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

The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations

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Introduction

We recall here the context of the Water Waves problem, and introduce all the meaningful dimensionless parameters of this problem. We then present the shallow water regime and more specifically the Boussinesq-Peregrine regime. We finally introduce the different results proved in this paper, which are local existence theorems for the Boussinesq-Peregrine equations on different time scales and with different conditions.

The Water Waves problem

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 commonly referred to as the Water Waves problem. It 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 , which is the domain occupied by the water. More precisely, 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)

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 " ζpXqu.

(1.2)

In these equations, V and w are the components of the velocity evaluated at the surface. The vector n in the last equation stands for the upward normal vector at the bottom pX, z " ´H0 `bpXqq, and e z is the unit upward vector in the vertical direction. We denote P atm the constant pressure of the atmosphere at the surface of the fluid, ρ the (constant) density of the fluid, and g the acceleration of gravity. 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.

The dimensionless parameters

Since the properties of the solutions of the Water Waves problem depend strongly on the characteristics of the flow, 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 , P 1 " P P 0 , where t 0 " L x ? gH 0 , P 0 " ρgH 0 .

After rescaling, four dimensionless parameters appear in the Euler 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. The free surface Euler equations (1.1) and (1.2) become after rescaling (we omit the "primes" for the sake of clarity):

# B t U `εpV ¨∇γ `1 µ wB z qU " ´1 ε ∇ γ P, B t ζ ´a1 `ε2 |∇ γ ζ| 2 U ¨n " 0 (1.3)
where we used the following notations:

∇ γ " t pB x , γB y q if d " 2 and ∇ γ " B x if d " 1, and we recall that the unknown is the velocity U " pV, wq where V and w are respectively the horizontal and vertical components of the velocity. The equations (1.3) with boundary conditions for the pressure and the velocity are commonly referred to as the "dimensionless free surface Euler equations".

The Shallow Water regime

When the shallowness parameter µ "

H 2 0 L 2
x is small, it is possible to use a simplified equation in order to study the Water Waves problem. More precisely, at first order with respect to µ, the horizontal velocity V becomes columnar, which means that V " V `µR (1.4) where V stands for the vertical average of the horizontal velocity V pt, Xq " 1 hpt, Xq ż εζpt,Xq 1´βbpXq V pt, X, zqdz and h is the height of the water hpt, Xq " 1 `εζpt, Xq ´βbpXq. We do not give precise estimate for the residual R of (1.4) in Sobolev norm here. Lagrange [START_REF] Louis | Mémoire sur la théorie du mouvement des fluides[END_REF], and later Saint-Venant [START_REF] De Saint-Venant | Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et a l'introduction des marées dans leur lit[END_REF] derived from the Euler equations and under the assumption that the pressure is hydrostatic1 the following Shallow-Water equation expressed in term of unknowns pV , ζq:

# B t ζ `∇γ ¨phV q " 0 B t V `∇γ ζ `εpV ¨∇γ qV " 0, (1.5)
with initial data pV , ζq |t"0 " pV 0 , ζ 0 q. The Shallow-Water equations (1.5) are a typical example of quasilinear symmetrizable system (the symmetrization is done by multiplying the second equation by h). The local existence result for such a system is classical, and is done for example in [START_REF] Michael | Partial differential equations[END_REF] Chapter XVI (see also [START_REF] Benzoni | Multidimensional hyperbolic partial differential equations[END_REF]). The Shallow-Water equation is said to be consistent at the first order in µ with the Water-Waves equations, which means that formally, one has water-waves equation = shallow-water equation `Opµq.

It formally means that if one is interested in working with a shallow water (i.e. with small values of µ), one can get rid of all the terms of size µ in the Water-Waves equations and obtain the simplified model of the Shallow-Water equation. Alvarez-Sameniago and Lannes [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF], and Iguchi [START_REF] Iguchi | A shallow water approximation for water waves[END_REF] fully justified the Shallow-Water model by proving the mathematical convergence of the Water-Waves equation to the Shallow-Water equation. More precisely, one has the following result:

Theorem 1.1 There exists N ě 1, such that for all pV 0 , ζ 0 q P H N pR d q d`1 , there exists T ą 0 such that:

(1) There exists a unique solution pζ E , U E q defined on r0; T r to the free surface Euler equation (1.1), (1.2) such that pζ E , V E q P H N pR d q d`1 , and pζ E , V E qp0q " pζ 0 , V 0 q.

(2) There exists a unique solution pζ SW , V SW q P Cpr0; T r; H N pR d q d`1 q to the Shallow-Water equation

(1.5) with initial conditions pζ 0 , V 0 q.

(3) One has, for all t P r0; T r:

|pζ E , V E qptq ´pζ SW , V SW qptq| H N ď Cp|pζ 0 , V 0 q| H N qµt.
This theorem implies that the error made by using the solutions of the simplified model of Shallow-Water instead of the solutions of the free-surface Euler equations is "of size µ".

At the second order with respect to µ, one can derive several models for the Water-Waves problem which are formally more precise than the Shallow-Water equation. We consider in this paper the case where ε " Opµq, which corresponds to a small amplitude model in the Shallow-Water regime. Boussinesq ([9], [START_REF] Boussinesq | Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal[END_REF]) derived the following model for flat bottoms, and later Peregrine [START_REF] Howell Peregrine | Long waves on a beach[END_REF] for nonflat bottoms:

# B t ζ `∇γ ¨phV q " 0 r1 `µT b sB t V `εpV ¨∇γ qV `∇γ ζ " 0 (1.6)
where T b is the following operator :

@V , T b V " ´1 3h b ∇ γ ph 3 b ∇ γ ¨V q `β 2h b r∇ γ ph 2 b ∇ γ b ¨V q ´h2 b ∇ γ b∇ γ ¨V s `β2 ∇ γ b∇ γ b ¨V , with the notation h b " 1 ´βb.
See also [START_REF] Lannes | The water waves problem[END_REF] for a complete proof of the formal derivation of this equation. This equation is known as the Boussinesq-Peregrine equation, and has been used a lot in applications to coastal flows. In the case of a flat bottom, one has β " 0 and T b " ´1 3 ∇ γ ∇ γ ¨. The equation (1.6) can then be seen as a particular case of a Boussinesq system (see [START_REF] Bona | Boussinesq equations and other systems for smallamplitude long waves in nonlinear dispersive media. i: Derivation and linear theory[END_REF] for the 1d case, and [START_REF] Bona | Long wave approximations for water waves[END_REF] for the 2d case).

The Boussinesq-Peregrine model is a good compromise for numerical simulation between the precision of the approximation of the Water-Waves problem, and the simplicity of the equations. Indeed, the Boussinesq-Peregrine equation preserves the dispersive nature of the Water-Waves equation. To understand this statement, one can look for plane wave solutions pζ, V q " pζ 0 , V 0 qe ipk¨X´ωpkqtq of the linearized Boussinesq-Peregrine equation (1.6) and finds solutions with a dispersive relation:

ωpkq 2 " |k γ | 2 1 `µ 3 |k γ | 2 ,
with k γ " t pk 1 , γk 2 q. Thus the group velocity c " ωpkq |k γ | of the water waves depends on the frequency, which is a definition of dispersion. The Shallow-Water equation, however, is not a dispersive equation, at least in dimension d " 1, since one would find a group velocity of the water waves equals to 1. One could also derive an even more precise model at the Opµ2 q order than the Boussinesq-Peregrine model, without any assumption on the smallness of ε, which is called the Green-Naghdi equation (see [START_REF] Serre | Contribution à l'étude des écoulements permanents et variables dans les canaux[END_REF] for the case d " 1, [START_REF] Su | Korteweg-de vries equation and generalizations. iii. derivation of the korteweg-de vries equation and burgers equation[END_REF], [START_REF] Fernando | Numerical and experimental study of the transformation of a solitary wave over a shelf or isolated obstacle[END_REF] for the case of non-flat bottom and also under the name of "fully nonlinear Boussinesq equations" in [START_REF] Wei | A fully nonlinear boussinesq model for surface waves. part 1. highly nonlinear unsteady waves[END_REF]; see also [START_REF] Miles | Weakly dispersive nonlinear gravity waves[END_REF]). It has a very similar form as one of the Boussinesq-Peregrine equation, but with h b replaced by h in the definition of the operator T b . Therefore, though the Green-Naghdi model should be more precise than the Boussinesq-Peregrine equation, its numerical implementation leads to the computation at each time step of the inverse of I `µT b (which now depends on h, and therefore on the time), which increases the computational complexity 2 .

Long time existence for the Water-Waves models

We are interested in the dependence of the existence time for the solutions of systems like (1.5) and (1.6) with respect to the parameter ε. For such quasilinear equations with an ε factor on the nonlinearity, the "good" time existence should be of size 1 ε . Let us explain this statement by studying the one dimensional Burgers equation:

# B t u `εuB x u " 0 up0, xq " u 0 pxq
where we assume some reasonable regularity on u 0 . Using the method of characteristics to solve it, we find that characteristics are of the form xptq " εu 0 px 1 qt `x1 with x 1 P R. The solutions do not exist globally in the general case because the characteristics intersect themselves. Let us estimate the time at which they intersect. Let us consider two characteristics xptq " εu 0 px 1 qt `x1 and εu 0 px 2 qt `x2 . They intersect at time t " ´x2 ´x1 εpu 0 px 2 q ´u0 px 1 qq .

Therefore, the existence time for the solutions is

T " min x1,x2PR ´x2 ´x1 εpu 0 px 2 q ´u0 px 1 qq " ´1 εmin xPR u 1 0 pxq
and therefore is of size 1 ε .

The standard theory for quasilinear symmetrizable systems gives the local existence of solutions pV , ζq of the Shallow-Water equations (1.5) on the space Cpr0; T s; H N pR d q d`1 q, with N large enough, and gives an explosion criterion: one has T ă 8 if and only if lim tÑT |pV , ζqptq| W for a smooth non decreasing function g, one would get by a continuity argument that the solutions exist on an interval of size 1 ε . An L 2 estimate of the form (1.7) is easy to obtain, because multiplying the second equation of (1.5) symmetrizes both first space derivatives order terms and quantities of size ε. Indeed, one can differentiate with respect to time the "energy"

Epζ, V q " 1 2 |ζ| 2 2 `1 2 phV , V q 2 to get dE dt " pB t ζ, ζq 2 `pB t V , hV q 2 `1 2 ppB t hqV , V q2
and replace B t pζ, V q by their expressions given in the equation (1.5). The terms of order 1 are transparent in the energy estimate because they cancel one another, thanks to the "symmetry" of the system, and one gets dE dt " 0 which is even better than needed. However, if one differentiates the equation (1.5) with respect to space variable, one finds a system of the form (B denotes here any space derivative of order one): # B t BV `∇γ ¨ppBhqV q `∇γ ¨phBpV qq " 0 B t Bζ `εpBV q ¨∇γ V `εV ¨∇γ BV `∇γ Bζ " 0.

