γ v h } L 8 H t 0 .
(3.48) Remark 3.7 We don't treat the case t 0 ă s ď t 0 `3{2, since we will obtain it by interpolation of the two cases above. One has to combine the difference in the product estimate of Proposition A.1 between the cases 0 ď s`1 ď t 0 and t 0 ă s, and the difference in Lemma 3.8 between the cases 0 ď s`1 ď t 0 `3{2 and t 0 `3{2 ă s. For this reason, we split the proof in only two cases, and get the third one by interpolation.

One has, using (3.43) and the embedding H s`1{2,1 in L 8 H s pR d q given by Proposition A.2:

}Q} L 8 H t 0 ď M 0 , }Λ s`1{2 Q} 2 ď M 0 |pεζ, βbq| H s`1 . (3.49)
We use Proposition 3.3 and Proposition A.2 to write: (3.52) where we recall the notation xay cond " a if cond is satisfied, and else 0. Note that we got the result for t 0 ă s ă t 0 `3{2 by interpolation. Remembering that v " B x ϕ, one gets the control of Proposition 3.5 for this term.

@0 ď s `1{2 ď t 0 , › › ∇ µ,γ v h › › H s`1{2,0 ď ? µM ps `1q|Pv| H s`1 @t 0 `3{2 ă s `1{2, › › ∇ µ,γ v h › › H s`1{2,0 ď ? µM p|Pv| H s`1{2 `|pεζ, βbq| H s`1 |Pv| H t 0 `3{2 q › › ∇ µ,γ v h › › L 8 H t 0 ď ? µM 0 |Pψ| H t 0 `1{2 . ( 3
We now focus on the most difficult term of (3.46) (the last term of (3.46) is estimated by a similar technique). Using Cauchy-Schwarz inequality, one gets:

| ż S ∇ µ,γ Λ s`1{2 u : ¨P pΣq ¨∇µ,γ `pxvq h ´xv h ˘| ď }Λ s`1 P pΣq∇ µ,γ `pxvq h ´xv h ˘}2 }Λ ´1{2 ∇ µ,γ u : } 2 .
Using the definition of u : given by (3.45), one has easily

}Λ ´1{2 ∇ µ,γ u : } 2 ď Cµ 1{4 |u| 2 . (3.53)
The product estimate of Proposition A.1 shows that Note that the case t 0 ď s ď t 0 `3{2 is obtained by interpolation. We now prove the estimate (3.54) for s `1 ď t 0 . The case s `1 ą t 0 `3{2 is estimated by the same technique, so we omit it for the sake of clarity. The case t 0 ď s ď t 0 `3{2 is obtained by interpolation. The quantity w " pxvq h ´xv h satisfies the following elliptic equation: # ∇ µ,γ ¨pP pΣq∇ µ,γ wq " ´∇µ,γ ¨pP pΣq∇ µ,γ xv h q ´P pΣq∇ µ,γ x ¨∇µ,γ v h w z"0 " 0, B n w z"´1 " 0.

@0 ď s `1 ď t 0 , }Λ s`1 P pΣq∇ µ,γ `pxvq h ´xv h ˘}2 ď Cp1 `}Q} L 8 H t 0 q}Λ s`1 ∇ µ,γ `pxvq h ´xv h ˘}2 @t 0 `3{2 ď s `1, }Λ s`1 P pΣq∇ µ,γ `pxvq h ´xv h ˘}2 ď Cp1 `}Q| L 8 H t 0 q}Λ s`1 ∇ µ,γ `pxvq h ´xv h ˘}2 `}Λ s`1 Q} 2 }∇ µ,
(3.55)

We now prove the following elliptic regularity type result:

Lemma 3.8 Let t 0 ą d{2, s ě 0 and Σ be the diffeomorphism from Ω to S defined by (1.10). Let ζ, b P H t0`1 X H s`1{2 pR d q be such that Dh min ą 0, @X P R d , 1 `εζpXq ´βbpXq ě h min .

