Florent Chazel
email: florent.chazel@math.u-bordeaux1.fr

Mathematical Modelling and Numerical Analysis Will be set by the publisher Modélisation Mathématique et Analyse Numérique INFLUENCE OF BOTTOM TOPOGRAPHY ON LONG WATER WAVES

Keywords: 1991 Mathematics Subject Classification. 76B15, 35L55, 35C20, 35Q35. Water waves, uneven bottoms, bottom topography, long-wave approximation, asymptotic expansion, hyperbolic systems, Dirichlet-Neumann operator

We focus here on the water waves problem for uneven bottoms in the long-wave regime, on an unbounded two or three-dimensional domain. In order to derive asymptotic models for this problem, we consider two different regimes of bottom topography, one for small variations in amplitude, and one for strong variations. Starting from the Zakharov formulation of this problem, we rigorously compute the asymptotic expansion of the involved Dirichlet-Neumann operator. Then, following the global strategy introduced by Bona, Colin and Lannes in [6], new symetric asymptotic models are derived for each regime of bottom topography. Solutions of these systems are proved to give good approximations of solutions of the water waves problem. These results hold for solutions that evanesce at infinity as well as for spatially periodic ones.

, nous obtenons ensuite de nouveaux modèles asymptotiques symétriques pour chaque régime de variation topographique du fond. Nous prouvons alors que les solutions de ces systèmes fournissent de bonnes approximations aux solutions des équations d'Euler surface libre. Ces résultats sont valables aussi bien pour des solutions évanescentes à l'infini que pour des solutions spatialement périodiques.

Introduction

Generalities

This paper deals with the water waves problem for uneven bottoms which consists in describing the motion of the free surface and the evolution of the velocity field of a layer of fluid, under the following assumptions : the fluid is ideal, incompressible, irrotationnal, and under the only influence of gravity. Earlier works have set a good theoretical background for this problem : its well-posedness has been discussed

Presentation and formulation of the problem

In this paper, we work indifferently in two or three dimensions. Let us denote by X ∈ R d the transverse variable, d being equal to 1 or 2. In the two-dimensional case, d = 1 and X = x corresponds to the coordinate along the primary direction of propagation whilst in the three-dimensional case, d = 2 and X = (x, y) represents the horizontal variables. We restrict our study to the case where the free surface and the bottom can be described by the graph of two functions (t, X) → η(t, X) and X → b(X) defined respectively over the surface z = 0 and the mean depth z = -h 0 both at the steady state, t corresponding to the time variable. The time-dependant domain Ω t of the fluid is thus taken of the form :

Ω t = {(X, z), X ∈ R d , -h 0 + b(X) ≤ z ≤ η(t, X)} .
In order to avoid some special physical cases such as the presence of islands or beaches, we set a condition of minimal water depth : there exists a strictly positive constant h min such that η(t, X) + h 0 -b(X) ≥ h min , (t, X) ∈ R × R 2 .

(0.1)

For the sake of simplicity, we assume here that b and all its derivatives are bounded.

n -• V | z=-h0+b(X) = 0 for t ≥ 0, X ∈ R d , (0.2)
where n + = 1 √ 1+|∇η| 2 (-∇η, 1) T denotes the outward normal vector to the surface and n -= √ 1+|∇b| 2 (∇b, -1) T denotes the outward normal vector to the bottom. The first equation corresponds to the Euler equation for a perfect fluid under the influence of gravity (which is characterized by the term -ge z where e z denotes the base vector along the vertical component). The second and third one characterize the incompressibility and irrotationnality of the fluid. The fourth and last ones deal with the boundary conditions at the surface and the bottom. These are given by the usual assumption that they are both bounding surfaces, i.e. surfaces across which no fluid particles are transported. As far as the pressure P is concerned, we assume that it is constant at the surface by neglicting the surface tension. Up to a renormalization, we can assume that it is equal to zero at the surface.

In this paper, we use the Bernoulli formulation of the water-waves equations. The conditions of incompressibility and irrotationnality ensure the existence of a potential flow φ such that V = ∇ X,z φ. From now on, we separate the transverse variable X ∈ R d and the vertical variable z ∈ R : the operators ∇ and ∆ act only on the transverse variable X ∈ R d so that we have V = ∇φ + ∂ 2 z φ. The use of the potential flow φ instead of the velocity V leads to the following formulation of (0.2) :

                     ∂ t φ + 1 2 |∇φ| 2 + |∂ z φ| 2 + gz = -P in Ω t , t ≥ 0 , ∆φ + ∂ 2 z φ = 0 in Ω t , t ≥ 0 , ∂ t η -1 + |∇η| 2 ∂ n + φ| z=η(t,X) = 0 for t ≥ 0, X ∈ R d , ∂ n -φ| z=-h0+b(X) = 0 for t ≥ 0, X ∈ R d , (0.3)
where we used the notations

∂ n -= n -• ∇ ∂ z and ∂ n + = n + • ∇ ∂ z .
Separating the variables X and z in the boundary conditions and taking the trace of (0.3) on the free surface thus leads to the system :

                   ∆φ + ∂ 2 z φ = 0 in Ω t , t ≥ 0 , ∂ t φ + 1 2 |∇φ| 2 + |∂ z φ| 2 + gη = 0 at z = η(t, X), X ∈ R d , t ≥ 0 , ∂ t η + ∇η • ∇φ -∂ z φ = 0 at z = η(t, X), X ∈ R d , t ≥ 0 , ∇b • ∇φ -∂ z φ = 0 at z = -h 0 + b(X), X ∈ R d , t ≥ 0 .
(0.4)

We now perform a non-dimensionalisation of these equations using the following parameters : λ is the typical wavelength, a the typical amplitude of the waves, h 0 the mean depth of the fluid, b 0 the typical amplitude of the bottom, t 0 = λ √ gh0 a typical period of time ( √ gh 0 corresponding to sound velocity in the fluid) and φ 0 = λa h0 √ gh 0 . Introducing the following parameters :

ǫ = a h 0 ; β = b 0 h 0 ; S = aλ 2 h 3 0 ,
and taking the Stokes number S to be equal to one, one gets for the non-dimensionnalized version of (0.4) :

                   ε ∆φ + ∂ 2 z φ = 0 -1 + β b ≤ z ≤ εη, X ∈ R d , t ≥ 0 , ∂ t φ + 1 2 ε|∇φ| 2 + |∂ z φ| 2 + gη = 0 at z = εη, X ∈ R d , t ≥ 0 , ∂ t η + ε ∇η • ∇φ -1 ε ∂ z φ = 0 at z = εη, X ∈ R d , t ≥ 0 , ∂ z φ -εβ ∇b • ∇φ = 0 at z = -1 + β b, X ∈ R d , t ≥ 0 .
(0.5)

The final step consists in recovering the Zakharov formulation by reducing the previous system (0.5) to a system expressed at the free surface. To this end, we introduce the trace of the velocity potential φ at the free surface, namely ψ : ψ(t, X) = φ(t, X, ε η(t, X)) , and the operator Z ε (εη, βb) which maps ψ to ∂ z φ| z=ε η . This operator is defined for any

f ∈ (C 1 ∩ W 1,∞ )(R d ) by : Z ε (εη, βb)f :           H 3 2 (R d ) -→ H 1 2 (R d ) f -→ ∂ z u |z=εη with u solution of : ε ∆u + ∂ 2 z u = 0, -1 + β b ≤ z ≤ εη , ∂ z u -εβ ∇b • ∇u = 0, z = -1 + βb , u(X, εη) = f, X ∈ R d .           . (0.6)
Using this operator and computing the derivatives of ψ in terms of ψ and η, the final formulation (S 0 ) of the water waves problem reads :

(S 0 )      ∂ t ψ -ε∂ t ηZ ε (εη, βb)ψ + 1 2 ε |∇ψ -ε ∇ηZ ε (εη, βb)ψ| 2 + |Z ε (εη, βb)ψ| 2 + η = 0 , ∂ t η + ε∇η • [ ∇ψ -ε∇ηZ ε (εη, βb)ψ ] = 1 ε Z ε (εη, βb)ψ . (0.7)

Organization of the paper

The aim of this paper is to derive and study two different asymptotic regimes based each on a specific assumption on the parameter β which characterizes the topography of the bottom. The first assumption deals with the case β = O(ε) which corresponds to the physical case of a bottom with small variations in amplitude. The second one deals with the more complex case β = O(1) which corresponds to the physical case of a bottom with high variations in amplitude.

The following part will be devoted to the asymptotic expansion of the operator Z ε (εη, βb) in the two regimes mentionned above. To this end, a general method is introduced and rigorously proved which aims at deriving asymptotic expansions of Dirichlet-Neumann operators for a large class of elliptic problems. This result is then applied in each regime, wherein a formal expansion is performed and an asymptotic Boussinesq-like model of (0.7) is derived. The second and third part are both devoted to the derivation of new classes of equivalent systems, following the strategy developped in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF]. In the end, completely symmetric systems are obtained for each bottom topography regime : convergence results are proved showing that solutions of these symmetric asymptotic systems tend to associated solutions of the water waves problem.

Asymptotic expansion of the operator Z ε (εη, βb)

This section is devoted to the asymptotic expansion of the operator Z ε (εη, βb) defined in the previous section as ε tends to zero, in both regimes β = O(ε) and β = O(1). To this end, we first enounce some general results on elliptic equations on a strip : the final proposition gives a general rigourously justified method for determining an approximation of Dirichlet-Neumann operators. This result is then applied to the case of the operator Z ε (εη, βb) and two asymptotic models with bottom effects are derived.

Elliptic equations on a strip

In this part, we aim at studying a general elliptic equation on a domain Ω given by :

Ω = {(X, z) ∈ R d+1 /X ∈ R d , -h 0 + B(X) < z < η(X)} ,
where the functions B and η satisfy the following condition :

∃ h min > 0 , ∀X ∈ R d , η(X) -B(X) + h 0 ≥ h min . (1.1)
Let us consider the following general elliptic boundary value problem set on the domain Ω :

-∇ X,z . P ∇ X,z u = 0 in Ω , (1.2) u | z=η(X) = f and ∂ n u | z=-h 0 +B(X) = 0 , (1.3)
where P is a diagonal (d + 1) × (d + 1) matrix whose coefficients (p i ) 1≤i ≤d+1 are constant and strictly positive.

Straightforwardly P is coercive. We denote by ∂ n u |z=-h0+B(X) the outward conormal derivative associated to P of u at the lower boundary {z = -h 0 + B(X)}, namely :

∂ n u | z=-h 0 +B(X) = -n -• P ∇ X,z u | z=-h 0 +B(X) ,
where n -denotes the outward normal vector to the lower boundary of Ω. For the sake of simplicity, the notation ∂ n will always denote the outward conormal derivative associated to the elliptic problem under consideration.