It is not possible anymore to make this system symmetric in order to cancel the terms that are not of size ε in the energy estimates. Indeed, the term Bh is not of size ε, since h " 1 `εζ ´βb, and thus also depends on β. The H N norms of the unknowns are then not easily controlled by terms of size ε, which prevent us from proving directly an estimate of the form (1.7). The long existence for this system is therefore tied to a singular perturbation problem with variable coefficients.

Long time existence results for similar types of equations have been proved for example by Schochet in [START_REF] Schochet | The compressible euler equations in a bounded domain: Existence of solutions and the incompressible limit[END_REF] for the compressible Euler equation in a bounded domain with well-prepared data, or by Schochet-Métivier in [START_REF] Métivier | The incompressible limit of the non-isentropic euler equations[END_REF] for the Non-isentropic Euler equation with general data. Alazard ( [START_REF] Alazard | Incompressible limit of the nonisentropic Euler equations with the solid wall boundary conditions[END_REF]) proved a long time existence result for the non-isentropic compressible Euler equation, in the case of bounded and unbounded domains. ) proved that for N ą d{2 `1 and pζ 0 , V 0 q P H N pR d q d`1 , there exists a unique solution pζ, V q P Cpr0; T ε s; H N pR d q d`1 q to the equation (1.5) with initial data pζ 0 , V 0 q where T only depends on the norm of the initial data, even if β is not assumed to be small.

In the case of a flat bottom, as explained in Section 1.3, the Boussinesq-Peregrine equation can be seen as a particular case of the Boussinesq systems. Saut and Li ([22]) proved the local existence on a large time interval of size 1 ε for most of these systems, when the bottom is flat. There is, in our knowledge, no local existence results in the literature in the case of non flat bottoms. A local existence result for the Boussinesq-Peregrine equation (1.6) on a time T ε with T independent on µ would be important to fully justify this model, and get a convergence result similar to Theorem 1.1.

Main result

In [START_REF] Mésognon-Gireau | The Cauchy problem on large time for the Water Waves equations with large topography variations[END_REF], a large time existence result is proved for the Water-Waves equations in presence of large topography, extending the result of [START_REF] Alvarez-Samaniego | Large time existence for 3D water-waves and asymptotics[END_REF] which holds for small topography variations 3 (β " Opεqq). Coupling this result with the one of [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF], one can prove that the Shallow-Water equations are consistent at order 1 in µ with the full Water-Waves equations, on a time interval of size 1 ε , even in the case of large topography variations (β " Op1q). A similar result between the Water-Waves equations and the Boussinesq-Peregrine equation (1.6) would be a new step in the mathematical justification of the Water-Waves models.

We first prove in this paper a well posedness result for the equation (1.6) on a time Op1q (Theorem 2.1 below). Though not optimal as we shall see, such a local well posedness result did not seem to be available in the literature for non flat bottoms. The proof consists in a particular adaptation of the proof of local existence for symmetrizable quasilinear systems. The difficulty is that this system is not easily symmetrizable. In the case of the Shallow-Water equations (1.5), as explained in Section 1.3, one should multiply the second equation by h to get a "symmetric" system. However, in the case of the Boussinesq-Peregrine equation (2.11), multiplying the second equation by h indeed symmetrizes the system with respect to order one space derivatives. But the operator hpI `µT b q is not symmetric. Therefore, phpI `µT b qB t V , V q 2 is not equal to d dt phpI `µT b qV , V q 2 . It implies the presence of some commutators terms between hT b and B t in the energy estimates which are difficult to control. However, the operator h b pI `µT b q is symmetric and we have the equivalence

ph b pI `µT b qV , V q 2 " |V | 2 2 `µ|∇ γ ¨V | 2 2 .
Therefore, a "good" energy for the Boussinesq-Peregrine equation seems to be

Epζ, V q " 1 2 |ζ| 2 2 `1 2 ph b pI `µT b qV , V q 2 . (1.8)
But multiplying the second equation by h b does not properly symmetrize the system with respect to space derivatives. More precisely, it symmetrizes them up to an εζ∇ γ ¨V factor. In the Boussinesq-Peregrine regime, one has ε " Opµq and therefore this term is actually controlled by the energy... It yields to a local existence result for a time interval of size Op1q and not Op 1 ε q. This is the purpose of Theorem 2.1.

One then looks for an improved time of existence for the Boussinesq-Peregrine equation. In order to do so, one could try to use an adaptation of the proof of the long time existence result by Bresch-Métivier [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF]. The idea of this proof is to have energy estimates of the form Eptq ď CpEqtε `C0 for some constant CpEq which depends on the energy, and C 0 which only depends on initial data, where E is an energy. One can then conclude by a continuity argument that the energy stays bounded on an interval of size vanishes, which is even better than being of size ε. However, it does not stand true for higher order 3 However this result needs the presence of a small surface tension in the model estimates. Looking for an estimate on d dt |uptq| H N for some N ą 0, one differentiates the equation (1.9) and finds a system of the form: B t B x u `εpuB x qB x u `εruB x , B x su `Lpεu, apxqqB x u `εpB x uqdL 1 pεu, apxqqu `pB x aqdL 2 pεu, apxqqu " 0, where we denoted dL i pεu, apxqq the differential of L with respect to the i ´th variable, at the point pεu, apxqq, for i " 1, 2. Due to an extra term pB x aqdL 2 pεu, apxqqu, the energy estimates involve terms which are not of size ε. This problem does not appear for time derivatives: if one differentiates the equation (1.9) with respect to time, one finds: B t pB t uq `εuB x pB t uq `εruB x , B t su `Lpεu, apxqqB t u `εpB t uqdL 1 pεu, apxqqu.

Therefore, one can find an energy estimate of the form

|pB k t uqptq| 2 ď |pB k t uqp0q| 2 `εtCp|pB k t uqptq| 2 q,
for all k ď N . In order to find a similar energy estimate in H N , one uses the equation, which gives an expression of space derivatives with respect to time derivatives:

Lpεu, apxqqu " ´Bt u ´εuB x u and using the previous estimate for time derivatives, and the ellipticity of L, one gets:

|uptq| H 1 ď Cp|uptq| H 1 qεt `C0
for some constant C 0 and a non decreasing smooth function C. One can do the same for higher order Sobolev estimates, by considering higher order time derivatives B k t and using a finite induction on k. By a continuity argument, an estimate of the form

|uptq| H N ď Cp|uptq| H 1 qεt `C0
implies that the H N norm of u stays bounded on an interval of size 1 ε .

This technique only works if time and space derivatives have the same "order". More precisely, for the Shallow-Water equation (1.5), the time derivatives are equal to sum of terms involving one space derivative. This is not the case for the Boussinesq-Peregrine equation (1.6). Indeed, in the second equation, pI `µT b qB t V is equal to one space derivative order terms, while pI `µT b q is of order two. It leads to issues if one tries to use the equation to control space derivatives by time derivatives and tries to recover an estimate of size ε for the space derivatives. For example, the second equation of (1.6) provides

∇ γ ζ " ´pI `µT b qB t V `ε ˆother terms ,
and T b is an order two operator with respect to space. It is therefore not clear that T b B t V is controlled by the energy (1.8).

To overcome this problem, we introduce a modified equation, which is consistent with the Boussinesq-Peregrine equation (1.6) at the Opµ 2 q order (and therefore with the Water-Waves equations). Such equation would have a proper structure adapted to the use of the method used by Bresch-Métivier in [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF]. The approach of modifying the equation without changing the consistency, in order to improve the structure of the equation has been used for example by Israwi in [START_REF] Israwi | Derivation and analysis of a new 2d green-naghdi system[END_REF] for the Green-Naghdi equation, or by Saut and Xu ([23]) for a model of full dispersion. In the Boussinesq-Peregrine case, a short study (see later Section 3 for more details) leads us to introduce the following modified equation:

$ & % B t ζ `∇γ ¨phV q " 0 pI `µpT b ´∇γ p 1 h b ∇ γ ¨ph b ¨qq ´1 h b ∇ γK ∇ γK ¨qqB t V `εV ¨∇γ V `pI ´µ∇ γ 1 h b ∇ γ ¨ph b ¨qq∇ γ ζ " 0
(1.10) The main result of this paper is the following (see later Theorem 3.7 for a precise statement): ( 2) In dimension 1, the equations (1.10) admit a unique solution on a time interval of the form r0; T ε s where T only depends on the initial data.

Remark 1.3 For technical reasons which are discussed further below, the Theorem we prove is only true in dimension 1. However, we precisely explain in this paper the difficulties raising for a proof in dimension 2.

Though this Theorem is proved by adapting the technique used by Bresch-Métivier in [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF], its adaptation to the case of a dispersive equation has not been done yet in the literature to our knowledge. As one shall see later in Section 3.2, this result is tied to a singular perturbation problem. The plan of the article is the following:

-In Section 2, we prove a local existence result for the Boussinesq-Peregrine equation in dimension

d " 1, 2,
-In Section 3 we introduce a modified Boussinesq-Peregrine equation and in Section 3.1 we prove its local well-posedness in dimension d " 1, 2,

-In Section 3.2, we prove the long time existence result for the modified Boussinesq-Peregrine equation, in dimension d " 1.

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, ∇ γK " t p´γB y , B x q if d " 2 and ∇ γK " 0 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, ie γ " 1. However, if one is not interested in the dependence of γ, it is possible to take γ " 1 in all the following proofs. A typical situation where γ ‰ 1 is for weakly transverse waves for which γ " ? µ; this leads to weakly transverse Boussinesq systems and the Kadomtsev-Petviashvili equation (see [START_REF] Lannes | Weakly transverse Boussinesq systems and the KP approximation[END_REF]). A byproduct of our results is therefore a generalization to the case of nonflat bottoms of the results on weakly transverse Boussinesq systems of [START_REF] Lannes | Weakly transverse Boussinesq systems and the KP approximation[END_REF]; this opens new perspectives towards the derivation and justification of Kadomtsev-Petviashvili equations for nonflat bottoms in the spirit of [START_REF] Israwi | Variable depth kdv equations and generalizations to more nonlinear regimes[END_REF] for the KdV equation.