We consider the following elliptic problem: # ∇ µ,γ ¨pP pΣq∇ µ,γ wq " ´∇µ,γ ¨g `f w z"0 " 0, B n w z"´1 " 0.

(3.56)

Then, there exists a unique variational solution w P H 1 0,surf pSq to the boundary value problem (3.56). Moreover, one has :

@0 ď s ď t 0 `3{2, ? µ@}Λ s ∇ µ,γ w} 2 ď M ps `1{2qp}g} H s,1 `}f } H s´1,0 q @t 0 `3{2 ď s, ? µ@}Λ s ∇ µ,γ w} 2 ď M p}g} H s,1 `}f } H s´1,0 `|pεζ, βbq| H s`1{2 p}g} H t 0 `1{2,1 `}f | H t 0 ´1{2,0 q.
Proof By definition, w is a variational solution to (3.56) if for all θ P H 1 0,surf pSq, one has ż

S ∇ µ,γ w ¨P ∇ µ,γ θ " ´żS p∇ µ,γ ¨gqθ `żS f θ " ż S g ¨∇µ,γ θ `żS f θ ´żz"´1 gθ.
(3.57)

The existence and uniqueness follows from the coercivity of P given by (3.44) and the Lax-Milgram Theorem. We now introduce Λ s δ " Λ s χpδΛq for δ ą 0 and χ a smooth and compactly supported function, equals to 1 in a neighbourhood of zero. If w P H 1 0,surf pSq is the variational solution of (3.56), then pΛ s δ q 2 w is also in H 1 0,surf pSq, and thus, taking θ " pΛ s δ q 2 w in (3.57) (recall that P " I `Q):

ż S ∇ µ,γ Λ s δ w ¨P ∇ µ,γ Λ s δ θ " ż S Λ s δ g ¨∇µ,γ Λ s δ w `żS Λ s δ f Λ s δ w ´żz"´1 Λ s δ gΛ s δ w `żS rQ, Λ s δ s∇ µ,γ w ¨∇µ,γ Λ s δ w.
We now use Cauchy-Schwarz inequality and the coercivity of P (see (3.44)) to get:

kpΣq}∇ µ,γ Λ s δ w} 2 2 ď }Λ s δ g} 2 }∇ µ,γ Λ s δ w} 2 `}Λ s´1 δ f } 2 }Λ s`1 δ w} 2 `|Λ s`1{2 δ wp., ´1q| 2 |Λ s´1{2 δ g| 2 `}rQ, Λ s δ s∇ µ,γ w} 2 }∇ µ,γ Λ s δ w} 2 . (3.58)
Since Λ s`1 δ w P H 1 0,surf pSq, one has using Poincaré's inequality (recall that S " R d ˆp´1, 0q and H 1 0,surf pSq is the set of H 1 functions of S with vanishing trace at the surface):

}Λ s`1 δ w} 2 ď }Λ s δ w} H 1,0 ď 1 ? µ }Λ s δ ∇ µ,γ w} L 2 , (3.59)
where the From now, the idea of the proof is to show a commutator estimate of the form

@0 ď s ď t 0 `3{2, }rΛ s δ , Qsp∇ µ,γ wq} 2 ď M ps `1{2q}Λ s´ε δ ∇ µ,γ w} 2 @t 0 `3{2 ď s, }rΛ s δ , Qsp∇ µ,γ wq} 2 ď M p|pεζ, βbq| H s`1{2 p}g} H t 0 `1{2,1 `}f } H t 0 ´1{2,0 q}Λ s´α δ ∇ µ,γ w} 2 , (3.61)
for some α ą 0.