Remark 1.1. When no confusion can be made, we denote ∇ X by ∇.

As in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF][START_REF] Lannes | Well-posedness of the water-waves equations[END_REF][START_REF] Nicholls | Reitich A new approach to analyticity of Dirichlet-Neumann operators[END_REF] we transform the boundary value problem (1.2)(1.3) into a new boundary problem defined over the flat band

S = {(X, z) ∈ R d+1 /X ∈ R d , -1 < z < 0} .
Let S be the following diffeomorphism mapping S to Ω :

S : S -→ Ω (X, z) -→ s(X, z) = (η(X) -B(X) + h 0 ) z + η(X)
.

(1.4)

Remark 1.2. As shown in [START_REF] Lannes | Well-posedness of the water-waves equations[END_REF], a more complex "regularizing" diffeomorphism must be used instead of (S) to obtain a shard dependence on η in terms of regularity, but since the trivial diffeomorphism (S) suffices for our present purpose, we use it for the sake of simplicity.

Clearly, if v is defined over Ω then v = v • S is defined over S. As a consequence, we can set an equivalent problem to (1.2)(1.3) on the flat band S using the following proposition (see [START_REF] Lannes | Sur le caractère bien posé des équations d'Euler avec surface libre[END_REF] for a proof) :

Proposition 1.3. u is solution of (1.2)(1.3) if and only if u = u • S is solution of the boundary value problem -∇ X,z . P ∇ X,z u = 0 in S , (1.5)
u |z=0 = f and ∂ n u |z=-1 = 0 , (1.6)
where P (X, z) is given by

P (X, z) = 1 η + h 0 -B M T P M , with M (X, z) = (η + h 0 -B)I d×d -(z + 1) ∇η + z ∇B 0 1 .
Consequently, let us consider boundary value problems belonging to the class (1.5) (1.6). From now on, all references to the problem set on S will be labelled with an underscore.

On the class (1.5)(1.6) of problems set on the flat band S, we have the following classical existence theorem : assuming that P and all its derivatives are bounded on S, if f ∈ H k+ 3 2 (R d ) then there exists a unique solution u ∈ H k+2 (S) to (1.5)(1.6). The proof is very classical and we omit it here.

As previously seen, we need to consider the following operator Z(η, B) which maps the value of u at the upper bound to the value of ∂ z u| z=η :

Z(η, B) :   H 3 2 (R d ) -→ H 1 2 (R d ) f -→ ∂ z u| z=η with u solution of (1.2)(1.3)   .
Remark 1.4. The operator Z ε defined in (0.6) corresponds to the operator Z in the case where P =

εI d 0 0 1 in (1.2)(1.3).
To construct an approximation of this operator Z(η, B), we need the following lemma which gives a coercitivity result taking into account the anisotropy of (1.2)(1.3).

Lemma 1.5. Let η ∈ W 1,∞ (R d ) and B ∈ W 1,∞ (R d ). Then for all V ∈ R d+1 : (V , P V ) ≥ c 0 ( ||η|| W 1,∞ , ||B|| W 1,∞ ) | √ P V | 2 ,
where c 0 is a strictly positive function given by

c 0 (x, y) = h min (d + 1) 2 min   1, 1 h min (x + h 0 + y) , min 1≤i≤d p d+1 p i (x + y) 2    .
Proof. Using Proposition 1.3 , we can write, with δ(X) = η(X) + h 0 -B(X) :

(V , P V ) = 1 δ V , M T P M V = 1 δ M V , P M V = 1 δ √ P M V , √ P M V = 1 √ δ M ( √ P V ) 2
where M = √ P M ( √ P ) -1 . Thanks to the condition (1.1), we deduce the invertibility of M and hence the invertibility of M. This yields the following norm inequality for all U ∈ R d+1 :

|U | ≤ (d + 1) √ δ M -1 ∞ 1 √ δ M U , with M -1 = 1 δ I d×d 1 δ √ p d+1 √ P d ((z + 1) ∇η -z ∇B) 0 1
.

where

|A| ∞ = sup 1≤i,j≤d+1 |a i,j | L ∞ (R d
) and P d is the d × d diagonal matrix whose coefficients are (p i ) 1≤i≤d .

If we apply the previous inequality to our problem, one gets :

(V , P V ) ≥ 1 (d + 1) 2 √ δ M -1 2 ∞ √ P V 2 .
Thanks to the expression of M -1 given above, we obtain the following inequality :

(V , P V ) ≥ c 0 ( ||η|| W 1,∞ , ||B|| W 1,∞ ) | √ P V | 2 ,
where c 0 as in the statement of the Lemma 1.5.

Let us introduce the space H k,0 (S) :

H k,0 (S) = {v ∈ L 2 (S), ||v|| H k,0 := 0 -1 |v(•, z)| 2 H k (R d ) dz 1 2 < +∞} .
The result of this subsection consists in the following theorem which aims at giving a rigourous method for deriving an asymptotic development of Z(η, B). Of course, P , and thus P , as well as the boundaries η and B, can depend on ε in the following theorem. In such cases, the proof can be easily adapted just by remembering that 0 < ε < 1.

Theorem 1.6. Let p ∈ N * , k ∈ N * , η ∈ W k+2,∞ (R d ) and B ∈ W k+2,∞ (R d ). Let 0 < ε < 1 and u app be such that -∇ X,z • P ∇ X,z u app = ε p R ε in S , (1.7)
u app |z=0 = f , ∂ n u app |z=-1 = ε p r ε , (1.8)
where (R ε ) 0<ε<1 and (r ε ) 0<ε<1 are bounded independently of ε respectively in H k+1,0 (S) and H k+1 (R d ).

Assuming that h min is independent of ε and that the coefficients (p i ) 1≤i≤d+1 of P are such that ( pi p d+1 ) 1≤i≤d are bounded by a constant γ independent of ε, we have

Z(η, B)f - 1 η + h 0 -B (∂ z u app ) |z=0 H k+ 1 2 ≤ ε p √ p d+1 C k+2 (||R ε || H k+1,0 + |r ε | H k+1 ) ,
where

C k+2 = C(|η| W k+2,∞ , |B| W k+2,∞
) and C is a non decreasing function of its arguments, independent of the coefficients (p i ) 1≤i≤d+1 .

Proof. In this proof, we often use the notation

C k = C(|η| W k,∞ , |B| W k,∞ , h 0 , h min , k, d, γ)
where C is an undefined non decreasing function of its arguments. The notation C k can thus refer to different constants, but of the same kind.

A simple calculus shows that Z(η, B) can be expressed in terms of the solution u of (1.5)(1.6) via the following relation :

Z(η, B)f = 1 η + h 0 -B ∂ z u |z=0 .
Using this fact, we can write

Z(η, B)f - 1 η + h 0 -B ∂ z u app |z=0 = 1 η + h 0 -B ∂ z (u -u app ) |z=0 .
Introducing ϕ := u app -u we use a trace theorem (see Metivier [START_REF] Métivier | Small Viscosity and Boundary Layer Methods: Theory, Stability Analysis, and Applications[END_REF] p. [START_REF] Peregrine | Long waves on a beach[END_REF][START_REF] Schneider | The long-wave limit for the water-wave problem. I. The case of zero surface tension[END_REF][START_REF] Wei | Kirby A time-dependent numerical code for extended Boussinesq equations[END_REF][START_REF] Wei | A fully nonlinear Boussinesq model for surface waves. I. Highly nonlinear, unsteady waves[END_REF][START_REF] Wu | Well-posedness in Sobolev spaces of the full water wave problem in 2-D[END_REF] to get

| Z(η, B)f - 1 η + h 0 -B (∂ z u app ) |z=0 | H k+ 1 2 ≤ C k+1 (||∂ z ϕ|| H k+1,0 + ||∂ 2 z ϕ|| H k,0 ) . (1.9)
It is clear that the proof relies on finding an adequate control of ||∂ z ϕ|| H k+1,0 and ||∂ 2 z ϕ|| H k,0 . The rest of this proof will hence be devoted to the estimate of both terms.

1. Let us begin with the estimate of ||∂ z ϕ|| H k+1,0 . To deal correctly with this problem, we introduce the following norm ||.|| Ḣ1 defined by :

||ϕ|| Ḣ1 := || √ P ∇ X,z ϕ|| L 2 (S) .
First remark that for all α ∈ N d such that |α| ≤ k, ∂ α ϕ solves :

-∇ X,z • P ∇ X,z ∂ α ϕ = ε p ∂ α R ε + ∇ X,z • [∂ α , P ] ∇ X,z ϕ , ∂ α ϕ |z=0 = 0 , ∂ n (∂ α ϕ) |z=-1 + ∂ [∂ α ,P ] n ϕ |z=-1 = ε p ∂ α r ε . (1.10)
In order to get an adequate control of the norm ||∂ z ϕ|| H k+1,0 , we prove the following estimate by induction on |α| ≤ k:

∀α ∈ N d / |α| ≤ k, ||∂ α ϕ|| Ḣ1 ≤ ε p √ p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) . (1.11)
The proof of (1.11) is hence divided into two parts : initialization of the induction and heredity.