We define a _ b for two real numbers a, b by:

a _ b " maxpa, bq.
For all α " pα 1 , ..., α d q P N d , we define B α the operator of S 1 pR d q by:

B α " B α1 x1 ...B α d x d .
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. We also use the following operators:

@V , T b V " ´1 3h b ∇ γ ph 3 b ∇ γ ¨V q `β 2h b r∇ γ ph 2 b ∇ γ b ¨V q ´h2 b ∇ γ b∇ γ ¨V s `β2 ∇ γ b∇ γ b ¨V
in all this paper, and the operators:

A " ∇ γ p 1 h b ∇ γ ¨ph b ¨qq, B " pI `µT b ´µ∇ γ p 1 h b ∇ γ ¨ph b ¨qq ´µ 1 h b ∇ γK ∇ γK ¨q
in Section 3.

Functional spaces

The standard scalar product on L 2 pR d q is denoted by p , q 2 and the associate norm | ¨|2 . We will denote the standard scalar product on Sobolev spaces H s pR d q by p , q H s and the associate norm by | ¨|H s . We denote the norm W k,8 pR d q by | ¨|W k,8 , and we use the notation | ¨|8 " | ¨|W 0,8 when no ambiguity is possible.

We also introduce in Section 3.2 for all s P R the Banach space

X s pR d q " tf P L 2 pR d q d , |f | X s ă 8u endowed with the norm |f | X s " |f | 2 X s " |f | 2 H s `µ|∇ γ ¨f | 2 H s .

Local existence for the Boussinesq-Peregrine equation

In this section, we prove a local existence result for the Boussinesq-Peregrine equation in dimension d " 1, 2, on a time interval independent on µ. We recall that the Boussinesq-Peregrine equation of unknowns V and ζ is:

# B t ζ `∇γ ¨phV q " 0 r1 `µT b sB t V `εpV ¨∇γ qV `∇γ ζ " 0 (2.11)
where h " 1 `εζ ´βb and where T b is the following operator :

@V , T b V " ´1 3h b ∇ γ ph 3 b ∇ γ ¨V q `β 2h b r∇ γ ph 2 b ∇ γ b ¨V q ´h2 b ∇ γ b∇ γ ¨V s `β2 ∇ γ b∇ γ b ¨V , (2.12)
with the notation h b " 1 ´βb, where b describes the variations of the bottom and is known. We also recall (see section 1.6) the notation

|f | X s " |f | 2 H s `µ|∇ γ ¨f | 2 H s .
Let us state a local existence result for the Boussinesq-Peregrine equation:

Theorem 2.1 Let t 0 ą d{2 and s ą t 0 `1. Let b P H s`2 pR d q be such that there exists h min ą 0 such that inf

XPR d h b pXq ě h min .
Let ε, β be such that 0 ď ε, β ď 1.

Let U 0 " pζ 0 , V 0 q P H s pR d q ˆXs pR d q. Then, there exists µ max ą 0 such that for all 0 ď µ ď µ max with ε " Opµq, there exists T ˚ą 0 and a unique solution U " pζ, V q P Cpr0; T ˚r; H s pR d q d`1 q of the equation (2.11) with initial condition U p0q " U 0 .

Moreover, for all T ă T ˚, if one chooses

λ ě sup tPr0;T s C 3 p|U | W 1,8 , |h b | H s`2 , µ max qptq,
the solution U satisfies the following inequality:

@t ď T, Eptq ď λ C 1 ph min q ż t 0 e λ C 1 ph min q pt´t 1 q C 4 p|U | H s qpt 1 qdt 1 `C2 p|h b | H t 0 q C 1 ph min q Ep0qe λ C 1 ph min q t (2.13) with Eptq " pµ|∇ γ ¨V | 2 H s `|U | 2 H s qptq (2.14)
and where C i are non decreasing smooth functions of their arguments, for i " 1, 2, 3, 4.

Remark 2.2 -It is very important to note that the energy estimate (2.13) implies that while |U | W 1,8 ptq is bounded, the solution U can be continued. More precisely, if one sets T " suptt, U exists on Cpr0; tr; H s pR d q d`1 qu then if T ă 8, one has sup tÑT |U | W 1,8 ptq " `8. Moreover, one has to notice that the energy E defined in the statement of the Theorem by (2.14) controls H s norms of both U and ∇ γ ¨V , while the energy estimate (2.13) only requires a bound of E by the H s norm of U . In particular, it suffices to bound the H s norm of U (instead of U and ∇ γ ¨V ) to use a continuity argument.

-Note that the time of existence T ˚defined by Theorem 2.1 is independent on µ. This is crucial in view of the proof of the consistency of the Boussinesq-Peregrine equation with the Water-Waves equations (see Theorem 1.1).

Proof of Theorem 2.1 The system (2.11) can be put under the form

BB t U `d ÿ j"1 A j pU qB j U " 0 where B " ˆ1 0 0 I `µT b ˙, A j pU q " ˆεV j h 1 εV j I d ˙@1 ď j ď d. (2.15)
The non linear terms of the equation can be symmetrized if we multiply the system by SpU q " ˆ1 0 0 hI d ḃut for the reasons explained in Section 1.5 we use rather the following matrix:

S " ˆ1 0 0 h b I d ˙.
(2.16)

Using this symmetrizer brings one difficulty: the operator h b pI `µT b q is symmetric, but multiplying the second equation of (2.11) by h b does not symmetrize the non linear terms of the form A j pU qB j U defined by (2.15), for j " 1, .., d. The proof of Theorem 2.1 is inspired of the classical existence result for quasilinear hyperbolic systems (see [START_REF] Michael | Partial differential equations[END_REF] Chapter XVI for instance). We follow the following steps:

-Step 1 : We solve a smoothed equation involving a mollifier p1 ´δ∆q.

-Step 2 : We prove that the existence time of the solution of the mollified equation does not depend on δ, and the uniform bound in H s norm of this solution.

-Step 3 : We pass to the limit δ goes to zero in the mollified equation to get a solution of the equation (2.11).

-Step 4 : We recover regularity for the solution of (2.11).

Step 1

We solve the Cauchy problem

# p1 ´δ∆qSBp1 ´δ∆qB t U δ `řd i"1 SA j pU δ qB j U δ " 0 U δ p0q " p1 ´δ∆q ´1U 0 , (2.17)
of unkown U δ in the Banach space H s pR d q d`1 . Recall that using the definition of B given in (2.15) and S given by (2.16), one has

SB " ˆ1 0 0 h b pI d `µT b q ˙.
In order to apply the Cauchy-Lipschitz Theorem, one must check that the application

H s pR d q d`1 ÝÑ H s pR d q d`1 U Þ ÝÑ p1 ´δ∆q ´1pSBq ´1p1 ´δ∆q ´1 ř d i"1 A j pU qB j U
is well defined and locally Lipschitz. The unique difficulty is to check that ph b pI `µT b qq ´1 is well defined from H s to H s . It is the point of the following Proposition (see [START_REF] Lannes | The water waves problem[END_REF] Chapter 5 Lemma 5.44 for a full proof). We first define the Banach space

X s " tV P H s pR d q d , ∇ γ ¨V P H s pR d qu endowed with the norm |V | 2 X s " |V | 2 H s `µ|∇ γ ¨V | 2 H s .
Proposition 2.3 Let t 0 ą d{2, β ď 1 and b P H t0`1 pR d q be such that there exists h min such that h b " 1 ´βb ě h min . Then the mapping

h b pI `µT b q : X 0 ÝÑ L 2 pR d q d `∇γ L 2 pR d q
is well defined, one-to-one and onto. One has, for all V P X 0 :

C 1 ph min q|V | 2 X 0 ď ph b pI `µT b qV, V q 2 ď C 2 p|h b | H t 0 `1 q|V | 2 X 0
where C i are non decreasing functions of its arguments. Moreover, for all s ě 0,

if b P H 1`s_t0 pR d q, then @W P H s pR d q d , |ph b pI `µT b qq ´1W | X s ď Cp 1 h min , |b| H 1`s_t 0 q|W | H s
where C is a non decreasing function of its arguments. Moreover, one has, for all s P R:

? µ|ph b pI `µT b qq ´1∇ γ W | H s ď Cp 1 h min , |b| H 1`|s|_t 0 q|W | H s .
Therefore, the Cauchy-Lipschitz Theorem applies and the equation (2.17) has a unique solution U δ P Cpr0; T δ r; H s pR d q d`1 q, and if T δ ă `8 one has

lim tÑT δ |U δ ptq| H s " `8.
Step 2

We now check that one can choose T δ independent of δ by comparing |U δ ptq| H s with a solution of an ODE independent of δ, and using a Gronwall Lemma. We define

U δ s " Λ s U δ .
The unknown U δ s satisfies the following system :

p1 ´δ∆qSBp1 ´δ∆qB t U δ s `d ÿ j"1 SA j pU δ qB j U δ s " F (2.18)
where we wrote the commutators under the form

F " p1 ´δ∆qrSB, Λ s sp1 ´δ∆qB t U δ `d ÿ j"1 rSA j pU δ q, Λ s sB j U δ . (2.19)
In order to estimate, |U δ ptq| H s , recall that

SB " ˆ1 0 0 h b pI d `µT b q ȧnd remark, using Proposition (2.3), that |p1 ´δ∆qV δ s | X 0 " ph b pI `µT b qp1 ´δ∆qV δ s , p1 ´δ∆qV δ s q 2
where the implicit constant only depend on h b . Therefore, one computes:

d dt 1 2 pp1 ´δ∆qSBp1 ´δ∆qU δ s , U δ s q 2 " pp1 ´δ∆qSBp1 ´δ∆qB t U δ s , U δ s q 2 .
Note that the symmetry of SB, and more precisely of h b pI `µT b q is crucial here. One uses the equation (2.18) to replace p1 ´δ∆qSBp1 ´δ∆qB t U δ s by its expression. One gets:

d dt 1 2 pp1 ´δ∆qSBp1 ´δ∆qU δ s , U δ s q 2 " ´d ÿ j"1 pSA j pU δ qB j U δ s , U δ s q 2 `pF, U δ s q 2 . (2.20)
Let us check that the first term of the rhs of (2.20) has a contribution of order zero to the energy estimate.