Putting together (3.59), (3.60) and (3.61) into (3.58), letting δ goes to zero and using a finite induction on s, one gets the result of Lemma 3.8. However the commutator estimate (3.61) is technical to obtain, and therefore we omit the proof for the sake of clarity (see [START_REF] Lannes | The water waves problem[END_REF] Lemma 2.38 for details).

l

We now go back to the proof of (3.54). For 0 ď s `1 ď t 0 , one has, using Lemma 3.8:

}Λ s`1 ∇ µ,γ ppxvq h ´xv h q} 2 ď M ps `1{2q 1 ? µ p}P pΣqp∇ µ,γ xqv h } H s`1,1 `}P pΣq∇ µ,γ x ¨∇µ,γ v h } H s,0 q ď M ps `1{2qp1 `}Q} H t 0 `1,1 qp}v h } H s`1,1 `}∇ µ,γ v h } H s,0

Local existence in weighted Sobolev Spaces

We prove here an existence result for the Water-Waves equation in weighted Sobolev spaces (see also [START_REF] Nguyen | Pseudo-local property of gravity water waves system[END_REF] for another use of weighted spaces for the Water-Waves). We recall that x denotes the identity of R d , and we define, for all N ě 2 the energy E N x by

E N x " E N pζ, ψq `ÿ αPN d ,1ď|α|ďN ´2 |xζ pαq | 2 2 `|Pxψ pαq | 2 2
where E N is the standard energy for the Water-Waves equations given by (3.38).

Theorem 3.9 Let us consider the assumptions of Theorem 3.1, and then consider T ą 0 and pζ, ψq the unique solution provided by the theorem on r0; T ε_β s of the Water-Waves equation (1.4). If pζ 0 , ψ 0 q P E N x , then one has pζ, ψq P L 8 pr0; T ε _ β s, E N x q, with pζ, ψq L 8 pr0; T ε_β s,E N

x q ď C 2 , where C 2 is a constant of the form C 2 " CpE N pU 0 q, 1 hmin , 1 a0 q with C a non decreasing continuous function of its arguments. than for the "Sobolev" norms E N . This is due to the presence of commutators of the form rG, xsψ pαq in the evolution equation for ψ pαq , which are of order 1 (at least) in ψ pαq .

-Note also that we control Pψ pαq , ζ pαq only for |α| ě 1. This is due to the fact that we only control terms of the form xB x ϕ.

Proof The proof is an adaptation of the proof of the Theorem 3.1 (see [START_REF] Lannes | The water waves problem[END_REF] Chapter 4 for a full proof). Therefore, we only give the main ideas and insist on the specificity of using weights. Considering the result given by Theorem 3.1, we only need to recover estimates for weighted norms (estimates for the "classical Sobolev" norms of E N are done in the proof of the local existence result of [START_REF] Lannes | The water waves problem[END_REF]). We recall (see for instance [START_REF] Lannes | The water waves problem[END_REF] Chapter 3 for reference) that one has: Therefore, we set, for all 0 ď |α| ď N ´2:

E α " 1 2µ
pGxψ pαq , xψ pαq q 2 `1 2 pxζ pαq , xζ pαq q and look for a control of E α . We now differentiate E α with respect to time and get, using the symmetry of G:

d dt E α " pGxψ pαq , B t xψ pαq q 2 `pdGpεB t ζqxψ pαq , xψ pαq q 2 `pB t xζ pαq , xζ pαq q 2 `1 2 pxζ pαq , pB t aqxζ pαq q 2 .

We now need an equation in terms of ζ pαq , ψ pαq . To this purpose, one computes B α of the equations (1.4). One gets in the first equation a term of the form B α Grεζ, βbsψ " Grεζ, βbsB α ψ `ÿ νăα,δ1`...`δm`l1`...`ln`ν"α dGpB δ1 εζ, ..., B δm εζ, B l1 βb, ..., B ln βbqB ν ψ where dG denotes the shape derivative of Grεζ, βbs with respect to the bottom b and the surface ζ. We therefore obtain, after computations, a system of the form (see [START_REF] Lannes | The water waves problem[END_REF] The following Proposition states a L 8 embedding result for the Beppo-Levi spaces:

Proposition A.2 For all s P R:

(1) The mapping u Þ Ñ u |z"0 extends continuously from H s`1,1 to H s`1{2 pR d q.

(2) The space H s`1{2,1 is continuously embedded in L 8 H s .

B 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.

Proposition B.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 .