• Initialization : |α| = 0 . Taking α = 0, multiplying (1.10) by ϕ and integrating by parts leads to :

( P ∇ X,z ϕ, ∇ X,z ϕ ) L 2 (S) + R d ∂ n ϕ |z=0 ϕ |z=0 - R d ∂ n ϕ |z=-1 ϕ |z=-1 = ( ε p R ε , ϕ ) L 2 (S) .
The boundary term at the free surface vanish because of the condition ϕ |z=0 = 0 and using the condition at the bottom leads to :

( P ∇ X,z ϕ, ∇ X,z ϕ ) L 2 (S) = ( ε p R ε , ϕ ) L 2 (S) + ε p R d r ε ϕ |z=-1 .
Finally, using Cauchy-Schwartz inequality, one gets :

( P ∇ X,z ϕ, ∇ X,z ϕ ) L 2 (S) ≤ ε p ||R ε || L 2 (S) ||ϕ|| L 2 (S) + ε p |r ε | L 2 (R d ) |ϕ |z=-1 | L 2 (R d ) .
(1.12)

Recalling that ϕ |z=0 = 0 and that the band S is bounded in the vertical direction, one can use Poincaré inequality so that ||ϕ||

L 2 (S) ≤ ||∂ z ϕ|| L 2 (S) and |ϕ |z=-1 | L 2 (R d ) ≤ ||∂ z ϕ|| L 2 (S)
. Therefore, (1.12) yields

( P ∇ X,z ϕ, ∇ X,z ϕ ) L 2 (S) ≤ ε p √ p d+1 ||R ε || L 2 (S) ||ϕ|| Ḣ1 + ε p √ p d+1 |r ε | L 2 (R d ) ||ϕ|| Ḣ1 . (1.13)
Using Lemma 1.5 to bound ( P ∇ X,z ϕ, ∇ X,z ϕ ) L 2 (S) from below, one finally gets :

c 0 ( |η| W 1,∞ , |B| W 1,∞ )||ϕ|| 2 Ḣ1 ≤ ε p √ p d+1 (||R ε || H 0,0 + |r ε | H 0 ) ||ϕ|| Ḣ1 . Since c 0 ( |η| W 1,∞ , |B| W 1,∞
) depends only on h min , d and γ through the quantity min 1≤i≤d p d+1 pi (by Lemma 1.5), and since the function c 0 is a decreasing function of its arguments (again by Lemma 1.5), we get the following desired estimate :

||ϕ|| Ḣ1 ≤ ε p √ p d+1 C 1 (||R ε || H 0,0 + |r ε | H 0 ) ,
which ends the initialization of the induction.

• Heredity : for m ∈ N * fixed such that m ≤ k, we suppose that (1.11) is verified for all α ∈ N d such that |α| ≤ m -1. Let α ∈ N d such that |α| = m.
Multiplying (1.10) by ∂ α ϕ and integrating by parts on S leads to :

( P ∇ X,z ∂ α ϕ, ∇ X,z ∂ α ϕ ) L 2 (S) + R d ∂ α ϕ |z=0 ∂ n ∂ α ϕ |z=0 - R d ∂ α ϕ |z=-1 ∂ n ∂ α ϕ |z=-1 = ( ε p ∂ α R ε , ∂ α ϕ ) L 2 (S) -( [∂ α , P ] ∇ X,z ϕ, ∇ X,z ∂ α ϕ ) L 2 (S) - R d ∂ α ϕ |z=0 ∂ [∂ α ,P ] n ϕ |z=0 + R d ∂ α ϕ |z=-1 ∂ [∂ α ,P ] n ϕ |z=-1 .
The boundary terms at z = 0 vanish because of the condition ∂ α ϕ |z=0 = 0, and using the second boundary condition

∂ n (∂ α ϕ) |z=-1 + ∂ [∂ α ,P ] n ϕ |z=-1 = ε p ∂ α r ε , one gets : ( P ∇ X,z ∂ α ϕ, ∇ X,z ∂ α ϕ ) L 2 = ( ε p ∂ α R ε , ∂ α ϕ ) L 2 + ε p R d ∂ α ϕ |z=-1 ∂ α r ε -( [∂ α , P ] ∇ X,z ϕ, ∇ X,z ∂ α ϕ ) L 2 ,
and with Cauchy-Schwartz :

( P ∇ X,z ∂ α ϕ, ∇ X,z ∂ α ϕ ) L 2 (S) ≤ ε p ||∂ α R ε || L 2 (S) ||∂ α ϕ|| L 2 (S) + ε p |∂ α r ε | L 2 (R d ) |∂ α ϕ |z=-1 | L 2 (R d ) + ( [∂ α , P ] ∇ X,z ϕ, ∇ X,z ∂ α ϕ ) L 2 (S)
.

By using the same method and arguments as in the initialization, the following inequality arises :

c 0 ( |η| W 1,∞ , |B| W 1,∞ )||∂ α ϕ|| 2 Ḣ1 ≤ ε p √ p d+1 (||R ε || H k,0 + |r ε | H k ) ||∂ α ϕ|| Ḣ1 + ( [∂ α , P ] ∇ X,z ϕ, ∇ X,z ∂ α ϕ ) L 2 (S)
.

(1.14) Let us now focus on the second term of the left hand side of (1.14). In order to get an adequate control of this term, we have to write explicitly the commutator [∂ α , P ] :

[∂ α , P ] ∇ X,z ϕ = α ′ +α ′′ =α α ′ =0 C(|α ′ |, |α ′′ |)∂ α ′ P ∇ X,z ∂ α ′′ ϕ ,
where C is a constant depending only on |α ′ | and |α ′′ |. This leads to the expression

( [∂ α , P ]∇ X,z ϕ, ∇ X,z ∂ α ϕ ) L 2 (S) = α ′ +α ′′ =α α ′ =0 C(|α ′ |, |α ′′ |) ∂ α ′ P ∇ X,z ∂ α ′′ ϕ , ∇ X,z ∂ α ϕ L 2 (S)
.

From now on, we just consider a single term of this sum. Using Proposition 1.3 we derive the explicit expression of P and deduce from it the explicit expression of ∂ α ′ P :

∂ α ′ P =    ∂ α ′ η -∂ α ′ B P d P d ∂ α ′ U P d ∂ α ′ U T ∂ α ′ p d+1 +U •P d U η+h0-B    ,
where P d is the diagonal (d × d) matrix whose coefficents are (p i ) 1≤i≤d , and U the vector defined by U = -(z + 1)∇η + z∇B. Using this expression, one easily gets (with ∇ = ∇ X ) :

∂ α ′ P ∇ X,z ∂ α ′′ ϕ , ∇ X,z ∂ α ϕ L 2 (S) = (∂ α ′ η -∂ α ′ B)P d ∇∂ α ′′ ϕ , ∇∂ α ϕ L 2 (S) + ∂ z ∂ α ′′ ϕP d ∂ α ′ U , ∇∂ α ϕ L 2 (S) + P d ∂ α ′ U • ∇∂ α ′′ ϕ , ∂ z ∂ α ϕ L 2 (S) + ∂ α ′ p d+1 +U •P d U η+h0-B ∂ z ∂ α ′′ ϕ , ∂ z ∂ α ϕ L 2 (S)
.

(

If we focus on the first term of the right hand side of this equality, we easily get the following intermediate control using Cauchy-Schwartz inequality and the definition of ||.|| Ḣ1 :

(∂ α ′ η -∂ α ′ B)P d ∇∂ α ′′ ϕ , ∇∂ α ϕ L 2 (S) ≤ (|η| W |α ′ |,∞ + |B| W |α ′ |,∞ ) || P d ∇∂ α ′′ ϕ|| L 2 (S) || P d ∇∂ α ϕ|| L 2 (S) ≤ (|η| W k,∞ + |B| W k,∞ ) ||∂ α ′′ ϕ|| Ḣ1 ||∂ α ϕ|| Ḣ1 ≤ ε p √ p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) ||∂ α ϕ|| Ḣ1 .
To derive the last inequality, we used the induction hypothesis on

||∂ α ′′ ϕ|| Ḣ1 since |α ′′ | ≤ m -1.
Let us now focus on the second term of the right hand side of (1.15). Using the same arguments as previously and Poincaré inequality, the following controls arise :

∂ z ∂ α ′′ ϕP d ∂ α ′ U , ∇∂ α ϕ L 2 (S) ≤ || P d ∂ α ′ U|| ∞ ||∂ z ∂ α ′′ ϕ|| L 2 (S) || P d ∇∂ α ϕ|| L 2 (S) ≤ ||P d || ∞ p d+1 (|η| W |α ′ |+1,∞ + |B| W |α ′ |+1,∞ ) ||∂ α ′′ ϕ|| Ḣ1 ||∂ α ϕ|| Ḣ1 ≤ ε p √ p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) ||∂ α ϕ|| Ḣ1 , since ||P d ||∞ p d+1 ≤ γ.
The control of the third term of the right hand side of (1.15) comes in the same way :

P d ∂ α ′ U • ∇∂ α ′′ ϕ , ∂ z ∂ α ϕ L 2 (S) ≤ ε p √ p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) ||∂ α ϕ|| Ḣ1 .
The next step consists in controling the last term of the right hand side of (1.15). We need to do some preliminary work on this term before attempting to estimate it adequately. A straightforward calculus gives us :

∂ α ′ p d+1 + U • P d U η + h 0 -B = β1+β2=α ′ β 1 = 0 C(|β 1 |, |β 2 |)∂ β1 (U • P d U) ∂ β2 1 η + h 0 -B + U • P d U ∂ α ′ 1 η + h 0 -B + p d+1 ∂ α ′ 1 η + h 0 -B .
We plug the previous writing and use the same tools as previously to get :

∂ α ′ p d+1 + U • P d U η + h 0 -B ∂ z ∂ α ′′ ϕ , ∂ z ∂ α ϕ L 2 (S) ≤ ||P d || ∞ C k+1 ||∂ z ∂ α ′′ ϕ|| L 2 (S) ||∂ z ∂ α ϕ|| L 2 (S) + C k+1 || √ p d+1 ∂ z ∂ α ′′ ϕ|| L 2 (S) || √ p d+1 ∂ z ∂ α ϕ|| L 2 (S) ≤ ||P d || ∞ p d+1 C k+1 ||∂ α ′′ ϕ|| Ḣ1 ||∂ α ϕ|| Ḣ1 +C k+1 ||∂ α ′′ ϕ|| Ḣ1 ||∂ α ϕ|| Ḣ1 ≤ ε p √ p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) ||∂ α ϕ|| Ḣ1 ,
where we once more used the induction hypothesis.

Gathering the four previous estimations of each term of the right hand side of (1.15) and using the explicit writing of the commutator [∂ α , P ] leads to the final estimate of ([

∂ α , P ] ∇ X,z ϕ , ∇ X,z ∂ α ϕ ) L 2 (S) : ([∂ α , P ] ∇ X,z ϕ , ∇ X,z ∂ α ϕ ) L 2 (S) ≤ ε p √ p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) ||∂ α ϕ|| Ḣ1 .
The last step simply consists in pluging this last estimation in the estimation (1.14), which gives :

c 0 ( |η| W 1,∞ , |B| W 1,∞ )||∂ α ϕ|| 2 Ḣ1 ≤ ε p √ p d+1 ||R ε || H k,0 ||∂ α ϕ|| Ḣ1 . (1.16)
As in the initialization, this last estimation leads to the desired result, which ends the heredity and hence the proof of (1.11).

To conclude this first part of the proof, we use the fact that :

||∂ z ϕ|| H k+1,0 ≤ C(k + 1) sup |α|≤k+1 ||∂ z ∂ α ϕ|| L 2 (S) ≤ C(k + 1) √ p d+1 sup |α|≤k+1 ||∂ α ϕ|| Ḣ1 ,
and the estimate (1.11) we just proved to finally get :

||∂ z ϕ|| H k+1,0 ≤ ε p p d+1 C k+2 (||R ε || H k+1,0 + |r ε | H k+1 ) , (1.17)
which ends the first part of the proof.