One uses the definition of A j given by (2.15) to put this matrix under the form A j pU δ q " Ãj pU δ q`CpU δ q with Ãj pU δ q "

˜εV δ j h b 1 εV δ j I d ¸, CpU δ q " ˆ0 εζ δ 0 0 ˙, (2.21)
for j " 1, .., d. Note that since S is not a true symmetrizer for the equation (2.11), the matrix SA j is not symmetric. The above decomposition allows us to write SA j under the form of a symmetric matrix (S Ãj ) plus a rest which we intend to control in the energy estimates. We now write:

d ÿ j"1 pSA j pU δ qB j U δ s , U δ s q 2 " d ÿ j"1 pSCpU δ qB j U δ s , U δ s q 2 `d ÿ j"1 pS Ãj pU δ qB j U δ s , U δ s q 2 . (2.22)
Using the definition of CpU δ q given by (2.21), the first term of the rhs of (2.22) is equal to

ż ζ δ ζ δ s ε∇ γ ¨V δ s
and therefore is controlled, using the Cauchy-Schwarz inequality by

| ż εζ δ ∇ γ ¨V δ s ζ δ s | ď |ζ δ | W 1,8 |µ∇ γ ¨V δ s | 2 |ζ δ s | 2 (2.23)
where we used the Boussinesq regime condition ε ď Cµ stated by the Theorem. For the second term of the rhs of (2.22), one can write, for all 1 ď j ď d, and using the symmetry of S Ãj pU δ q (recall the definition of the symmetrizer S given by (2.16)):

pS Ãj pU δ qB j U δ s , U δ s q 2 " pB j U δ s , S Ãj pU δ qU δ s q 2 " ´`U δ s , B j pS Ãj pU δ qU δ s q

˘2

by integrating by parts. Now, one has

´`U δ s , B j pS Ãj pU δ qU δ s q ˘2 " ´`U δ s , B j pS Ãj pU δ qqU δ s ˘2 ´pU δ s , S Ãj pU δ qB j U δ s q 2

and thus one has

pS Ãj pU δ qB j U δ s , U δ s q 2 " ´1 2 `U δ s , B j pS Ãj pU δ qqU δ s ˘2. (2.24)
Using the definition of S given by (2.16), and the definition of Ãj given by (2.21), one has

S Ãj " ˜εV δ j h b h b εh b V δ j I d
for all j " 1, .., d and thus (2.24) is controlled, using Cauchy-Schwarz inequality:

|pS Ãj U δ B j U δ s , U δ s q 2 | ď |U δ s | 2 2 c 2 p|U δ | W 1,8 , |h b | W 1,8 q, (2.25)
where c 2 is a non decreasing and smooth function of its arguments.

We now control the second term of the right hand side of the energy estimate (2.20). Using the definition of F given by (2. [START_REF] Miles | Weakly dispersive nonlinear gravity waves[END_REF]), one has that pF, U δ s q 2 " A 3 `A4

where

A 3 " pp1 ´δ∆qrSB, Λ s sp1 ´δ∆qB t U δ , U δ s q 2 and A 4 " prSA j pU δ q, Λ s sB j U δ , U δ s q 2 . -Control of A 3
We start by replacing p1 ´δ∆qB t U δ by its expression given in the equation (2.17):

pp1 ´δ∆qrSB, Λ s sp1 ´δ∆qB t U δ , U δ s q " ´d ÿ j"1 pp1 ´δ∆qrSB, Λ s spSBq ´1p1 ´δ∆q ´1SA j pU δ qB j U δ , U δ s q 2 .
(2. for a smooth function f . One has to recall that

B " ˆ1 0 0 I `µT b ṡo that rSB, Λ s s " ˆ0 0 0 rh b pI `µT b q, Λ s s ˙.
Using the definition of T b given by (2.12), one writes this operator under the form:

I `µT b " I `µpA `B `C `Dq, where A " ´∇γ ph 3 b ∇ γ ¨q, B " β∇ γ ph 2 b ∇ γ b¨q, C " ´βh 2 b ∇ γ b∇ γ ¨, D " β 2 ∇ γ b∇ γ ¨(2.27)
One expands the commutator (2.26) with respect to A, B, C, D. We set

A 31 " pp1 ´δ∆qµrA, Λ s sph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ , V δ s q 2
and we now control this latter term. For all 1 ď j ď d, using the definition of A given by (2.27), one has, integrating by parts:

A 31 " pp1 ´δ∆qrh 3 b , Λ s s ? µ∇ γ ¨ph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ , ? µ∇ γ ¨V δ s q 2 .
Using Cauchy-Schwarz inequality and splitting p1 ´δ∆q, one gets:

|A 31 | ď ? µ|∇ γ ¨V δ s | 2 `|rh 3 b , Λ s s ? µ∇ γ ¨ph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ | 2 `δ|rh 3 b , Λ s s ? µ∇ γ ¨ph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ | H 2 ˘.
We now use the Kato-Ponce estimate of Proposition A.1 to control rh 3 b , Λ s s, using the fact that s ą d{2`1 (and thus H s´1 pR d q and H s pR d q are respectively continuously injected into L 8 pR d q and W 1,8 pR d q):

|A 31 | ď ? µ|∇ γ ¨V δ s | 2 C s p|∇ γ h b | H s`1 q `|? µ∇ γ ¨ph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ | H s´1 `δ| ? µ∇ γ ¨ph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ | H s`1 ˘, (2.28)
where C s is a smooth non decreasing function which only depends on s. We now control the operator ? µ∇ γ ¨ph b pI `µT b qq ´1 in H s´1 and H s`1 norms by using the last part of Proposition 2.3 and a duality argument. One has, for all k ě t 0 and all u P H k pR d q d , using the symmetry of h b pI `µT b q : ? µ|∇ γ ¨hb pI `µT b q ´1u| H k " ? µ sup

vPH ´k pR d q |v| H ´k "1 p∇ γ ¨ph b pI `µT b qq ´1u, vq 2 " ? µ sup vPH ´k pR d q |v| H ´k "1 ´pu, ph b pI `µT b qq ´1∇ γ vq 2 ď sup vPH ´k pR d q |v| H ´k "1 Cp 1 h min , |b| H k`1 q|u| H k |v| H ´k ď Cp 1 h min , |b| H k`1 q|u| H k , (2.29)
where C is a smooth non decreasing function of its arguments, and where we used the fact that k ě t 0 . Using (2.29) with k " s ´1 and k " s `1 in (2.28) (recall that s ą t 0 `1), one gets:

|A 31 | ď C ? µ|∇ γ ¨V δ s | 2 Cp 1 h min , |b| H s`2 q `|p1 ´δ∆q ´1SA j pU δ qB j V δ | H s´1 `δ|p1 ´δ∆q ´1SA j pU δ qB j V δ | H s`1 ď C ? µ|∇ γ ¨V δ s | 2 Cp 1 h min , |b| H s`2 q `|SA j pU δ qB j V δ | H s´1 `Cδ 1 δ |SA j pU δ qB j V δ | H s´1
where we used the estimates

|p1 ´δ∆q ´1f | H s´1 ď |f | H s´1 and |p1 ´δ∆q ´1f | H s`1 ď C δ |f | H s´1 with C independent on δ. We recall that SA j pU δ q " ˜εV δ j h h b εh b V δ j for j " 1, .
., d and we use the Moser estimate of Proposition A.2 and the fact that s´1 ą d{2 to conclude:

|A 31 | ď Cp 1 h min , |b| H s`2 q ? µ|∇ γ ¨V δ s | 2 |V δ s | 2 2
with C a smooth, non decreasing function of its arguments.

To control the term of (2.26) involving B, using the definition of B given by (2.27), we write, integrating by parts:

d ÿ j"1 pp1 ´δ∆qµrB, Λ s sph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ , V δ s q 2 " ´µpp1 ´δ∆qrh 3 b ∇ γ b¨, Λ s sph b pI `µT b qq ´1p1 ´δ∆q ´1SA j pU δ qB j V δ , ∇ γ ¨V δ s q 2
and we use exactly the same techniques as used for the control of A 31 to get the same control. The terms of (2.26) involving C and D are easily controlled by

Cp 1 h min , |b| H s`2 q ? µ|V δ s | 3 2 .
We finally proved that:

A 3 ď c 3 p|U δ | W 1,8 |, |∇ γ b| H s`2 q|U δ s | 2 2 p|U δ s | 2 `?µ|∇ γ ¨V δ s | 2 q (2.30)
where c 3 is a smooth positive non decreasing function of its argument and independent of δ.

-Control of A 4 Let us now control A 4 by using the Kato-Ponce estimate of Proposition A.1 and the Moser estimate of Proposition A.2, using again that s ą d{2 `1:

prSA j pU δ q, Λ s sB j U δ , U δ s q 2 ď Cp|∇ γ SA j pU δ q| H s´1 |B j U δ | L 8 `|∇ γ SA j pU δ q| L 8 |B j U δ | H s´1 q|U δ s | 2 ď pCp|U δ | 8 q|U δ | H s |B j U δ | 8 |h b | H s `Cp|U δ | W 1,8 , |h b | W 1,8 q|U δ | H s q|U δ s | 2 ď c 4 p|U δ | W 1,8 , |h b | H s q|U δ s | 2 2 . ( 2
d dt pp1 ´δ∆qSBp1 ´δ∆qU δ s , U δ s q 2 ď c 5 p|U δ | W 1,8 , |h b | H s`2 , µ max q `|U δ s | 2 2 `µ|∇ γ ¨V δ s | 2 2 `F p|U δ s | 2 q
(2.32)

with c 5 and F some smooth non decreasing functions of their arguments, independent of δ. At this point, recalling the equivalence |V | X 0 " ph b pI `µT b qV, V q 2 stated by Proposition 2.3, we proved that

d dt pp1 ´δ∆qSBp1 ´δ∆qU δ s , U δ s q 2 ď F pp1 ´δ∆qSBp1 ´δ∆qU δ s , U δ s q 2 q
where F is a Lipschitz function which does not depend on δ. By Cauchy-Lipschitz theorem, there exists T ˚ą 0 such that the Cauchy problem

# d dt gptq " F pgptqq gp0q " |U p0q| H s
admits a unique solution g on a time interval r0; T ˚s. By Gronwall's lemma, one has for all t ă T ˚that pp1 ´δ∆qSBp1 ´δ∆qU δ s , U δ s q 2 ď gptq and consequently, using again the equivalence |V | X 0 " ph b pI `µT b qV, V q 2 stated by Proposition 2.3:

@0 ď t ď T ˚, |p1 ´δ∆qU δ s | 2 2 `µ|p1 ´δ∆q∇ γ ¨V δ s | 2 2 ď 1 C 1 ph min q gptq. (2.33)
This proves that the H s norm of U δ does not explode as t goes to T ˚, and then T δ ą T ˚, which give us a uniform time of existence for U δ independent of δ.