(

5 Figure 1 :

 51 Figure 1: Graph of Φ 2

Remark 3. 10 -

 10 Note that there are less space derivatives for the weighted norms |xζ pαq | 2 2 `|Pxψ pαq | 2 2

  |xB x ϕ| 2 qq.The Theorem 2.6 gives a better decay than Theorem 2.5 for the linear operator of the Water-Waves, however its use in view of long time results for the full Water-Waves equation (1.4) may require to prove that solutions are bounded in L 1 norm, or in |x ¨|L 2 norm, which is difficult. Indeed, the proof of local existence for this equation in weighted spaces requires the control of the commutator rG, xs, which is difficult to get (while the one for rG, xsB x is less difficult to get, see later Section 3.2). consists in the precise study of where the phase Φ may be stationary. Here, the derivative of the phase Φpξq " tpωpξq `xξ{tq(2.14) may vanish and we are therefore led to study the behaviour of the second derivative Φ 2 . However, |Φ 2 pξq| " ξÑ8 C |ξ| 3{2 and therefore Φ 2 cannot be bounded from below by a constant and one cannot apply Van der Corput's Lemma 2.4. We therefore need to compensate the bad bound of Φ 2 with good weighted estimates on p ϕ.

	The proof is based on a stationary phase result and a use of the Littlewood decomposition. More
	precisely, the control of oscillatory integrals of the form
	ż b
	e itΦpξq dξ
	a

  As suggested by the behaviour of ω 2 , we split the study of the linear operator e itωpDq into low and high frequencies cases. To this purpose, one can define χ a smooth compactly supported function equals to 1 in the neighbourhood of the origin, and write ϕ " χpDqϕ`p1´χpDqqϕ, with as usual { χpDqϕpξq " χpξq p ϕpξq for all ϕ P SpRq. Note that the estimate of Theorem 2.5 stay true if we prove it for χpDqp ? µ|D γ |qϕ or p1 ´χp ? µ|D γ |qqϕ instead of ϕ.

		1 ? µ gp ? µξq with gpξq "	a |ξ| tanhp|ξ|q. It is easy to show that
			|g 2 pξq| " ξÑ0	|ξ|,	|g 2 pξq| " ξÑ`8	|ξ| ´3{2 .	(2.15)
	We write:	ż
	e itωpDq ϕ "	R ż	e iptωpξq`xξq p ϕpξqdξ
	"	|ξ|ďy0{ ? µ	e iptωpξq`xξq p ϕpξqdξ	`ż|ξ|ąy0{ ? µ	e iptωpξq`xξq p ϕpξqdξ.	(2.16)

  We now give a control of the oscillatory integral of (2.19). Recall that ω " 1 Therefore, there exists C ą 0 independent on µ such that |Φ 2 psq| ě Cµ|ξ| on r0; y 0 {

											? µ gp ? µξq with (2.15). ? µs (see also Figure
	1). Therefore, one has:							@δ ď s ď	y 0 ? µ	,	|Φ 2 psq| ě µs ě µδ.
	Using Van der Corput's Lemma 2.4, one gets the control:
											sup δďξď y 0 ? µ	|	ż ξ δ	e itΦpsq ds| ď	C ? µδt	.	(2.20)
	Putting together (2.20) into (2.19), one gets:
						|	ż δďξď	y 0 ? µ	e itΦpξq p ϕpξqdξ| ď	C δµ 1{4	1 µδt ?	p|ϕ| 2 `|xB x ϕ| 2 q.	(2.21)
	Combining the estimates of (2.18) and (2.21), one obtains:
		|		ż	|ξ|ď	y 0 ? µ	e itΦpξq p ϕpξqdξ| ď p ? 2δ	`C δµ 1{4	1 µδt ?	qp|ϕ| 2 `|xB x ϕ| 2 q.
	1 µ 3{8 t 1{4 and therefore, we finally get: |ξ|ď The above quantity is minimal for δ " | ż y 0 ? µ e itΦpξq p ϕpξqdξ| ď 1 µ 3{16 t 1{8 .	(2.22)
											y0 ? µ r):
		|	ż δďξď	y 0 ? µ	e itΦpξq p ϕpξqdξ| ď |	´żδďξď y 0 ? µ	ż ξ δ	e itΦpsq ds	d dξ	p ϕpξqdξ|.
	Using Cauchy-Schwarz inequality, one gets:
	|	ż δďξď	y 0 ? µ	e itΦpξq p ϕpξqdξ| ď ď	? y 0 µ 1{4 sup δďξď y 0 µ ? δµ 1{4 sup δďξď y 0 | ? µ C	ż ξ δ | ż ξ e itΦpsq ds| ˆ|1 δď|ξ|	1 ξ	pξ	d dξ	p ϕqpξq| 2