2.

In this second part, we aim at controlling the quantity ||∂ 2 z ϕ|| H k,0 . To this end, we prove with a direct method the following estimate :

||∂ 2 z ϕ|| H k,0 ≤ ε p p d+1 C k+2 (||R ε || H k+1,0 + |r ε | H k+1 ) . (1.18)
We first use the equation satisfied by ϕ in order to express ∂ 2 z ϕ in terms of other derivatives of ϕ such as ∇ϕ, ∂ z ϕ, ∇∂ z ϕ or ∆ϕ. There comes the following expression :

∂ 2 z ϕ = η + h 0 -B p d+1 + U • P d U [-∇ X,z • Q ∇ X,z ϕ - ∂ z (U • P d U) η + h 0 -B ∂ z ϕ -ε p R ε ] ,
where

Q = (η + h 0 -B) P d P d U (P d U) T 0 .
The following estimates arise (using

||u|| H k,0 ≤ C(k) sup |α|≤k ||∂ α u|| L 2 (S) ) : ||∂ 2 z ϕ|| H k,0 ≤ 1 p d+1 C k ||∇ X,z • Q ∇ X,z ϕ|| H k,0 + C k+1 ||P d || ∞ ||∂ z ϕ|| H k,0 + ε p ||R ε || H k,0 , ≤ 1 p d+1 C k+1 C(k) sup |α|≤k ||∂ α (∇ X,z • Q ∇ X,z ϕ) || L 2 (S) + ||P d || ∞ √ p d+1 C(k) sup |α|≤k ||∂ α ϕ|| Ḣ1 +ε p ||R ε || H k,0 , ≤ 1 p d+1 C k+1 sup |α|≤k ||∂ α (∇ X,z • Q ∇ X,z ϕ) || L 2 (S) + ε p p d+1 ||P d || ∞ p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) + ε p p d+1 C k+1 ||R ε || H k,0 , ≤ 1 p d+1 C k+1 sup |α|≤k ||∂ α (∇ X,z • Q ∇ X,z ϕ) || L 2 (S) + ε p p d+1 C k+1 (||R ε || H k,0 + |r ε | H k ) ,
where we used the result (1.11) and the fact that

||P d ||∞ p d+1 ≤ γ.

The last part of the initialization aims at correctly estimating the norm ||∂

α (∇ X,z • Q ∇ X,z ϕ) || L 2 (S) . The explicit writing ∂ α (∇ X,z • Q ∇ X,z ϕ) = α ′ +α ′′ =α C(|α ′ |, |α ′′ |) ∇ X,z • ∂ α ′ Q ∇ X,z ∂ α ′′ ϕ ,
and the expression of Q furnishes us with the following estimates :

||∇ X,z • ∂ α ′ Q ∇ X,z ∂ α ′′ ϕ|| L 2 (S) ≤ C |α ′ | ||∇ • P d ∇∂ α ′′ ϕ|| L 2 (S) +C |α ′ |+1 || P d || ∞ || P d ∇∂ α ′′ ϕ|| L 2 (S) +C |α ′ |+2 ||P d || ∞ ||∂ z ∂ α ′′ ϕ|| L 2 (S) +C |α ′ |+1 ||P d || ∞ ||∂ z ∇∂ α ′′ ϕ|| L 2 (S) ) ≤ C k+2 ( ||∇ • P d ∇∂ α ′′ ϕ|| L 2 (S) +|| P d || ∞ ||∂ α ′′ ϕ|| Ḣ1 + ||P d || ∞ √ p d+1 ||∂ α ′′ ϕ|| Ḣ1 ) ≤ C k+2 ||∇ • P d ∇∂ α ′′ ϕ|| L 2 (S) + ε p C k+1 (||R ε || H k,0 + |r ε | H k ) . (1.19)
We estimate the term ||∇ • P d ∇∂ α ′′ ϕ|| L 2 (S) using the following technique :

||∇ • P d ∇∂ α ′′ ϕ|| L 2 (S) ≤ || P d || ∞ 1≤i≤d || √ p i ∂ 2 xi ∂ α ′′ ϕ|| L 2 (S) ≤ || P d || ∞ 1≤i≤d ||∂ xi ∂ α ′′ ϕ|| Ḣ1 ≤ d || P d || ∞ sup |m|=|α ′′ |+1 ||∂ m ϕ|| Ḣ1 ≤ d || P d || ∞ sup |m|≤k+1 ||∂ m ϕ|| Ḣ1 ≤ ε p C k+2 (||R ε || H k+1,0 + |r ε | H k+1 ) .
We plug this result in (1. [START_REF] Madsen | A new form of the Boussinesq equations with improved linear dispersion characteristics (part 1)[END_REF]) to obtain

||∇ X,z • ∂ α ′ Q ∇ X,z ∂ α ′′ ϕ|| L 2 (S) ≤ ε p C k+2 (||R ε || H k+1,0 + |r ε | H k+1 ) ,
which finally leads to

||∂ α (∇ X,z • Q ∇ X,z ϕ) || L 2 (S) ≤ ε p C k+2 (||R ε || H k+1,0 + |r ε | H k+1 ) .
This way, we get our desired estimation of ||∂ 2 z ϕ|| H k,0 :

||∂ 2 z ϕ|| H k,0 ≤ ε p p d+1 C k+2 (||R ε || H k+1,0 + |r ε | H k+1 ) .
Gathering (1.17) and (1.18) in (1.9) ends the proof of the theorem.

Application

We recall that by definition, Z ε (εη, βb)f = ∂ z u |z=εη where u is solution of the boundary value problem

ε ∆u + ∂ 2 z u = 0 in Ω , (1.20)
u |z=εη = f , (∂ z u -εβ∇b • ∇u) | z=-1+βb = 0 , (1.21)
This elliptic problem (1.20)(1.21) belongs to the class of general elliptic problems (1.2)(1.3) defined in the previous subsection. The corresponding matrix P is here designed by P ε :

P ε = ε I d×d 0 0 1 . (1.22)
The upper boundary of Ω is here defined by {z = εη} and the lower one by {z = -1 + βb}. We make the additionnal assumption that ε and β are bounded in the following sense : 0 < ε < 1 and there exists a strictly positive constant β 0 such that and 0 < β < β 0 . Furthermore, condition (1.1) is here verified thanks to condition (0.1). And finally, we remark that ( pi p d+1 ) 1≤i≤d are bounded by 1 since 0 < ε < 1. Our goal is here to apply the previous theorem to get asymptotic estimates on Z ε (εη, βb).

We recall that we are here interested in two differerent regimes depending on the β parameter. The first one, namely β = O(ε), refers to the physical case of a bottom with variations of slow amplitude. The second one, namely β = O(1), refers on the contrary to variations of high amplitude of the bottom. In order to improve the readability, we take β 0 = 1 : we thus write β = ε for the first regime and β = 1 for the second one.

The regime β = ε : small variations of bottom topography

The boundaries of the domain Ω are here defined by {z = εη} and {z = -1 + εb} while matrix P ε remains as in (1.22). Thanks to Proposition 1.3 we are able to set an equivalent problem to (1.20)(1.21) defined over the flat band S : u = u • S then solves the problem :

-∇ X,z • P ε ∇ X,z u = 0 in S , (1.23
)

u |z=0 = f , ∂ n u |z=-1 = 0 . (1.24)
where the matrix P ε is given by

P ε =   ε(1 + ε(η -b)) I d×d -ε 2 [(z + 1)∇η -z∇b] -ε 2 [(z + 1)∇η -z∇b] T 1+ε 3 |(z+1)∇η-z∇b| 2 1+ε(η-b)   .
The following result gives a rigourously justified asymptotic expansion of Z ε (εη, βb)f as ε goes to 0 :

Proposition 1.7. Let k ∈ N, η ∈ W k+2,∞ (R d ) and b ∈ W k+2,∞ (R d ).
Then for all f such that ∇f ∈ H k+6 (R d ), we have :

Z ε (εη, βb)f -(εZ 1 + ε 2 Z 2 ) H k+1/2 ≤ ε 3 C k+2 |∇f | H k+6 ,
with :

Z 1 := -∆f , Z 2 := -1 3 ∆ 2 f -(η -b)∆f + ∇b • ∇f .
Proof. To prove this proposition, we use essentially Theorem 1.6 with p = 3. We know that pi p d+1 1≤i≤d are bounded by 1. Thus, in order to derive an asymptotic expansion of Z ε (εη, βb)f , we only need to compute an approximate solution u app which satisfies the hypothesis of Theorem 1.6 for p = 3. This approximate solution u app can be constructed as in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF] using a classical WKB method, which consists in looking for u app under the form u app = u 0 + εu 1 + ε 2 u 2 . We want this function to verify the properties required by Theorem 1.6, that is to say :

-∇ X,z . P ε ∇ X,z u app = ε p R ε in S , (1.25)
u app |z=0 = f , ∂ n u app |z=-1 = ε p r ε . (1.26)
where (R ε ) 0<ε<1 and (r ε ) 0<ε<1 are bounded independently of ε respectively in H k+1,0 (S) and H k+1 (R d ).

We decompose the matrix P ε under the form P ε = P 0 + εP 1 + ε 2 P 2 + ε 3 P ε where P 0 , P 1 , P 2 are independent of ε, and if we plug the desired expression of u app into this problem, we get

R ε = ∇ • T ε and r ε = e z • T ε |z=-1
where

T ε = P 2 ∇ X,z u 1 + P 1 ∇ X,z u 2 + P ε ∇ X,z (u o + u 1 + u 2 )
, and the following system of equations and boundary conditions on u 0 , u 1 , u 2 :

             ∂ 2 z u 0 = 0 , ∂ 2 z u 1 + ∆ -(η -b)∂ 2 z u 0 = 0 , ∂ 2 z u 2 + ∆ -(η -b)∂ 2 z u 1 + (η -b)∆u 0 -2 [(z + 1)∇f -z∇b] • ∇∂ z u 0 -[(z + 1)∆f -z∆b] • ∂ z u 0 -(η -b) 2 ∂ 2 z u 0 = 0 , with            u 0|z=0 = f , u i|z=0 = 0, 1 ≤ i ≤ 2 , ∂ z u i|z=-1 = 0, 0 ≤ i ≤ 1 , ∂ z u 2|z=-1 -∇b • ∇u 0|z=-1 = 0 .
We can verify that the following values of u 0 , u 1 , u 2 satisfy the previous equations and boundary conditions :

u 0 = f , u 1 = 1 2 - (z + 1) 2 2 ∆f , u 2 = (z + 1) 4 24 - (z + 1) 2 4 + 5 24 ∆ 2 f + 1 -(z + 1) 2 (η -b)∆f + z∇b • ∇f .
Using these values of u 0 , u 1 and u 2 , one can easily obtain the following estimates of R ε and r ε :

||R ε || H k+1,0 ≤ C k+2 |∇f | H k+6 , |r ε | H k+1 ≤ C k+2 |∇f | H k+3 .
Thus u app satisfies the properties required to apply Theorem 1.6. The last steps of the proof consists in computing 1 1+ε(η-b) ∂ z u app|z=0 using the explicit expression of u app previously determined, and then apply Theorem 1.6. An easy Taylor expansion of 1 1+ε(η-b) ∂ z u app|z=0 then yields the result.