We can be more precise for all 0 ă T ă T ˚if one chooses

λ ě sup tPr0;T s c 5 p|U δ | W 1,8 , |h b | H s`2 , µ max qptq
then one has the following inequality, using estimate (2.32):

d dt pp1 ´δ∆qSBp1 ´δ∆qU δ s , U δ s q 2 ď λp|U δ s | 2 2 `µ|∇ γ ¨V δ s | 2 2 `F p|U δ s | 2 qq
and by integrating in time and using one last time the equivalence |V | X 0 " ph b pI `µT b qV, V q 2 stated by Proposition 2.3, one gets, for all 0 ď t ď T :

pµ|∇ γ ¨p1 ´δ∆q 1{2 V δ s | 2 2 `|p1 ´δ∆q 1{2 U δ s | 2 2 qpt 1 q ď λ C 1 ph min q ż t 0 F p|U δ s | 2 qpt 1 qdt 1 `λ C 1 ph min q ż t 0 p|p1 ´δ∆q 1{2 U δ s | 2 2 `µ|∇ γ ¨p1 ´δ∆q 1{2 V δ s | 2 2 qpt 1 qdt 1 `C2 p|h b | H t 0 q C 1 ph min q Ep0q
where we recall that the energy E is defined by (2.14). One can conclude by Gronwall Lemma that:

pµ|∇ γ ¨p1 ´δ∆q 1{2 V δ s | 2 2 `|p1 ´δ∆q 1{2 U δ s | 2 2 qpt 1 q ď λ C 1 ph min q ż t 0 e λ C 1 ph min q pt´t 1 q F p|U δ s | 2 qpt 1 qdt 1 `C2 p|h b | H t 0 q C 1 ph min q Ep0qe λ C 1 ph min q t .
(2.34)

Step 3-4

The inequality (2.33) and the equation (2.17) prove that pU δ q δ is bounded in the space L 8 pr0; T ˚s; H s pR d qq X W 1,8 pr0; T ˚s; H s´1 q. By compact embedding in H s 1 pR d q for all s 1 ă s, one has the strong convergence of pU δ q δ in Cpr0; T ˚s; H s 1 loc pR d qq to a function U . If one chooses s 1 close enough to s, H s 1 pR d q is embedded in C 1 pR d q and one can pass to the limit in the non-linear terms of (2.17). The linear terms do not raise any difficulty. It gives us a solution U of the problem. A short analysis as in [START_REF] Michael | Partial differential equations[END_REF] Proposition XVI.1.4 shows that U is in fact Cpr0; T ˚s; H s pR d qq. One can pass the limit δ goes to zero in the estimate (2.34) and recovers the estimate (2.13) stated in the Theorem. l

Modified equation

As explained in the Introduction, the Boussinesq-Peregrine equation (2.11) does not have the proper structure to apply the technique used by Bresch-Métivier in [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF]. One time derivative of ζ is not equal to sum of terms of one space derivative order of V .

In order to implement the technique used by [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF], we modify a bit the equation without changing the consistency with the Water-Waves equation. More precisely, the Boussinesq-Peregrine equation is consistent at order Opµ 2 q with the Water-Waves equation and therefore we look for a new equation consistent with the Boussinesq-Peregrine equation at a Opµ 2 q order. In this new equation, one space derivative of ζ should have the "same order" as one time derivative of V . For this purpose, we use the following formal consideration:

B t V " ´pI `µT b q ´1pεV ¨∇γ V `∇γ ζq " ´∇γ ζ `µR
where R is of order 0 or more in µ (recall that ε " Opµq in the Boussinesq-Peregrine regime). Therefore, one can add to the Boussinesq-Peregrine equation any expression of the form µA∇ γ ζ and the corresponding term µAB t V , where A is an operator independent on µ, without changing the consistency:

µAB t V " ´µA∇ γ ζ `µ2 R.
The operator A should respect the following constraints:

-the operator h b A should be symmetric, since multiplying the second equation by h b V should give the time derivative of a positive quantity, such as |V | 2 2 or |∇ γ ¨V | 2 2 ; -the whole system must conserve a certain symmetry and be of the form: BB t U `εU ¨∇γ U `LU " 0, where U " pζ, V q, B is symmetric and where L is an anti-symmetric operator;

-the operator I `µA should be elliptic and of order at least two; therefore, one would have

∇ γ ζ " ´pI `µAq ´1ppI `µT b `µAqB t V `εRq
and a good control for B t V would provide a good control for ∇ γ ζ.

The two first constraints ensure the local existence for the new equation, while the third one ensures the large existence time. A short study shows that one should consider the following operator for A:

A∇ γ W " ´∇γ p 1 h b ∇ γ ¨ph b W qq
and the following symmetrizer for the equation:

S " ˆI ´µ∇ γ ¨ph b ∇ γ ¨q 0 0 h b ˙, (3.35)
with an adapted change of unknown (see later) inspired by [START_REF] Bresch | Anelastic limits for euler-type systems[END_REF]. However, with the consideration B t V " ´∇γ ζ `Opµq, one can for free make the operator h b pI `µT b `µAq elliptic by addition of the operator µ∇ γK ∇ γK ¨, which provides a total control of a full derivative:

ph b pI `µT b `AqV , V q 2 " |V | 2 2 `µ|∇ γ ¨V | 2 2 `µ|∇ γK ¨V | 2 2 ,
where ∇ γK ¨V " p´γBV x `Bx V y q if d " 2, and ∇ γK ¨" 0 if d " 1. Remark that ∇ γK ¨∇γ ζ " 0, and since d " 1, 2, the operator ∇ γK ¨acts like the curl operator in dimension 3.

Remark 3.1 The operator I `µA " I ´µ∇ γ p 1 h b ∇ γ ¨ph b ¨qq is not elliptic, but it is invertible and its inverse gives precise control of the H 1 norm of the divergence, which is enough to control ∇ζ in H 1 norm, since its curl is zero:

∇ γ ζ " ´ph b pI `µAqq ´1ph b pI `µT b `µA ´∇γ ∇ γK ¨qB t V `εRq.
We are therefore led to consider the following equation:

$ & % B t ζ `∇γ ¨phV q " 0 " I `µpT b ´∇γ p 1 h b ∇ γ ¨ph b ¨qq ´1 h b ∇ γK ∇ γK ¨q‰ B t V `εV ¨∇γ V `pI ´µ∇ γ 1 h b ∇ γ ¨ph b ¨qq∇ γ ζ " 0
(3.36) However, the symmetrizer S defined by (3.35) does not symmetrize properly the equation (3.36): there is a residual term in the first equation ∇ γ ¨pεζV q which is not canceled in the time derivative of the energy pSU, U q 2 (with U " pζ, V q). To overcome this problem, we use the following change of variable inspired by Brech-Métivier:

q " 1 ε logp1 `εζ h b q. (3.37)
The following Proposition is the key point of this change of variable, and states a precise relation between q and ζ:

Proposition 3.2 Let N ą d{2 `1.
Let also h b P H N pR d q be such that there exists h min ą 0 such that:

@X P R d , h b pXq ě h min .
Then, for ε small enough, the quantity q defined by (3.37) is well defined. Moreover, one has:

q " Qpζqζ, with Qpζq ą 0. More precisely, for all α P N d , 1 ď |α| ď N , one has

B α q " Q 1 pζ, h b qζ α `Pα pζ, h b q with Q 1 pζ, h b q " ż t 0 h b h b `εtζ dt and P α pζ, h b q " ε ÿ βą0 Q β pζ, h b q `ÿ 0ăβďα R β pζ, h b qB β h b
with Q β , R β smooth functions of their arguments, and

|P α pζ, h b q| H 1 ď Cph min , |h b | H N q|B α ζ| 2 ,
where C is a smooth non decreasing function of its arguments.

Remark 3.3 The Proposition 3.2 states that at the leading order, a derivative of q is equal to a derivative of ζ up to a positive factor. Moreover, if one differentiates q only with respect to time, since B t h b " 0 this equality is true up to an ε factor.

Proof Just notice that the definition of q (3.37) implies

q " Qpζqζ, (3.38)
with

Qpζq "

ż 1 0 1 h b `εtζ dtζ.
For a given ζp0q, the quantity q is well-defined if there exists h min ą 0 such that the following condition is satisfied:

@X P R d , h 0 pXq ą h min .
Indeed, for ε small enough one has h b `εtζp0q ą 0 and this condition stay satisfied for t small enough. Moreover, one has Qpζq ą 0.

We now differentiate one time (3.38) (the notation B stands for any derivative of order one):

Bq "

ż t 0 h b h b `tεζ dtBq (3.39)
which gives the expression of Q 1 pζ, h b q given by the Proposition. The end of the proof is done by differentiating (3.39). l

Local existence for the modified Boussinesq-Peregrine equation

We prove in this Section a local existence result for the modified Boussinesq-Peregrine equation (3.36) introduced previously. We recall that we consider the change of unknown

q " 1 ε logp1 `εζ h b q. ( 3
$ ' ' ' ' & ' ' ' ' % h b `Bt Qpζqζ `εV ¨∇γ pQpζqζq ˘`∇ γ ¨ph b V q " 0 h b pI `µpT b ´∇γ p 1 h b ∇ γ ¨ph b ¨qq ´1 h b ∇ γK ∇ γK ¨qqB t V `hb εV ¨∇γ V `hb pI ´µ∇ γ 1 h b ∇ γ ¨ph b ¨qq∇ γ ζ " 0. (3.41)
For the sake of clarity, we will use the following notations:

B " pI `µT b ´µ∇ γ p 1 h b ∇ γ ¨ph b ¨qq ´µ 1 h b ∇ γK ∇ γK ¨q, A " pI ´µ∇ γ 1 h b ∇ γ ¨ph b ¨qq, q " Qpζqζ. (3.42)
For all N P N, we define the following space:

E N " tpV , ζq P H N pR d q d ˆHN pR d q | E N pV , ζq ă 8u (3.43)
where

E N pζ, V q " |ζ| H N `?µ|∇ γ ζ| H N `|V | H N `?µ|∇ γ V | H N . (3.44)
We denoted ∇ γ V the differential of V . The space E N endowed with the norm E N is a Banach space. We prove in this Section the following local existence result:

Theorem 3.4 Let N P N be such that N ą d{2 `2. Let h b P H N `1pR d q be such that there exists h min ą 0 such that @X P R d , h b pXq ě h min .