δ e itΦpsq ds|p|ϕ| 2 `|xB x ϕ| 2 q.

(2.19)

  The following lines are an adaptation of[START_REF] Bulut | An optimal decay estimate for the linearized water wave equation in 2d[END_REF]. Using the Littlewood-Paley decomposition, we split e itωpDq ϕ into

	with:
			S 1 " S 2 " S 3 "	ÿ 2 k ďλptq ÿ |P k e itωpDq ϕpxq|, |P k e itωpDq ϕpxq|, λptqď2 k ďΛptq ÿ 2 k ěΛptq
	where supp p ϕ Ă r y0 ? µ ; `8r. e itωpDq ϕ " 2 k ďλptq ÿ kPZ	P k e itωpDq ϕ	`ÿ λptqď2 k ďΛptq	P k e itωpDq ϕ	norm the quantity `ÿ 2 k ěΛptq P k e itωpDq ϕ ş R e itΦpξq p ϕpξqdξ
	where	λptq " C 1 p1 `|t{	? µ|q ´1,	Λptq "	1 C 1	p1 `|t{	? µ|q,

with C 1 ą 1. We therefore have |pe itωpDq ϕqpxq| ď S 1 `S2 `S3 ,

  We are therefore led to control the oscillatory integral ş ξ 2 k´1 e itΦpsq ds with ξ P suppψ k X supp p ϕ. We put this integral under the form ż ξ itppx{tqs`ωpsqq ds and we use the following lower bound for the derivative of the phase:|x{t `ω1 psq| ě |ω 1 psq| ´|x{t|. Now, recall that k P I 1 and therefore |x{t| ď 2 ´k{2 C 2 . Moreover, recall that ωpξq " 1

									ż ξ
				2 k´1	e itΦpsq ds "	2 k´1
	? µ gp ? µξq with µ; `8r. Therefore, the derivative of ω is bounded from ? µ 1{4 |s| ´1{2 where C 3 is independent on µ. We therefore get: |ξ| ´1{2 , and that supp p ϕ Ă ry 0 { below on supp p |g 1 pξq| " ξÑ`8 ϕ by C3
				|x{t `ω1 psq| ě	C 3 µ 1{4 |s| ´1{2 ´2´k{2 C 2 .
										e itΦpsq dsq y P k ϕpξqdξ
					"	´żR	k´1 2 k´1 ż ξ e itΦpsq ds	d dξ	y P k ϕpξqdξ
	by integrating by parts, and recalling that supp y P k ϕ Ă suppψ k Ă t2 k´1 ď |ξ| ď 2 k`1 u. We now use Cauchy-Schwarz inequality to get:
	|	ż	R	e itΦpξq y P k ϕpξqdξ| ď C2 k{2	sup ξPsuppψ k Xsupp p ϕ	|	ż ξ 2 k´1	e itΦpsq ds|	ˆ| d dξ	y P k ϕ| 2 .	(2.25)

e Now, since s P suppψ k , one has

  2δ2 ´sk p|ϕ| H s `|xB x ϕ| 2 q (2.31)where we used Lemma 2.3 to derive the last inequality, with s ą 1{2 to be set. We now focus on the control of the first integral of the right hand side of (2.30) (the last integral is controlled by using the same technique). Integrating by parts, one gets: 1{4 |ξ| 3{2 and therefore, one has:

	R	e itΦpξq y P k ϕpξqdξ "	ż c´δ 2 k´1	e itΦpξq y P k ϕpξqdξ	`ż c`δ c´δ	e itΦpξq y P k ϕpξqdξ	c`δ `ż 2 k`1	e itΦpξq y P k ϕpξqdξ.	(2.30)
	The second integral of the right hand side of (2.30) is estimated as follows:
						|	ż c`δ c´δ	e itΦpξq y P k ϕpξqdξ| ď 2δ| y P k ϕpξq| 8
		|	ż c´δ 2 k´1	e itΦpξq y P k ϕpξqdξ| " | ď 2 k{2 ´ż c´δ 2 k´1 ξPsupp p ż ξ 2 k´1 sup ϕXr2 k´1 ;c´δs e itΦpsq ds | ż ξ d dξ 2 k´1 P k ϕqpξqdξ| p y e itΦpsq ds||	d dξ	y P k ϕ| 2 .	(2.32)
	One now estimates |	ş ξ 2 |Φ 1 psq| " ě	ż s c ż s c	|Φ 2 psq|ds `Φ1 pcq C µ 1{4 ξ 3{2
									ě Cµ ´1{4 p	1 ? c	´1 ? s	q
									ě Cµ ´1{4	? c	c sp ? ? ´s c `?sq	ě Cµ ´1{4 δ 2 3k{2

ď k´1 e itΦpsq ds| for ξ P r2 k´1 ; c ´δs X supp p ϕ. Recall that ωpξq " 1 ? µ gp ? µξq with (2.15), and that supp p ϕ Ă ry 0 { ? µ; `8r. Therefore, for ξ P r2 k´1 ; c ´δs X supp p ϕ, there exists C ą 0 such that |Φ 2 pξq| ě C µ

  We use the same notations as ones of the proof of Theorem 2.5. One can control the first integral of the right hand side of (2.16) differently:One gets the first estimate of Theorem 2.6.Another way to control the first integral of the right hand side of (2.16) is by integrating by parts:

										1 p1 `t{ ? µq 1{8 `1 p1 `t{ ? µq 1{2 qp|ϕ| H 1 `|xB x ϕ| 2 q.
			ż							ż
			|	|ξ|ď	y 0 ? µ	e iptωpξq`xξq ϕpξqdξ| ď |	|ξ|ď	y 0 ? µ
	We now control the integral	ş	|ξ|ď	y 0 ? µ
										|ξ|ď	y 0 ? µ	e itΦpξq dξ ď	C pµtq 1{3 .
		|	ż |ξ|ď	y 0 ? µ	e iptωpξq`xξq ϕpξqdξ| " |	ż |ξ|ď	y 0 ? µ	ż ξ 0	e iptωpsq`xsq ds	d dξ	p ϕpξqdξ|
	and thus, using Cauchy-Schwarz inequality, one gets:
	|	ż |ξ|ď	y 0 ? µ	e iptωpξq`xξq ϕpξqdξ| ď	? y 0 µ 1{4 sup ξPr0; y 0 ? µ r |	ż ξ 0	e iptωpsq`xsq ds||	d dξ	p ϕ| 2 .
	One estimates as above							sup ξPr0; y 0 ? µ r |	ż ξ 0	e iptωpsq`xsq ds| ď	C pµtq 1{3

l We now give a short proof for a better decay estimate, but with other norms of control: Proof of Theorem 2.6 e iptωpξq`xξq dξ| 8 |ϕ| L 1 . e itΦpξq dξ in L 8 norm. One has Φ 3 pξq " ω 3 pξq " µg 3 p ? µξq with the notations of the proof of Theorem 2.5. One can check by computation that g 3 pξq " ξÑ0 ´1 and therefore, using Van der Corput's Lemma 2.4, one gets: ż

  H s`1{2 `|Pv| H t 0 `1 |pεζ, βbq| H s`1 q. P pΣq∇ µ,γ v h ¨p∇ µ,γ xqΛ s`1{2 u : | ď µM ps `1q|Pv| H s`1{2 `x|Pv| H t 0 `1 |pεζ, βbq| H s`1 qy s`1{2ąt0 |u| 2