Remark 1.8. The method developped here to get and prove our asymptotic expansion is improved compared to the one developped in BCL since we do not need here to compute the term u 3 .

Remark 1.9. If we take b = 0 -i.e. if we consider a flat bottom -, we of course get the same expansion as the ones proved in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF].

1.2.2. The regime β = 1 : strong variations of bottom topography

The boundaries of the domain Ω are here defined by {z = εη} and {z = -1 + b}. Using again Proposition 1.3, we set an equivalent problem to (1.20)(1.21) defined over the flat band S : this new problem is the same as the one defined in the first regime, at the exception of the matrix P ε which is now given by

P ε =   ε(1 + εη -b) I d×d -ε[ε(z + 1)∇η -z∇b] -ε[ε(z + 1)∇η -z∇b] T 1+ε|ε(z+1)∇η-z∇b| 2 1+εη-b   .
As in the first regime we give a rigourously justified asymptotic expansion of Z ε (εη, βb)f in the present regime.

Proposition 1.10. Let k ∈ N, η ∈ W k+2,∞ (R d ) and b ∈ W k+2,∞ (R d ).
Then for all f such that ∇f ∈ H k+6 (R d ), we have :

Z ε (εη, βb)f -(εZ 1 + ε 2 Z 2 ) H k+1/2 ≤ ε 3 C k+2 |∇f | H k+6 , with :      Z 1 := -∇ • (1 -b) ∇f , Z 2 := 1 2 ∇ • 1 3 (1 -b) 3 ∇∆f -(1 -b) 2 ∇∇ • (1 -b) ∇f -η∆f .
Proof. The proof of this proposition follows exactly the same steps as the proof of Proposition 1.7. The following values of u 0 , u 1 , u 2 are found :

u 0 = f , u 1 = (1 -b) 2 2 (1 -(z + 1) 2 )∆f + z (1 -b)∇b • ∇f , u 2 = (1 -b) 4 24 ∆ 2 f z 4 + (1 -b) 3 6 ∆∇ • (1 -b)∇f z 3 -(1 -b)η∆f z 2 + (1 -b) 2 ∇ • (1 -b) 3 3 ∇∆f -(1 -b) 2 ∇∇ • (1 -b)∇f -η 2(1 -b)∆f + ∇b • ∇f z .
The error bound is derived in the same way and the previous values lead to the result.

Remark 1.11. By formally taking b = εb, we recover the result of Proposition 1.7.

Derivation of Boussinesq-like models for uneven bottoms

We recall the Zakharov formulation of the water waves equations, from which we intend to derive new systems using results of the previous subsection :

(S 0 )      ∂ t ψ -ε∂ t ηZ ε (εη, βb)ψ + 1 2 ε |∇ψ -ε ∇ηZ ε (εη, βb)ψ| 2 + |Z ε (εη, βb)ψ| 2 + η = 0 , ∂ t η + ε∇η • ∇ψ -ε∇ηZ ε (εη, βb)ψ = 1 ε Z ε (εη, βb)ψ .
As in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF], we introduce the notion of consistency.

Definition 1.12. Let σ, s ∈ R,

ε 0 > 0, T > 0 and let (V ε , η ε ) 0<ε<ε0 be bounded in W 1,∞ ([0, T ε ]; H σ (R d ) d+1
) independently of ε. This family is called consistent (at order s) with a system (S) if it is solution of (S) with a residual of order

ε 2 in L ∞ ([0, T ε ]; H s (R d ) d+1 ).
Thanks to the previous results, we are now able to enounce the following propositions which show the consistency of two Boussinesq-like systems with the system (S 0 ). We introduce here the quantity h = 1-b which corresponds to the non-dimensional still water depth. From now on, this quantity is considered as a bottom term since it only depends on b.

Proposition 1.13 (Small variations regime β = ε). Let T > 0, s ≥ 0, σ ≥ s and (ψ ε , η ε ) 0<ε<ε0 be a family of solutions of (0.7) such that

(∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to ε in W 1,∞ ([0, T ε ]; H σ (R d ) d+1
) with σ large enough. We define V ε := ∇ψ ε . Then the family (V ε , η) 0<ε<ε0 is consistent with the following system :

(B 1 )      ∂ t V + ∇η + ε 2 ∇|V | 2 = 0 , ∂ t η + ∇ • V + ε ∇ • (η -b) V + 1 3 ∆∇ • V = 0 .
Proof. This is clear thanks to the asymptotic expansion of the operator Z ε (εη, βb) : plugging this in system (0.7), neglecting the terms of order O(ε 2 ), and taking the gradient yields the result.

Proposition 1.14 (Strong variations regime β = 1). Let T > 0, s ≥ 0, σ ≥ s and (ψ ε , η ε ) 0<ε<ε0 be a family of solutions of (0.7) such that (∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to

ε in W 1,∞ ([0, T ε ]; H σ (R d ) d+1
) with σ large enough. We define V ε := ∇ψ ε . Then the family (V ε , η) 0<ε<ε0 is consistent with the following system (with h = 1 -b) :

(B 2 )      ∂ t V + ∇η + ε 2 ∇|V | 2 = 0 , ∂ t η + ∇ • (h V ) + ε ∇ • (ηV ) -1 2 ∇ • h 3 3 ∇∇ • V -h 2 ∇∇ • (h V ) = 0 .
These results close the first part. From now on, our work is divided into two parts, each corresponding to one of our two regimes β = ε and β = 1. Each analysis starts from the two previous Boussinesq-like models.

The regime of small topography variations

We recall our previously derived Boussinesq-like system (B 1 ) on which we base our analysis :

(B 1 )      ∂ t V + ∇η + ε 2 ∇|V | 2 = 0 , ∂ t η + ∇ • V + ε ∇ • (η -b) V + 1 3 ∆∇ • V = 0 .
We now follow the method put forward in [START_REF] Bona | Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media[END_REF] and [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF] to derive equivalent systems to (B 1 ) in the meaning of consistency. The rigourous justifications of the derivation of these systems is adressed in section 2 in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF].

A first class of equivalent systems

As in [START_REF] Bona | Boussinesq Equations and Other Systems for Small-Amplitude Long Waves in Nonlinear Dispersive Media[END_REF] (for the 1D case) and in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF], we define :

V θ = 1 + ε 2 (1 -θ 2 )∆ V ,
which corresponds to the approximation at the order ε 2 of the horizontal component of the velocity field at height -1

+ θ for θ ∈ [0, 1]. If we remark that V θ = 1 + ε 2 (θ 2 -1)∆ -1 V + O(ε 2
), the expression of V θ in terms of V comes in the following way by supposing V regular enough :

V = 1 + ε 2 (θ 2 -1)∆ V θ + O(ε 2 ) , where O(ε 2 ) is to be taken in the L ∞ ([0, T ε ], H s (R d )) norm.
Plugging this relation into the system (B 1 ) leads to :

   ∂ t V θ + ∇η + ε 2 ∇|V | 2 + (θ 2 -1)∆∂ t V θ = O(ε 2 ) , ∂ t η + ∇ • V θ + ε ∇ • (η -b) V + θ 2 2 -1 6 ∆∇ • V θ = O(ε) 2 .
At this point we use the classical BBM trick which consists in writing the following approximations at order O(1) , coming from the previous equations :

∂ t V θ = -∇η + O(ε) = (1 -µ)∂ t V θ -µ∇η + O(ε) , ∇ • V θ = -∂ t η + O(ε) = λ∇ • V θ -(1 -λ)∂ t η + O(ε) ,
where λ and µ are two real parameters.

We plug these relations into the dispersive terms of the last system to get :

     ∂ t V θ + ∇η + ε 2 ∇|V θ | 2 -µ(θ 2 -1)∆∇η -(µ -1)(θ 2 -1)∆∂ t V θ = O(ε 2 ) , ∂ t η + ∇ • V θ + ε ∇ • (η -b)V θ + λ θ 2 2 -1 6 ∆∇ • V θ -(1 -λ) θ 2 2 -1 6 ∆∂ t η = O(ε 2 ) .
We then introduce the class S of all the systems of the previous form : these systems are denoted by S 1 θ,λ,µ and can be rewritten in the compact form :

(S 1 θ,λ,µ )      (1 -εa 2 ∆)∂ t V θ + ∇η + ε 1 2 ∇|V θ | 2 + a 1 ∆∇η = 0 , (1 -εa 4 ∆)∂ t η + ∇ • V θ + ε ∇ • (η -b) V θ + a 3 ∆∇ • V θ = 0 . with a 1 = -µ θ 2 -1 2 , a 2 = (µ -1) θ 2 -1 2 , a 3 = λ θ 2 2 -1 6 , a 4 = (1 -λ) θ 2 2 -1 6
.

On this class S, the previous computations give us the following two results of consistency :

Proposition 2.1. Let θ ∈ [0, 1] and (ψ ε , η ε ) 0<ε<ε0 be a family of solutions of (0.7) such that

(∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to ε in W 1,∞ ([0, T ε ]; H σ (R d ) d+1
) with σ large enough. We define V ε = ∇ψ ε and

V ε θ = 1 + ε 2 (1 -θ 2 )∆ V ε .
Then for all (λ, µ) ∈ R 2 , the family (V ε θ , η ε ) 0<ε<ε0 is consistent with the system (S 1 θ,λ,µ ).

Proof. We saw in the previous section that if (ψ ε , η ε ) 0<ε<ε0 is a family of solutions of (0.7), then the family (∇ψ ε , η ε ) 0<ε<ε0 is consistent with the system (B 1 ). Thanks to the previous computations, and since the choice of the parameters (λ, µ) is totally free, it is clear that (V ε θ , η ε ) 0<ε<ε0 is consistent with any system (S 1 θ,λ,µ ).