Let pζ 0 , V 0 q P E N . Then, there exists T ˚ą 0 and a unique solution pV , ζq in Cpr0; T ˚r; E N q to the equation (3.41). Moreover, one has:

@T ă T ˚, @λ ě sup tPr0;T s Cp 1 h b , |h b | H N `1 , |ζ, V | W 2,8 q, E N ptq ď E N p0qe λt . (3.45)
Even if at first sight the proof seems to follow the lines of the proof of local existence for a standard quasilinear hyperbolic system, as done for the Boussinesq-Peregrine equation in Section 2, we give here a detailed proof. Indeed, the energy (3.44) is defined in terms of ζ, V , while the equation (3.41) is expressed in terms of unknowns ζ, q, V , and the dependence of q with respect to ζ is not trivial. This leads to technical complications that must be handled carefully.

Remark 3.5 In Theorem 2.1 which states the local existence for the initial Boussinesq-Peregrine equation, we used fractional order Sobolev spaces to define the energy of solutions, while we use here integer order Sobolev spaces. The reason is to have a coherent notation with the long time existence Theorem of Section 3.2, which can only be proved with an integer number of space derivatives, due to the method used.

Proof As usual, the local existence follows the steps used for the quasilinear hyperbolic systems:

-Step 1 : We solve a smoothed equation involving a mollifier p1 ´δ∆q.

-Step 2 : We prove that the existence time of the solution of the mollified equation does not depend on δ, and the uniform bound in H s norm of this solution.

-Step 3 : We pass to the limit δ goes to zero in the mollified equation to get a solution of the equation (3.41).

-Step 4 : We recover regularity for the solution.

Step 1 One sets δ P R be such that 0 ă δ ă 1, and considers the following equation:

$ & % pI ´δ∆q 2 B t q δ `εV ¨∇γ q δ `1 h b ∇ γ ¨ph b V q " 0 pI ´δ∆qh b BpI ´δ∆qB t V `εh b V ¨∇γ V `hb A∇ γ ζ δ " 0. (3.46)
We first solve (3.46) in the Banach space E N defined by (3.43). To this purpose, we note that the linear applications pI `δ∆q ´1 and pI `δ∆q ´1ph b Bq ´1pI `δ∆q ´1 are respectively continuous from H N ´1 to H N , and from H N ´3 to H N , using the following Proposition:

Proposition 3.6 Let N ě 0, t 0 ą d{2, and β ď 1. Let b P H N `1pR d q be such that there exists h min ą 0 such that h b " 1 ´βb ě h min . The operator

h b B : H N `2pR d q d ÝÑ H N pR d q d
is one-to-one and onto. One has, for all V P H 1 pR d q d :

C 1 ph min q|V | 2 H 1 ď ph b BV, V q 2 ď C 2 p|h b | H t 0 `1 q|V | 2 H 1 ,
where the C i are non decreasing functions of their arguments. Moreover, one has, if b P H 1`N _t0 pR d q and for all f P H N pR d q d :

|ph b Bq ´1f | H N `?µ|ph b Bq ´1f | H N `1 `µ|ph b Bq ´1f | H N `2 ď Cp 1 h b , |h b | H 1`N _t 0 q|f | H N
where C is a non decreasing continuous function of its arguments.

The proof is an easy adaptation of the proof of the invertibility of the operator I `µT b stated by Proposition 2.3 (see [START_REF] Lannes | The water waves problem[END_REF] Chapter 5 for a full proof). Just note that if W P L 2 pR d q d , pI `µT b q ´1W is only controlled with its divergence in L 2 pR d q norm, while ph b Bq ´1W is controlled in a full H 1 norm, due to the presence of the orthogonal gradient ∇ γK in the operator B (see definition (3.42)). Therefore, using Cauchy-Lipschitz theorem, there exists T δ ą 0 and a unique solution pζ δ , V δ q P Cpr0; T δ r, E N q to the equation (3.46) (just replace ζ δ by h b ε pe εq δ ´1q in the second equation, to have an ODE in terms of the unknowns pq δ , V δ q). Moreover, T δ ă 8 if and only if

lim tÑT δ |pζ δ , V δ qptq| E N " `8.
Step 2

We now want to bound uniformly with respect to δ the energy E N defined by (3.44) of the unknowns. We use the following notation: for all α P N d , |α| ď N , for all distribution f ,

f pαq " B α f. ( 3
$ ' & ' % pI ´δ∆q 2 B t q δ pαq `εV δ ¨∇γ q δ pαq `1 h b ∇ γ ¨ph b V δ pαq q " R α 1 pI ´δ∆qh b BpI ´δ∆qB t V δ pαq `εh b V δ ¨∇γ V δ pαq `hb A∇ γ ζ δ pαq " R α 2 (3.48) where R α 1 " ´ε ÿ 0ăβďα B β V δ ¨∇γ B α´β q δ ´ÿ 0ăβ`νďα B β p 1 h b q∇ γ ¨pB ν ph b qB α´β´ν V δ q and R α 2 " pI ´δ∆qrh b B, B α spI ´δ∆qB t V δ `εrh b V δ ¨∇γ , B α sV δ `rh b A, B α s∇ γ ζ δ .
As explained at the beginning of this Section, the system (3.48) can be made symmetric by multiplying it by the following operator: ˆhb ´µ∇ γ ph b ∇ γ q 0 0 I ˙.

Note that according to Proposition 3.6, one has

|ζ δ | 2 H N `µ|∇ γ ζ δ | 2 H N `ph b BV, V q H N " E N pζ δ , V q
and thus it is equivalent to control the E N norm of the unknown and the quantity

|ζ δ | 2 H N `µ|∇ γ ζ δ | 2 H N ph b BV, V q H N .
Following these considerations, one takes the L 2 scalar product of the first equation of (3.48) with ph b ´µ∇ γ ¨ph b ∇ γ ¨qqζ δ pαq and the scalar product of the second equation with V δ pαq . We obtain the following equality:

pT q `pV q `pZq " pR 1 α , h b ζ δ pαq ´µ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 `pR 2 α , V δ pαq q 2 , (3.49)
where the time derivatives are

pT q " ph b BpI ´δ∆qB t V δ pαq , pI ´δ∆qV δ pαq q 2 `ppI ´δ∆qB t q δ pαq , h b ζ δ pαq ´µ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 , (3.50)
the vanishing terms are

pV q " p 1 h b ∇ γ ¨ph b V δ pαq q, h b ζ δ pαq ´µ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 `ph b A∇ γ ζ δ pαq , V δ pαq q 2 , (3.51)
and the terms of order zero to the contribution of the energy estimate:

pZq " εph b V δ ¨∇γ V δ pαq , V δ pαq q 2 `εpV ¨∇γ q δ pαq , h b ζ δ pαq ´µ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 . (3.52)
-Control of the vanishing terms pV q

All has been made to conserve a certain symmetry in the equation, which is crucial here. Using the definition of A in the expression (3.51), one has

pV q " p∇ γ ¨ph b V δ pαq q, ζ δ pαq q 2 `ph b ∇ γ ζ δ pαq , V δ pαq q 2 ´µp 1 h b ∇ γ ¨ph b V δ pαq q, ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 ´µp∇ γ p 1 h b ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 , h b V δ pαq q 2 .
By integrating by parts, the first two terms cancel one another, and the last two terms cancel one another. Therefore, pV q actually vanishes.

-Control of the terms of order zero pZq We start to control the easiest term of (3.52), which is the first one, by a classical symmetry trick:

εph b V δ ¨∇γ V δ pαq , V δ pαq q 2 " ε d ÿ j"1 pV δ j B j V δ pαq , V δ pαq q 2 " ´ε d ÿ j"1 pV δ pαq , pB j V j qV δ pαq q 2 ´ε d ÿ j"1 pV δ pαq , V δ j B j V δ pαq q 2
by integrating by parts, and therefore

εph b V δ ¨∇γ V δ pαq , V δ pαq q 2 " ´1 2
ε d ÿ j"1 pV δ pαq , pB j V δ j qV δ pαq q 2 .
Using Cauchy-Schwarz inequality, one gets:

|εph b V δ ¨∇γ V δ pαq , V δ pαq q 2 | ď ε 1 2 |V δ | W 1,8 |V δ pαq | 2 2 . (3.53)
Note that the symmetry of this term is crucial here. For the other terms of pZq given in (3.52), the symmetry is less clear, since q δ pαq is not exactly ζ δ pαq . We use the Proposition 3.2 to compute:

εpV δ ¨∇γ q δ pαq , h b ζ δ pαq q 2 " εpV δ ¨∇γ pQ 1 pζ δ , h b qζ δ pαq q, h b ζ δ pαq q 2 `ε2 pV δ ¨∇γ pP α pζ δ , h b qq, h b ζ δ pαq q 2 . (3.54)
The second term of the right hand side of (3.54) is bounded using Cauchy-Schwarz inequality and Proposition 3.2 by

ε 2 Cp 1 h b , |h b | H N q|ζ δ pαq | 2 2 |V δ | 8 |h b | 8 ,
where C is a smooth non decreasing function of its arguments. The first term of the right hand side of (3.54) is bounded using the same symmetry trick as for the first term of pV q:

εpV δ ¨∇γ pQ 1 pζ δ , h b qζ δ pαq q, h b ζ δ pαq q 2 " ε d ÿ j"1 pV δ j B j pQ 1 pζ δ , h b qζ δ pαq q, h b ζ δ pαq q 2 " ε d ÿ j"1 pV δ j pB j Q 1 pζ δ , h b qqζ δ pαq , h b ζ δ pαq q 2 ´ε d ÿ j"1 pζ δ pαq , B j pQ 1 pζ δ , h b qζ δ pαq qV δ j h b q 2 ´ε d ÿ j"1 pζ δ pαq , B j ph b V δ j qQ 1 pζ δ , h b qζ δ pαq q 2
by integrating by parts. Therefore, using Cauchy-Schwarz's inequality, we get the bound:

|εpV δ ¨∇γ pQ 1 ζ δ pαq q, h b ζ δ pαq q 2 | ď ε 2 |Q 1 pζ δ , h b q| W 1,8 |h b | W 1,8 |V δ | W 1,8 |ζ δ pαq | 2
and finally, using the definition of Q 1 given by (3.39):

εpV δ ¨∇γ q δ pαq , h b ζ δ pαq q 2 ď εCp 1 h b , |V δ | W 1,8 , |ζ δ | W 1,8 , |h b | H N q|ζ δ pαq | 2 2 . (3.55)
The third term of the right hand side of the symmetric term (3.52) is controlled with a similar technique:

µp∇ γ pV δ ¨∇γ q δ pαq q, h b ∇ γ ζ δ pαq q 2 " µp∇ γ pV δ ¨∇γ pQ 1 pζ δ , h b qζ δ pαq qq, h b ∇ γ ζ δ pαq q 2 `µp∇ γ pV δ ¨∇γ pεP α pζ δ , h b qqq, h b ∇ γ ζ δ pαq q 2 . (3.56)
We recall the identity:

∇ γ pA ¨Bq " pA ¨∇γ qB `pB ¨∇γ qA `B∇ γK ¨A `A∇ γK ¨B.