	Combining (3.49) and (3.50) in (3.48), one finally gets:
	@0 ď s `1{2 ď t 0 , @t 0 `3{2 ă s `1{2,	› › ›Λ s`1{2 P pΣq∇ µ,γ v h › › ›Λ s`1{2 P pΣq∇ µ,γ v h	› › › 2 › › › 2	ď	ď µM p|Pv| (3.51) ? µM ps `1q|Pv| H s`1{2 ?
	Combining (3.51) with (3.47), one finally gets:
		ż
	|	S
						.50)

  γ `pxvq h ´xv h ˘}L 8 H t 0 .One has }Q} L 8 H t 0 ď M 0 and }Λ s`1 Q} 2 ď M 0 |pεζ, βbq| H s`3{2 . The proof of Proposition 3.5 is completed if one can prove:

@s ě ´1{2, }Λ

s`1 ∇ µ,γ `pxvq h ´xv h ˘}2 ď µM ps `1{2q|Pϕ| H s`1 `x|Pϕ| H t 0 `2 |pεζ, βbq| H s`1 qy s`1ąt0 . (3.54)

  Chapter 4 for details): $ & % B t ζ pαq `εV ¨∇γ ζ pαq ´1 µ Gψ pαq " R α B t ψ pαq `aζ pαq `εV ¨∇γ ψ pαq " S α (3.70) and therefore, using Proposition (B.1) again to control |V | H t 0 by the energy, one finally gets | 1 µ pGxψ pαq , pV ¨∇γ xqψ pαq q 2 | ď CpE N qE α . (3.76) Putting together (3.75) and (3.76), one proved| 1 µ pGxψ pαq , xV ¨∇γ ψ pαq q 2 | ď CpE N qE α . (3.77)The first term of the third line of (3.72) is estimated by using Proposition B.3. The only non trivial remaining term to control in (3.72) is the last one, which is the commutator 1 µ prG, xsψ pαq , xaζ pαq q 2 . Recall that |α| ą 1 and that:ψ pαq " B α ζ ´εwB α ψand one gets therefore, using Proposition 3.5, one can control both of these terms by CpE N qE α . One can obtain by summing on all α, 1 ď |α| ď N ´1 the following energy estimate: CpE N qE N x with C a continuous function of its arguments. Using a Gronwall's Lemma, one can conclude and end the proof of the Theorem.l.We recall the notation a _ b " maxpa, bq and we define L 8 H s " L 8 pp´1; 0q; H s pR d qq and use the notation xay sąt0 " a if s ą t 0 and else 0 . Proposition A.1 Let t 0 ą d{2. If s ě ´t0 , f, g P L 8 H t0 X H s,0 , one has f g P H s,0 and }f g} H s,0 ď C }f } L 8 H t 0 }g} H s,0 `x}f } H s,0 }g} L 8 H t 0 y sąt0 .

	d dt	E N x ď

A Estimates on the flat strip S

  )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| H t 0 `1 , |b| H t 0 `1 , |ζ| H s`1{2 , |b| H s`1{2 q.H s`1 pR d q into H s´1{2 pR d q and one has|Gψ| H s´1{2 ď µM ps `1q|Pψ| H s`1{2 ,where M ps `1q is a constant of the form Cp 1 h0 , |ζ| H t 0 `1 , |b| H t 0 `1 , |ζ| H s`1 , |b| H s`1 q.

	( 2)	The operator G maps continuously

.

where we used the product estimate of Proposition A.1 to derive the last inequality, and where the 1 ? µ factor has been canceled by ∇ µ,γ x which has a ? µ factor (recall the definition of ∇ µ,γ ). Using Lemma 3.3, we get the bound }∇ µ,γ v h } H s,0 ď ? µ|Pv| H s .