Proposition 2.2. Up to a change of variables, all the systems belonging to the class S are equivalent in the meaning of consistency.

Proof. Let (θ, λ, µ) ∈ [0, 1] × R 2 and (V ε θ , η ε ) 0<ε<ε0 a family consistent with (S 1 θ,λ,µ ). We then define, for

θ 1 ∈ [0, 1], V ε θ1 = 1 + ε 2 (1 -θ 2 1 )∆ 1 - ε 2 (1 -θ 2 )∆ V ε θ ; using the fact that 1 -ε 2 (1 -θ 2 )∆ V ε θ = V ε + O(ε 2
) and the previous proposition, we easily deduce that the family (V ε θ1 , η ε ) 0<ε<ε0 is consistent with any system (S 1 θ1,λ1,µ1 ) for any (λ 1 , µ 1 ) ∈ R 2 .

Remarks 2.3.

• By taking θ = 1, λ = 1, µ = 0, we remark that the previously derived Boussinesq-like system B 1 is actually a member of the class S • By taking λ = µ = 1/2 and θ 2 = 2/3, we get a 1 = a 2 = a 3 = a 4 = 1 12 , so that the dispersive part of the correponding system (S 1 θ,λ,µ ) is symmetric. However, the nonlinear terms, that are not affected by the choice of θ, λ, µ, are not symmetric : this problem is adressed in the next section.

• In [START_REF] Chen | Equations for bi-directional waves over an uneven bottom[END_REF], Chen formally studied in 1D the case of slowly variating bottoms and derived the same class of systems at the exception that she considered time-dependent bottoms : her systems thus contain additionnal time derivative terms on the bottom that does not appear here but could be easily obtained for a time dependent bottom.

A second class of equivalent systems

Adapting the nonlinear change of variables of [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF] to the present case of varying depth, we introduce Ṽ :

Ṽ = 1 + ε 2 (η -b) V .
This nonlinear change of variable symetrizes the nonlinear part of the equations.

This change of variables only affects the nonlinear terms and not the dispersive terms. If (V ε , η ε ) 0<ε<ε0 is consistent with a system (S 1 θ,λ,µ ) of the class S, then Ṽ ε = (1 + ε 2 η ε ) V ε and η ε satisfy the following equations :

     (1 -εa 2 ∆) ∂ t Ṽ ε + ∇η ε + ε 1 4 ∇|η ε | 2 + 1 2 ∇| Ṽ ε | 2 + 1 2 Ṽ ε ∇ • Ṽ ε -1 2 b ∇η ε + a 1 ∆∇η ε = O(ε 2 ) , (1 -εa 4 ∆) ∂ t η ε + ∇ • Ṽ ε + ε 1 2 ∇ • (η ε -b) Ṽ ε + a 3 ∆∇ • Ṽ ε = O(ε 2 ) .
As observed in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF], if we consider a two-dimensional domain, that is to say d = 1, the nonlinear terms are actually symmetric. But this is not the case in a three-dimensional domain. However we can deal with this problem for d = 2 using the following remark coming from [START_REF] Lannes | Sur le caractère bien posé des équations d'Euler avec surface libre[END_REF] :

1 2 ∇| Ṽ ε | 2 = 1 4 ∇| Ṽ ε | 2 + 1 2 ( Ṽ ε • ∇) Ṽ ε + 1 2 Ṽ ε ∧ (∇ × Ṽ ε )
Assuming that ∇ × Ṽ ε = O(ε), one formally derives the following system :

           (1 -εa 2 ∆) ∂ t Ṽ ε + ∇η ε + ε 1 4 ∇|η ε | 2 + 1 4 ∇| Ṽ ε | 2 + 1 2 ( Ṽ ε • ∇) Ṽ ε + 1 2 Ṽ ε ∇ • Ṽ ε -1 2 b ∇η ε + a 1 ∆∇η ε = O(ε 2 ) , (1 -εa 4 ∆) ∂ t η ε + ∇ • Ṽ ε + ε 1 2 ∇ • (η ε -b) Ṽ ε + a 3 ∆∇ • Ṽ ε = O(ε 2 ) .
The nonlinear terms of the previous system are now symmetric regardless of the dimension. This previous computations are summed up in the following proposition :

Proposition 2.4. Let (V ε , η ε ) 0<ε<ε0 be a family consistent with a system (S 1 θ,λ,µ ) and

Ṽ ε = 1 + ε 2 η ε V ε . If ∇ × Ṽ ε = O(ε)
, then the family ( Ṽ ε , η ε ) 0<ε<ε0 is consistent with the following system :

(T 1 θ,λ,µ )      (1 -εa 2 ∆) ∂ t V + ∇η + ε 1 4 ∇|η| 2 + 1 4 ∇|V | 2 + 1 2 (V • ∇)V + 1 2 V ∇ • V -1 2 b ∇η + a 1 ∆∇η = 0 , (1 -εa 4 ∆) ∂ t η + ∇ • V + ε 1 2 ∇ • (η -b) V + a 3 ∆∇ • V = 0 .
We introduce the class T composed with the systems of the form (T 1 θ,λ,µ ) for any (θ, λ, µ) ∈ [0, 1] × R 2 . Using this result, we prove the following proposition :

Proposition 2.5. Let θ ∈ [0, 1] and (ψ ε , η ε ) 0<ε<ε0 be a family of solutions of (0.7) such that (∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to ε in W 1,∞ ([0, T ε ]; H σ (R d ) d+1 ) with σ large enough. We define Ṽ ε = 1 + ε 2 (η -b) 1 + ε 2 (1 -θ 2 )∆ ∇ψ ε . Then for all (λ, µ) ∈ R 2 , the family ( Ṽ ε , η ε ) 0<ε<ε0 is consistent with the system (T 1 θ,λ,µ ).
Proof. Thanks to Proposition 2.1, the family (V ε θ , η ε ) 0<ε<ε0 is consistent with the system (S 1 θ,λ,µ ) for all (λ, µ) ∈ R 2 , where

V ε θ = 1 + ε 2 (1 -θ 2 )∆ V ε and V ε = ∇ψ ε .
We then use the following remark : by hypothesis, the velocity field V is irrotationnal, thus V ε θ = O(ε) and Ṽ ε = O(ε). Applying the previous proposition yields the result.

A new class of completely symmetric systems

We remarked in the first section that there exists values of (θ, λ, µ) such that the dispersive terms are symmetric. Consequently, the corresponding system (T 1 θ,λ,µ ) of the class T is completely symmetric since both its dispersive terms and nonlinear terms are symmetric. We thus introduce the non-empty subclass of T denoted by Σ composed with the systems of the form (T 1 θ,λ,µ ) for which we have a 1 = a 3 , a 2 ≥ 0, a 4 ≥ 0. The first condition a 1 = a 3 symetrizes the nonlinear terms and the last ones a 2 ≥ 0, a 4 ≥ 0 ensure the well-posedness of these completely symmetric systems. Indeed, one of the great advantages of these systems belonging to Σ is that we have a well-posedness over a long time scale : Proposition 2.6. Let s > d 2 + 1 and (θ, λ, µ) be such that the system (T 1 θ,λ,µ ) belongs to the class Σ. Then for all (V 0 , η 0 ) ∈ H s (R d ) d+1 , there exists a time T 0 ≥ 0 independent of ε and a unique solution

(V, η) ∈ C([0, T0 ε ]; H s (R d ) d+1 ) ∩ C 1 ([0, T0 ε ]; H s-3 (R d ) d+1
) to the system (T 1 θ,λ,µ ) such that (V, η) |t=0 = (V 0 , η 0 ). Furthermore, this unique solution is bounded independently of ε in the following sense : there exists a constant C 0 independent of ε such that for all k verifying s -3k > d 2 + 1, we have :

|(V, η)| W k,∞ ([0, T 0 ε ];H s-3k (R d ) d+1 ) ≤ C 0 .
Proof. This theorem is a very classical result on hyperbolic symmetric quasilinear systems, and we omit the proof here.

As in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF], we are now able to rigorously construct approximate solutions to the water waves problem from the solutions of any of these symmetric systems. More precisely, let us consider a solution (ψ ε , η ε ) to the initial system (0.7) with initial data (ψ ε 0 , η ε 0 ) such that (∇ψ ε 0 , η ε 0 ) ∈ H s (R d ) d+1 for a suitably large value of s. We define V ε = ∇ψ ε and V ε 0 = ∇ψ ε 0 . From this solution of the water waves problem, we construct an approximate solution as follows :

• We first construct what we call here approximate initial data, by applying the two successive changes of variable on the data (V ε 0 , η ε 0 ) :

V ε Σ,0 = 1 + ε 2 (η ε 0 -b) 1 + ε 2 (1 -θ 2 )∆ V ε 0 , η ε Σ,0 = η ε 0 .
• We then choose the parameters (θ, λ, µ) ∈ [0, 1] × R 2 such that the system (T 1 θ,λ,µ ) belongs to the class Σ of completely symmetric systems (this choice is always possible as we saw previously). Using Proposition 2.6, we know that there exists a unique solution to this system with initial data (V ε Σ,0 , η ε Σ,0 ) : we denote this solution by (V ε Σ , η ε Σ ). • From this exact solution of the symmetric system (T 1 θ,λ,µ ), we finally construct an approximate solution of the water waves problem by successively and approximatively inverting the two changes of variable as shown below :

   V ε app = 1 -ε 2 (1 -θ 2 )∆ 1 -ε 2 (η ε Σ -b) V ε Σ , η ε app = η ε Σ .
This formal construction of an approximate solution founds its mathematical justification in the following theorem which is the last result of this section.

Theorem 2.7. Let T 1 ≥ 0, s ≥ d 2 + 1, σ ≥ s + 3 and (∇ψ ε 0 , η ε 0 ) be in H σ (R d ) d+1
. Let (ψ ε , η ε ) 0<ε<ε0 be a family of solutions of (0.7) with initial data (∇ψ ε 0 , η ε 0 ) and such that

(∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to ε in W 1,∞ ([0, T1 ε ]; H σ (R d ) d+1
). We define V ε = ∇ψ ε and choose (θ, λ, µ) ∈ [0, 1] × R 2 such that the system (T 1 θ,λ,µ ) ∈ Σ. Then for all ε < ε 0 , there exists T ≤ T 1 such that we have :

∀t ∈ [0, T ε ] , |V ε -V ε app | L ∞ ([0,t];H s ) + |η ε -η ε app | L ∞ ([0,t];H s ) ≤ C ε 2 t .
Proof. We follow in this proof the strategy put forward in [START_REF] Bona | Long Waves Approximations for Water Waves[END_REF]: estimates are done on the symmetric system that provides the approximate solution rather than on the initial system (0.7).