The first term of the right hand side of (3.56) can be expanded using this last identity:

µp∇ γ pV δ ¨∇γ q δ pαq q, h b ∇ γ ζ δ pαq q 2 " µpV δ ¨∇γ ∇ γ pQ 1 pζ δ , h b qζ δ pαq q, h b ∇ γ ζ δ pαq q 2 `µpV δ ∇ γK ¨∇γ pQ 1 pζ δ , h b qζ δ pαq q, h b ∇ γ ζ δ pαq q 2 `µp∇ γ pQ 1 pζ δ , h b qζ δ pαq q ¨∇γ V δ , h b ∇ γ ζ δ pαq q 2 .
(3.57)

The first term of (3.57) is a symmetric term, controlled by the same technique as before. The second term vanishes, and the last one with the last term of the right hand side of (3.56) are easily controlled, and one gets:

|µp∇ γ pV δ ¨∇γ q δ pαq q, h b ∇ γ ζ δ pαq q 2 | ď µCp|h b | H N `1 , |V δ | W 2,8 , |ζ δ | W 2,8 , 1 h b qp|∇ γ ζ δ pαq | 2 `|ζ δ pαq | 2 q|∇ γ ζ δ pαq | 2 .
(3.58) To conclude, putting together (3.53),(3.55) and (3.58), we proved that

|pZq| ď εCp|h b | H N `1 , |V δ | W 2,8 , |ζ δ | W 2,8 , 1 h b qE N pζ δ , V δ q. (3.59)
-Control of the time derivatives pT q The terms of pT q involve time derivatives, and should be, up to terms controlled by the energy E N , the time derivatives of the energy E N . The first term of (3.50) is already symmetric, using the symmetry of h b B (which is crucial here):

ph b BpI ´δ∆qB t V δ pαq , pI ´δ∆qV δ pαq q 2 " B t 1 2 ph b BpI ´δ∆qV δ pαq , pI ´δ∆qV δ pαq q 2 . (3.60)
For the second term of (3.50), we use again Proposition 3.2 to write:

ppI ´δ∆q 2 B t q δ pαq , h b ζ δ pαq q 2 " ppI ´δ∆q 2 B t pQ 1 ζ δ pαq q, h b ζ δ pαq q 2 `ppI ´δ∆q 2 B t pεP α ζ δ q, h b ζ δ pαq q 2 . (3.61)
For the first term of the right hand side of (3.61), one computes:

ppI ´δ∆q 2 B t pQ 1 ζ δ pαq q, h b ζ δ pαq q 2 " ppI ´δ∆q 2 B t pQ 1 qζ δ pαq , h b ζ δ pαq q 2 `ppI ´δ∆q 2 Q 1 B t ζ δ pαq , h b ζ δ pαq q 2 " ppI ´δ∆q 2 B t pQ 1 qζ δ pαq , h b ζ δ pαq q 2 `prQ 1 , ´δ∆sB t ζ δ pαq , pI ´δ∆qh b ζ δ pαq q 2 `pQ 1 pI ´δ∆qB t ζ δ pαq , r´δ∆, h b sζ δ pαq q 2 `Bt 1 2 pQ 1 pI ´δ∆qζ δ pαq , h b pI ´δ∆qζ δ pαq q 2 .
(3.62)

All these computations are made to obtain the time derivative of a symmetric term with respect to ζ δ pαq . The first term of the right hand side of (3.62) is easily controlled by

|ppI ´δ∆q 2 B t pQ 1 qζ δ pαq , h b ζ δ pαq q 2 | ď |B t Q 1 | W 1,8 |pI ´δ∆qζ δ pαq | 2 2 |h b | W 2,8 .
In order to control the second term of the right hand side of (3.62), we replace B t ζ δ pαq by its expression given by the equation:

B t ζ δ pαq " ´Bα p∇ γ ¨ph δ V δ qq
and we notice that

|δpI ´δ∆q ´2r∆, Q 1 sB α ∇ γ ¨u| 2 ď Cp|Q 1 | W 2,8 q|B α u| 2
for all u in H N . Therefore, one gets:

ppI ´δ∆q 2 B t pQ 1 ζ δ pαq q, h b ζ δ pαq q 2 " B t 1 2 pQ 1 pI ´δ∆qζ δ pαq , h b pI ´δ∆qζ δ pαq q 2 `R (3.63) with |R| ď Cp|h b | H N , 1 h b , |ζ δ | W 2,8 , |V δ | W 2,8 qp|pI ´δ∆qζ δ pαq | 2 `|V δ pαq | 2 q|pI ´δ∆qζ δ pαq | 2 .
The same technique can be used for the control of the second term of (3.61) and one gets finally, combining (3.60) and (3.63):

pT q " B t 1 2 ph b BpI ´δ∆qV δ pαq , pI ´δ∆qV δ pαq q 2 `Bt 1 2 pQ 1 pI ´δ∆qζ δ pαq , h b pI ´δ∆qζ δ pαq q 2 `R (3.64) with |R| ď µCp|h b | H N , 1 h b , |ζ δ | W 2,8 , |V δ | W 2,8 qE N ppI ´δ∆qζ δ , pI ´δ∆qV δ q. (3.65)
-Control of the residual terms We now control the terms involving the residuals that appear in (3.49). One has:

R α 1 " ´ε ÿ 0ăβďα B β V δ ¨∇γ B α´β q δ ´ÿ 0ăβ`νďα B β p 1 h b q∇ γ ¨pB ν ph b qB α´β´ν V δ q
and thus one has, using a Kato-Ponce type estimate (of the form of Proposition A.1):

|R α 1 | 2 `?µ|∇ γ R α 1 | 2 ď Cp 1 h b , |h b | H N `1 q|pζ δ , V δ q| W 2,8 p|ζ δ pαq | 2 `?µ|∇ γ ζ δ pαq | 2 `|V δ pαq | 2 `?µ|∇ γ V δ pαq | 2 q.
(3.66)

It is very important to have ∇ γ ¨ph b V δ pαq q instead of ∇ γ ¨phV δ pαq q in the equation (3.48), because the term ∇ γ ¨pεζ δ pαq V δ pαq q would not be properly symmetrized and thus would not be controlled by the energy. One has easily, integrating by parts and using Cauchy-Schwarz inequality:

|pR α 1 , h b ζ δ pαq ´µ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 | ď |R α 1 | 2 |h b | 8 |ζ δ pαq | 2 `µ|∇ γ R α 1 | 2 |h b | 8 |∇ γ ζ δ pαq | 2 (3.67)
and thus, using (3.66), one gets:

|pR α 1 , h b ζ δ pαq ´µ∇ γ ¨ph b ∇ γ ζ δ pαq qq 2 | ď Cp 1 h b , |h b | H N `1 , |pζ δ , V δ q| W 2,8 qpE N q 2 . (3.68) Recall that R α 2 " pI ´δ∆qrh b B, B α spI ´δ∆qB t V δ `εrh b V δ ¨∇γ , B α sV δ `rh b A, B α s∇ γ ζ δ (3.69)
To control the first term of (3.69), as usual one replaces pI ´δ∆qB t V δ by its expression given by the equation (3.48), and uses the definition of T b given by (2.12):

pI ´δ∆qrh b B, B α spI ´δ∆qB t V δ " ´pI ´δ∆qrh b µT b , B α sph b Bq ´1pI ´δ∆q ´1`ε h b V δ ¨∇γ V δ `hb A∇ γ ζ δ ˘. (3.70) One has: @k ě 2, µ|ph b Bq ´1u| H k ď Cp 1 h b , |h b | H N `1 q|u| H k´2 (3.71)
using Proposition 3.6, for C a smooth non decreasing function of its arguments. One has also:

@k ě 0, |pI ´δ∆qu| H k ď Cp|u| H k `δ|u| H k`2 q (3.72)
with C independent on δ, and:

@k ě 0, 1 δ |pI ´δ∆q ´1u| H k`2 `|pI ´δ∆q ´1u| H k ď |u| H k . (3.73)
Using the definition of T b (see (2.12)), one has:

@k ě 0, µ|rh b T b , B α su| ď µCp|h b | H N `1 q|u| H k`α`1 . ( 3
h b V δ ¨∇γ V δ q| 2 ď Cp 1 h b , |h b | H N `1 q|V δ pαq | 2 .
One has to be more careful for the second term of the right hand side of (3.1), because the expression A∇ γ ζ δ is of order 2 in µ∇ γ ζ δ . One writes:

|ppI ´δ∆qrh b µT b , B α sph b Bq ´1pI ´δ∆q ´1ph b A∇ γ ζ δ q, V δ pαq q 2 | ď ? µ|ppI ´δ∆qrh b µT b , B α sph b Bq ´1pI ´δ∆q ´1ph b A∇ γ ζ δ q| H ´1 ? µ|V δ pαq | H 1
and we use the same controls (3.72), (3.73), (3.71), (3.1) as before. Finally, one gets:

|pR α 2 , V δ pαq q| ď Cp 1 h b , |h b | H N `1 qp|V δ pαq | 2 2 `µ|∇ γ V δ pαq | 2 |∇ γ ζ δ pαq | 2 q. ( 3
q 2 ď Cp 1 h b , |h b | H N `1 , |ζ δ , V δ | W 2,8
q ˆEN ppI ´δ∆qζ δ , pI ´δ∆qV δ q.

We recall that using Proposition 3.6, one has

ph b BV, V q 2 " |V | 2 2 `µ|∇ γ V | 2 2
and using Proposition 3.2, one has |Q 1 | 8 ě Cp 1 h b q. Therefore, we obtained:

B t E N ppI ´δ∆qζ δ , pI ´δ∆qV δ q ď Cp 1 h b , |h b | H N `1 , |ζ δ , V δ | W 2,8 qE N ppI ´δ∆qζ δ , pI ´δ∆qV δ q (3.76)
where C is a non decreasing continuous function of its arguments, independent on δ. Therefore, using Gronwall's Lemma, T δ does not depends on δ.