(3.62)

To control v h in H s`1,1 norm, we recall that v " B x ϕ and we notice that pB x ϕq h ´Bx pϕq h P H 1 0,surf pSq and we write }v h } H s`1,1 ď }pB x ϕq h ´Bx pϕq h } H s`1,1 `}B x pϕq h } H s`1,1 .

(3.63)

To control the first term of the right hand side of (3.63), we use the Poincaré's inequality on the flat strip S:

Now, if one defines w " pB x ϕq h ´Bx pϕq h then w satisfies the following boundary problem: # ∇ µ,γ ¨pP pΣq∇ µ,γ wq " ´∇µ,γ ¨g w z"0 " 0, B n w z"´1 " ´g ¨ez (3.64) with g " rP pΣq, B x s∇ µ,γ ϕ h , and e z the unit normal vector in the vertical direction. Adapting the proof of Lemma 3.8 (see also Lemma 2.38 in [START_REF] Lannes | The water waves problem[END_REF]), one can prove

and using Proposition 3.3, one finally gets

To control the second term of the rhs of (3.63), one uses Proposition 3.3 again:

which is the desired result (3.54). It concludes the proof of the first point of Proposition 3.5.

The proof of the second point of Proposition 3.5 only requires a small adaptation of the proof above. The only technical change is the control of }v h } H s`1,1 . We write, with v " f B x ϕ:

The second term of the right hand side of (3.68) is controlled using Proposition A.1, and the control of pB x ϕq h proved above. To control the first term of the right hand side of (3.68), one remarks that w " pf B x ϕq h ´f pB x ϕq h P H 1 0,surf pSq solves the following boundary problem: # ∇ µ,γ ¨pP pΣq∇ µ,γ wq " ´∇µ,γ f P pΣq∇ µ,γ pB x ϕq h ´∇µ,γ ¨pP pΣqpB x ϕq h ∇ µ,γ f q w z"0 " 0,

and we use the Poincaré's inequality on the flat strip S to control }pf B x ϕq h ´f pB x ϕq h } H s`1,1 by }∇ µ,γ w} H s,0 , and adapt the proof of Lemma 3.8 above to get the control of this latter term. l with

with C a continuous function of its arguments. In order to get a control of the form (3.71), one can adapt the proof of the control for the shape derivatives of G given in Proposition 3.28 of [START_REF] Lannes | The water waves problem[END_REF] (we do not detail this proof here). We therefore have, replacing B t pζ pαq , ψ pαq q by their expression given by (3.70):

(3.72)

The first two terms of (3.72) are the one of order 1 with respect to the unknowns xζ pαq , xψ pαq but cancel one another, thanks to the symmetry of the equation.

The two terms of the second line of (3.72) are of contributions of order 0 to the energy estimate, with respect to the unknowns, thanks to the symmetry. More precisely, one computes, integrating by parts:

pxV j B j ζ pαq , xaζ pαq q 2 ´pB j pV j aqxζ pαq , xζ pαq q 2 ´ppB j xqV j aζ pαq , xζ pαq q 2

and therefore one has where C is continuous and non decreasing. For the control of the second term of the second line of (3.72), one writes:

We use Proposition B.5 to write (recall the notations of M given by (3.41)):

and again, using the Proposition (B.1) one can control the W 1,8 norm of V by the energy and get

We now use Proposition B.3 with s " 1 to compute:

and one can prove, using the definition of P and standard Sobolev estimates:

Moreover, it is possible to replace Gψ 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).

Remark B.2

In all this paper, we consider the Water-Waves problem in finite depth. This is crucial for all these regularity results on G. For instance, in the linear case ζ " b " 0, the Dirichlet-Neumann operator is |D γ | tanhp|D γ |q in finite depth, while it is |D γ | in infinite depth. The low frequencies are therefore affected differently.

Proposition B.3 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 Proposition gives estimates of the shape derivatives of G.

Proposition B.4 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,

The following commutator estimate is useful (see [START_REF] Lannes | The water waves problem[END_REF] Proposition 3.30):

Proposition B.5 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,

where M is a constant of the form Cp 1 h0 , |ζ| H t 0 `2 , |b| H t 0 `2 q.

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