To this end, we take (θ, λ, µ) ∈ [0, 1] × R 2 such that the system T 1 θ,λ,µ belongs to the class Σ of completely symmetric systems. Since (ψ ε , η ε ) 0<ε<ε0 is a family of solutions of (0.7) such that (∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to ε in

W 1,∞ ([0, T1 ε ]; H σ (R d ) d+1 ), using Proposition 2.1 implies that (V ε θ , η ε ) 0<ε<ε0 where V ε θ = 1 + ε 2 (1 -θ 2 )∆ V ε is consistent with the system S 1
θ,λ,µ . Moreover, Proposition 2.4 states that any family (V ε , η ε ) 0<ε<ε0 consistent with the system S 1 θ,λ,µ is, up to the aforementionned nonlinear change of variables, consistent with the system T 1 θ,λ,µ . Applying this result to (V ε θ , η ε ) 0<ε<ε0 shows that the family ( Ṽ ε θ , η ε ) 0<ε<ε0 , where

Ṽ ε θ = 1 + ε 2 (η ε -b) V ε θ
, is actually consistent with the symmetric system T 1 θ,λ,µ . Thanks to Proposition 2.6, we know that there exists a time T 0 such that there exists a unique solution (V ε Σ , η ε Σ ) to this system with initial data (V ε Σ,0 , η ε Σ,0 ) (defined in the previous formal construction of the approximate solution). We are now interested in computing the error estimates between ( Ṽ ε θ , η ε ) and (V ε Σ , η ε Σ ). To this end we define

V = Ṽ ε θ -V ε Σ and η = η ε -η ε Σ .
Writing the equations satisfied by V and η and performing standard energy estimates on it leads to the following estimate :

∀t ∈ [0, T 1 ε ] , |V | L ∞ ([0,t];H s ) + |η| L ∞ ([0,t];H s ) ≤ C ε 2 t
where T = min(T 0 , T 1 ). Inverting the nonlinear change of variables and the pseudo-differential one yields the final result.

Remarks 2.8.

• The construction of the approximated solution of the water waves problem relies on the choice of the three parameters θ, λ, µ such that the system T 1 θ,λ,µ is completely symmetric. A great advantage of this method is that this choice is totally free : we are indeed allowed to choose any suitable triplet (θ, λ, µ) we want, and contruct our approximate solution from the exact solution of the system T 1 θ,λ,µ . In other words, approximate solutions of the water waves problem can be constructed starting from the exact solution of any symmetric system of the class Σ.

• Our theorem relies implicitly on the existence of a family (ψ ε , η ε ) 0<ε<ε0 of solutions to the water waves problem in Sobolev spaces : in 1D-surface, this existence have been already proved by Craig [START_REF] Craig | An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits[END_REF], Schneider-Wayne [START_REF] Schneider | The long-wave limit for the water-wave problem. I. The case of zero surface tension[END_REF] thus this implicit hypothesis of existence of solutions is actually a fact. However, in 2D-surface, we have no existence result for the water waves problem on a long time scale. Lannes proved reccenlty in [START_REF] Lannes | Well-posedness of the water-waves equations[END_REF] the existence of solutions to this problem in Sobolev spaces in 2D-surface, but we do not know if these solutions persist on a long time scale. Consequently, the analysis is not totally complete in 2D-surface.

The regime of strong topography variations

In this section, attention is given to the regime of strong variations of the bottom topography : these variations are here of order O(1). We recall the Boussinesq-like system (B 2 ) derived in the previous section, and the fact that solutions of the water waves problem are consistent with this system.

(B 2 )      ∂ t V + ∇η + ε 2 ∇|V | 2 = 0 , ∂ t η + ∇ • (h V ) + ε ∇ • (ηV ) -1 2 ∇ • h 3 3 ∇∇ • V -h 2 ∇∇ • (h V ) = 0 .
Like in the previous section, we aim at deriving asymptotic models, constructing approximate solutions of the water waves problem, and justifying these approximations. However, the method introduced in the previous section cannot be applyied in the exact same way : this regime is indeed much more complex since the bottom terms have here a greater influence than in the first regime. These bottom terms introduce new difficulties which compell us to revise and adapt our strategy.

A first equivalent system

First remark that the bottom term h (recall that h = 1 -b is the non-dimensional still water depth) appears in the first order term of the second equation of (B 2 ) whereas it is not present in the first one : we have here a dissymetry of the order one terms. This fact becomes important when it comes to the BBM trick which is unlikely to symetrize these terms. To correctly deal with this regime, we have to invert the order of the change of variables, and proceed with an adapted nonlinear change of variables first that symetrizes both order one terms and non-linear terms.

Taking into account the fact that we have to symmetrize both order one terms and nonlinear terms, we introduce the following change of variables :

Ṽ = √ h + ε 2 η √ h V .
so that

V = 1 √ h - ε 2 η h √ h Ṽ + O(ε 2 ) .
Assuming that ∇ × Ṽ = O(ε), we formally derive the following system of equations satisfied by Ṽ and η :

                     ∂ t Ṽ + √ h∇η + ε 2 √ h 1 4 ∇η 2 + 1 2 ∇| Ṽ | 2 + ( Ṽ • ∇) Ṽ + Ṽ ∇ • Ṽ + 1 h 1 2 (∇h • Ṽ ) Ṽ -| Ṽ | 2 ∇h = O(ε 2 ) , ∂ t η + ∇( √ h • Ṽ ) + ε 2 √ h ∇ • (η Ṽ ) - √ h∇ • h 3 3 ∇∇ • ( Ṽ √ h ) -h 2 ∇∇ • ( √ h Ṽ ) -η 2h ∇h • Ṽ = O(ε 2 ) .
We introduce the system (T b ) that corresponds to the homogeneous version of the previous system :

(T b )          ∂ t V + √ h ∇η + ε 2 F h V η = 0 , ∂ t η + ∇( √ h • V ) + ε 2 f h V η -∇ • h 3 3 ∇∇ • ( V √ h ) -h 2 ∇∇ • ( √ h V ) = 0 , where        F h V η = 1 √ h 1 4 ∇η 2 + 1 2 ∇|V | 2 + (V • ∇)V + V ∇ • V + 1 h 1 2 (∇h • V )V -|V | 2 ∇h , f h V η = 1 √ h ∇ • (ηV ) -η 2h ∇h • V .
On this new system (T b ), we have the following result of consistency :

Proposition 3.1. Consider a family (ψ ε , η ε ) 0<ε<ε0 of solutions of (0.7) such that (∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to ε in W 1,∞ ([0, T ε ]; H σ (R d ) d+1
) with σ large enough. Then the family (V ε , η ε ) 0<ε<ε0 is consistent with the system (T b ), where V ε = ∇ψ ε .

Proof. First remark that since the velocity field V ε is irrotationnal, we have ∇ × Ṽ ε = O(ε). And since (∇ψ ε , η ε ) 0<ε<ε0 is consistent with the Boussinesq-like system (B 2 ), the previous computations yield directly the result.

Derivation of a class of equivalent systems

In the previous section, we saw that a suitable change of variable comes from considering V θ , the horizontal component of the velocity at the height -1 + θ (θ ∈ [0, 1]), instead of the horizontal component of the velocity field at the free surface. We can remark that the link between these two variables (and hence the adequate change of variables) can be derived from the expression of u app computed during the asymptotic expansion process of the operator Z ε (εη, βb), which implies that we must adapt our change of variable for our regime of strong variations since the expression of u app relies on the considered regime of bottom topography. Indeed, we saw in the previous section that the computation of the asymptotic developpment of Z ε (η, b)ψ relies on finding an approximate solution of the elliptic problem (H) on the band S = [-1, 0] × R 2 . Starting from the truncation of the computed value of u app at the order O(ε 2 ),

u app = ψ + ε 1 2 - (z + 1) 2 2 h 2 ∆ψ -zh∇h • ∇ψ + O(ε 2 ) ,
where ψ is the value of the velocity potential at the free surface, shows that ∇u app (•, z) gives an approximation at order ε 2 of the horizontal component of the velocity field, namely

V (•, z) = ∇φ(•, z) at height z ∈ [-1, 0].
Consequently, in presence of huge bottom variations, the adequate change of variables is given by :

V θ = 1 - ε 2 (θ -1)(θ∇(h 2 ∇• ) + ∇∇ • (h 2 . )) V , so that V = 1 + ε 2 (θ -1)(θ∇(h 2 ∇• ) + ∇∇ • (h 2 . )) V θ + O(ε 2 ) .
From this change of variables, we easily compute the expressions of ∂ t V and ∇ • √ hV which we plug into the system (T b ). By rewriting carefully the bottom terms in order to make the quantity √ h appear, one gets the following system :

               ∂ t V θ + √ h ∇η + ε 2 F h V θ η + ∇ (θ 2 -1)h 2 ∇ • ∂ t V θ + 2(θ -1)h∇h • ∂ t V θ = O(ε 2 ) , ∂ t η + ∇( √ h • V θ ) + ε 2 f h V θ η -∇ • (θ 2 -1 3 )h 2 ∇∇ • ( √ hV θ ) + ( 3 2 θ 2 -7 6 )h∇h∇ • ( √ hV θ ) -(θ-2) 2 2 √ h∇h(∇h • V θ ) -( θ 2 2 -2θ + 7 6 )h √ h∇(∇h • V θ ) = O(ε 2 ) .
At this point a new problem arises. Applying the BBM trick in the exact same way as in the previous regime leads to a system that is never symmetric for any values of the parameters θ, λ and µ. Indeed, it implies to solve a numerical system on the unknowns θ, λ and µ which is over-determined. To deal with this problem, we simply introduce an additionnal unknown during the BBM trick process : we remark that the term ∂ t V θ appears twice in the dispersive terms of the first equation, we can then use two different expressions of ∂ t V θ , each with a different unknowns. This process is summed up in the following relations where we introduce the parameters λ 1 , λ 2 and µ :

         ∂ t V θ = (1 -λ 1 )∂ t V θ -λ 1 √ h∇η + O(ε) , ∂ t V θ = (1 -λ 2 )∂ t V θ -λ 2 √ h∇η + O(ε) , ∇ • ( √ hV θ ) = µ∇ • ( √ hV θ ) -(1 -µ)∂ t η + O(ε) ,
where we use the first relation on the term (θ 2 -1)h 2 ∇•∂ t V θ and the second relation on the term 2(θ-1)h∇h•∂ t V θ . The last difficulty stands in the possibility to lose the possibly symmetric structure during the derivation of the final system. The key point is to rely on the quantity √ h which is crucial to write a symetric form of the dispersive terms. Finally, we formally derive a new class (S b ) of systems, and we can prove that if a family (V ε , η ε ) 0<ε<ε0 is consistent with the system Γ b then (V ε θ , η ε ) 0<ε<ε0 where

V ε θ = 1 -ε 2 (θ -1)(θ∇(h 2 ∇• ) + ∇∇ • (h 2 .
)) V ε is consistent with any of the following systems (S θ,λ1,λ2,µ ) :

(S θ,λ1,λ2,µ )                                1 -ε 2 P 1 h ∂ t V + √ h ∇η + ε 2 F h V η + b 1 √ h∇∇ • (h 2 ∇η) + b 2 √ h∇(h∇h • ∇η) +b 3 ∇h∇ • (h √ h∇η) + b 4 √ h∇h(∇h • ∇η) = 0 , 1 -ε 2 P 2 h ∂ t η + ∇( √ h • V ) + ε 2 f h V η + c 1 ∇ • h 2 ∇∇ • ( √ hV ) + c 2 ∇ • (h∇h∇ • ( √ hV )) +c 3 ∇ • (h √ h∇(∇h • V )) + c 4 ∇ • ( √ h∇h(∇h • V )) = 0 .
where the operators P 1 h and P The previous computations are summed up in the following proposition.

Proposition 3.2. Let θ ∈ [0, 1] and (ψ ε , η ε ) 0<ε<ε0 a family of solutions of (0.7) such that (∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to ε in W 1,∞ ([0, T ε ]; H σ (R d ) d+1 ) with σ large enough. We define V ε = ∇ψ ε and

Ṽ ε = 1 -ε 2 (θ -1)(θ∇(h 2 ∇• ) + ∇∇ • (h 2 . )) √ h + ε 2 η √ h V ε .
Then for all (λ 1 , λ 2 , µ) ∈ R 3 , the family ( Ṽ ε , η ε ) 0<ε<ε0 is consistent with the system (S θ,λ1,λ2,µ ).

Moreover, we have the following proposition on the existence of a subclass of (S b ) composed with fully symmetric systems.

Proposition 3.3. There exists at least one value of (θ, λ 1 , λ 2 , µ) such that the system (S θ,λ1,λ2,µ ) is fully symmetric.

Proof. We are concerned here with the resolution of the following system :

       b 1 = c 1 , b 2 = -c 2 , b 3 = c 3 , b 4 = -c 4 .
This system on (θ, λ 1 , λ 2 , µ) have at least one solution that gives the following approximate values :

      
θ ≈ 0.6318 , λ 1 ≈ -0.3416 , λ 2 ≈ -2.8209 , µ ≈ -3.1157 , which ends the proof.

From now on, we only consider this solution and its approximate values.

The fully symmetric systems

Thanks to Proposition 3.3, we know that some of the systems (S θ,λ1,λ2,µ ) of the class (Λ b ) are completely symmetric : we hence denote by Σ b the non-empty subclass of (Λ b ) composed with these symmetric systems. Unfortunately, we do not have the same kind of existence theory on these systems as in the previous regime. Indeed, the main difference consists in the order one terms of the two equations √ h∇η ∇ • ( √ hV ) . In order to focus on the problem, we rewrite these terms : A(X, ∂ X ) V η where A(X, ∂

X ) = 0 √ h∇ X ∇ X • ( √ h× ) 0 .
The proof of the existence of solutions on a short time scale is not modified by these terms, the classical proof is still valid. However, the fact that the matrix A depends on the bottom term h is a real problem as far as the long time existence is concerned : indeed, one crucial point of the proof here relies on the size of the quantity ∇h ε on which we have no piece of information. The only case wherein we are surely able to demonstrate the long time existence is the case where ∇h is of order O(ε) : the term ∇h ε is then of order O(1) and we can conclude. In all other cases, the classical proof fails to provide a rigourous demonstration of the long time existence of solutions to these symmetric systems. Nevertheless, we are able to enounce the following proposition : Proposition 3.4. Let s > d 2 + 1 and (θ, λ 1 , λ 2 , µ) be such that the system (S θ,λ1,λ2,µ ) belongs to the class Σ b . Then for all (V 0 , η 0 ) ∈ H s (R d ) d+1 , there exists a time T 0 independant of ε and a unique solution (V, η) ∈ C([0, T 0 ]; H s (R d ) d+1 ) ∩ C 1 ([0, T 0 ]; H s-3 (R d ) d+1 ) to the system (S θ,λ1,λ2,µ ) such that (V, η) |t=0 = (V 0 , η 0 ). Furthermore, this unique solution is bounded independently of ε in the following sense : there exists a constant C 0 independent of ε such that for all k verifying s -3k > d 2 + 1, we have :

|(V, η)| W k,∞ ([0,T0];H s-3k (R d ) d+1 ) ≤ C 0 .
Besides, if we suppose that ∇h = O(ε), the previous result becomes valid on the long time interval [0, T0 ε ].

Proof. The key point of the proof is to demonstrate that the elliptic operator 1 -ε 2

P 1 h P 2
h is a positive one.

We first focus on P 1 h :

(1 -

ε 2 P 1 h V, V ) = |V | 2 2 + ε 2 (1 -θ 2 )(1 -λ 1 )|h∇ • V | 2 2 + ε(1 -θ)(1 -λ 2 )(∇h • V, h∇ • V )
Using the following inequality (satisfied for all a ∈ R) :

(∇h • V, h∇ • V ) ≤ a 2 2 |h∇ • V | 2 2 + 1 2a 2 |∇h • V | 2 2 ,
and taking a 2 = (1+θ)(1-λ1)

1-λ2 leads to :

(1 -

ε 2 P 1 h V, V ) ≥ |V | 2 2 - ε 2 (1 -θ)(1 -λ 2 ) 2 (1 + θ)(1 -λ 1 ) |∇h • V | 2 2
Using the classical Cauchy-Schwartz inequality leads finally to :

(1 -

ε 2 P 1 h V, V ) ≥ 1 - ε 2 (1 -θ)(1 -λ 2 ) 2 (1 + θ)(1 -λ 1 ) |∇h| 2 2 |V | 2 2 ,
At this point, if we take a small enough value of ε, f.e. ε ≤ 2(1+θ)(1-λ1)

(1-θ)(1-λ2) 2 |∇h| 2 2
, it ensures the global positivity of P 1 h . On P 2 h , we use the same method :

(1 -

ε 2 P 2 h η, η) = |η| 2 2 + ε 2 (1 -µ)(θ 2 - 1 3 )|h∇η| 2 2 + ε 2 (1 -µ)( 3 2 θ 2 - 7 6
)(η∇h, h∇η)

Using the same ideas as previously, one gets :

(1 - ε 2 P 2 h η, η) ≥ 1 - ε 8 (1 -µ)( 3 2 θ 2 -7 6 ) 2 θ 2 -1 3 |∇h| 2 2 |η| 2 2 ,

b 1 = λ 1 ( 1 -

 111 1 -θ) (1 -λ 1 )(θ + 1)∇(h 2 ∇• ) + 2(1 -λ 2 )∇(h∇h• ) , P 2 h = (1 -µ) (θ 2 -1 3 )∇ • (h 2 ∇ ) + ( 3 2 θ 2 -7 6 )∇ • (h∇h× ) ,and the parameters (a i ) 1≤i≤4 , (b i ) 1≤i≤4 have the following expressions : θ 2 );c 1 = µ(θ 2 -1 3 ); b 2 = (1 -θ)(2λ 2 -3 2 λ 1 (1 + θ)); c 2 = µ( 3 2 θ 2 -7 6 ); b 3 = λ1 2 (1 -θ 2 ); c 3 = -1 2 θ 2 + 2θ -7 6 ; b 4 = (1 -θ)(λ 2 -λ12 (1 + θ)); c 4 = 1 2 (θ -2) 2 ;

This work was supported by the ACI Jeunes chercheurs du ministère de la Recherche "Dispersion et nonlinéarités".

Once more, if we take f.e. ε ≤

, we have the global positivity of P 2 h .

Consequently, taking ε ≤ min( 2(1+θ)(1-λ1)

) ensures that the operator 1 -

At this point, using this result and performing usual energy estimates on the system proves the existence of a time T such that there exists an unique solution (V, η)

This result gives us an efficient theoretical background to contruct approximate solutions of the water waves problem on a time scale O(1), and O( 1 ε ) in the case ∇h = O(ε). This contruction follows the same steps -but in a different order -as the contruction of approximate solutions for the first regime : we consider a solution (ψ ε , η ε ) to the formulation (0.7) of the water waves problem. We take initial data (

for a suitably large value of s. We then define V ε = ∇ψ ε and V ε 0 = ∇ψ ε 0 : we first contruct the data (V ε Σ,0 , η ε Σ,0 ) by applying the two successive changes of variable on the data (V ε 0 , η ε 0 ). We then choose the parameters (θ, λ 1 , λ 2 µ) ∈ [0, 1] × R 2 such that the system (T 1 θ,λ1,λ2,µ ) is completely symmetric. Using Proposition 3.4, we know that there exists a unique solution to this system with initial data (V ε Σ,0 , η ε Σ,0 ) : we denote this solution by (V ε Σ , η ε Σ ). From this exact solution of the symmetric system (S θ,λ1,λ2,µ ), we finally construct an approximate solution of the water waves problem by successively and approximatively inverting the two changes of variable as shown below (which is possible if ε is small enough) :

We are now able to enounce our final result :

. Let (ψ ε , η ε ) 0<ε<ε0 be a family of solutions of (0.7) with initial data (∇ψ ε 0 , η ε 0 ) and such that (∇ψ ε , η ε ) 0<ε<ε0 is bounded with respect to

θ,λ1,λ2,µ ) ∈ Σ. Then for all ε < ε 0 , there exists a time T ≤ T 1 such that we have :

Besides, if we suppose that ∇h = O(ε) then (V ε app , η ε app ) approximates the water waves solutions on a long time scale :

Proof. The proof is an adaptation of the one of Theorem 2.7, and we omit it here.

Remark 3.6. In the general case, where we have no piece of information on the size of the quantity ∇h ε , our analysis is complete on a short time scale. We have indeed an approximation on this interval of time, and we know from Lannes [START_REF] Lannes | Well-posedness of the water-waves equations[END_REF] the existence of solutions to the water waves problem on a short time scale in 2D and