Step 3-4

The rest of the proof is exactly the same as for the local existence of the standard Boussinesq-Peregrine equation, and one gets a solution to (3.46) on a time interval r0; T ˚r. One gets however from (3.76) that:

@T ă T ˚, @λ ě sup tPr0;T s Cp 1 h b , |h b | H N `1 , |ζ, V | W 2,8 q, E N ptq ď E N p0qe λt .

Long time existence in dimension 1 for the modified Boussinesq-Peregrine equation

We now make the scaling t 1 " εt on the equation (3.41), and we obtain the equation (we get rid of the "primes" in the notation t 1 for the sake of clarity):

$ ' & ' % B t q `V ¨∇γ q `1 ε 1 h b ∇ γ ¨ph b V q " 0 h b BB t V `hb V ¨∇γ V `1 ε h b A∇ γ ζ " 0. (3.77)
This change of variable is not necessary on a mathematical point of view, but it allows to highlight the singular terms that must be canceled in the energy estimates in order to prove the result, which are the large terms of size 1 ε . Moreover, it allows the equation (3.77) to be seen as a singular perturbation problem. Note that a time existence of size 1 ε for the equation (3.41) is equivalent to a time existence independent on ε for (3.77).

We recall that for all N P N, we define the following space:

E N " tpV , ζq P H N pR d q d ˆHN pR d q | E N pV , ζq ă 8u where E N pζ, V q " |ζ| H N `?µ|∇ γ ζ| H N `|V | H N `?µ|∇ γ V | H N .
We prove in this section the following result:

Theorem 3.7 Let d " 1.
Let N P N be such that N ą d{2 `2. Let h b P H N `1pR d q be such that there exists h min ą 0 such that @X P R d , h b pXq ě h min .

Let pζ 0 , V 0 q P E N . Then, there exists T ą 0 and a unique solution pV , ζq in Cpr0; T r; E N q to the equation (3.77), with

T " C 1 pE N pζ 0 , V 0 q, 1 h b , |h b | H N `1 q,
where C 1 is a non decreasing continuous function of its arguments.

In particular, the time of existence does not depend on ε, µ.

Remark 3.8 It is very important to note that d " 1 here. In d " 2, there is an extra difficulty due to the need of a good estimate for ∇ γK ¨V . However, since this is the only difficulty that could prevent a similar result in dimension d " 2 to hold, we keep the notations of the multidimensional equation, and we specifically highlight at the end of the proof the difficulty that one must overcome to prove the result in dimension 2.

Let us consider pV , ζq the unique solution of (3.77) given by Theorem 3.4 on a time interval r0; T ε s. We set

K " sup tPr0;T ε s E N pζ, V q.
We use the notation u k " pεB t q k u for all distribution u (thus u k corresponds to the time derivative of u in the original time variables).

The idea of the proof is to obtain a "good" energy estimate of the form

Eptq ď CpKqpt `εq `C0 ,
where C is non decreasing and smooth, and where C 0 only depends on the initial data. Such estimate would allow us to get by a continuity argument a time existence uniform with respect to ε. There are two main ideas in the proof:

-The system is still symmetric with respect to singular terms if we differentiate it with respect to time. It allows us to get the "good estimate" for the time derivatives ζ k , V k .

-Using the equation, one can control the space derivatives by the time derivatives, and recover the "good estimate" for the full energy E N of the solutions.

The following Proposition states that the time derivatives of the solutions pV , ζq have the same regularity as the space derivatives: Proposition 3.9 One has, for all 0 ď k ď N ,

|pV k , ζ k q| H N ´k `?µ|pV k , ζ k q| H N ´k`1 ď CpKq,
where C is a smooth, non decreasing function of its argument.

Proof For k " 0, it is clear. Suppose it is true for k ě 0. One commutes pεB t q k with the equation (3.77). One gets, since B t h b " 0:

# q k`1 `ε ř k j"0 V j ¨∇γ q k´j `∇γ ¨ph b V k q " 0 V k`1 " ´ph b Bq ´1pεh b ř k j"0 V j ¨∇γ V k´j `hb A∇ γ ζ k q.
We only prove the most difficult estimate which is the following, in order to prove that the induction hypothesis is true at rank k `1:

|ph b Bq ´1ph b A∇ γ ζ k q| H N ´k´1 ď CpKq.
We recall that

A " I ´µ∇ γ 1 h b ∇ γ ¨ph b ¨q
and therefore, using Proposition (3.6):

|ph b Bq ´1ph b A∇ γ ζ k q| H N ´k´1 ď |ph b Bq ´1ph b ∇ γ ζ k q| H N ´k´1 `µ|ph b Bq ´1p∇ γ 1 h b ∇ γ ¨ph b ∇ γ ζ k qq| H N ´k´1 ď Cp 1 h b , |h b | H N `1 q|ζ k | H N ´k
and one gets the desired control by using the induction hypothesis. The other controls are done similarly, using Proposition 3.6, and the relation between q k and ζ k given by Proposition 3.2 and Remark 3.

l

The key point of the proof of Theorem 3.7 is the following Lemma, which states a "good estimate" for the unknowns:

Lemma 3.10 One has E N pζ, V q ď CpKqpt `εq `C0
where C is a non decreasing function of its arguments, and C 0 is a constant which only depends on the initial data.

Proof There are two ideas in the proof of this lemma:

-the time derivatives of the unknowns satisfy a system which is still symmetric with respect to singular terms of size 1 ε ; -the space derivatives are related to time derivatives by the equation.

The unknowns pζ k , V k q satisfy the following equation:

$ ' & ' % B t q k `V ¨∇γ q k `1 ε 1 h b ∇ γ ¨ph b V k q " R 1 k h b BB t V k `hb V ¨∇γ V k `1 ε h b A∇ γ ζ k " R 2 k (3.78) where R 1 k " rV , pεB t q k sq, R 2
k " rV ¨∇γ , pεB t q k sV .

(3.79)

The symmetry with respect to large terms of size 1 ε is conserved, which allows to get the following result: Lemma 3.11 One has, for all 0 ď k ď N ,

E 0 pζ k , V k q ď CpKqt `C0 .
Proof The equation (3.78) is still symmetric with respect to large terms of size 1 ε . More precisely, if one multiplies the first equation by h b ζ k ´µ∇ γ ¨ph b ∇ γ ζ k q, one finds exactly as in the proof of Theorem 3.4 an expression of the form: pT q `pV q `pZq " pR

1 k , h b ζ k ´µ∇ γ ¨ph b ∇ γ ζ k qq 2 `pR 2 k , V k q 2 with
exactly the same terms for pT q, pV q and pZq as in (3.50), (3.52) with pζ pαq , V δ pαq q replaced by pζ k , V k q. The vanishing terms are exactly ones of size 1 ε and the others are controlled exactly with the same techniques, using Proposition 3.9 for the regularity of the time derivatives. l Now, we recover the "good estimate" of Lemma 3.10 for the space derivatives of the unknowns, using the equation.

Lemma 3.12 One has, for all 0 ď k ď N ,

E N ´kpζ k , V k q ď CpKqpt `εq `C0 .
Proof We prove it by backward finite induction on k. For k " N , it is Lemma 3.11. Suppose it is true for k `1 with k ď N ´1. Let us prove it is true for k.

∇ γ ζ k " ´ph b Aq ´1ph b BV k`1 `εh b V ¨∇γ V k ´εR 2 k q. (3.80)
Recall that the operator A is given by

A " pI ´µ∇ γ 1 h b ∇ γ ¨ph b ¨qq.
We also recall that we defined |f | X N for f P L 2 pR d q d by:

|f | 2 X N " |f | 2 H N `µ|∇ γ ¨f | 2 H N .
We used the following Proposition that states the invertibility of h b A to derive the equality (3.80):

Proposition 3.13 Let N P N and let h b P H N pR d q be such that there exists h min ą 0 such that @X P R d , h b pXq ě h min .

We set, for all f P L 2 pR d q :

|f | 2 X N " |f | 2 H N `µ|∇ γ ¨f | 2 H N . The operator h b A is invertible on H N pR d q d .
Moreover, the following estimates stand.

( 1) For all f P H

N pR d q d , |ph b Aq ´1f | X N ď Cp 1 h b , |h b | H N `1 q|f | H N .
( 2) For all g P H

N pR d q, ? µ|ph b Aq ´1∇ γ g| X N ď Cp 1 h b , |h b | H N `1 q|g| H N .
We postpone the proof of Proposition 3.13 to Appendix B for the sake of clarity. Now, in order to use the relation (3.80), one takes the H N ´k´1 scalar product of (3.80) with h b pI ´µ∇ γ ∇ γ ¨q∇ γ ζ, and gets, using the notations of Proposition 3.13 :

|∇ γ ζ k | 2 X N ´k´1 " ´ph b pI ´µ∇ γ ∇ γ ¨q∇ γ ζ k , ph b Aq ´1ph b BV k`1 `εh b V ¨∇γ V k ´εR 2 k qq H N ´k´1 . (3.81)
Now, one has by definition of R 2 k given by (3.79) :

h b V ¨∇γ V k ´R2 k " pε t q k pV ¨∇γ V q
and thus this term is sum of terms of the form

V l ¨∇γ V k´l ,
with 0 ď l ď k and therefore one has, using Proposition 3.13:

|ph b Aq ´1ph b V ¨∇γ V k ´R2 k q| X N ´k´1 ď Cp 1 h min , |h b | H N `1 q|h b V ¨∇γ V k ´R2 k | H N ´k´1 ď CpKq. (3.82)
We now focus on the control of

pph b Aq ´1ph b BqV k`1 , h b pI ´µ∇ γ ∇ γ ¨q∇ γ ζ k q H N ´k´1 . Recall that B " pI `µT b ´µ∇ γ p 1 h b ∇ γ ¨ph b ¨qq ´µ 1 h b ∇ γK ∇ γK ¨q, A " pI ´µ∇ γ 1 h b ∇ γ ¨ph b ¨qq.
Lemma 3.14 One has, for all V, W P H k`1 pR d q d , all 0 ď k ď N :

pph b Aq ´1h b BV, h b ∇ γ W q H k ď Cp 1 h b , |h b | H N `1 q ? µ|∇ γ V | H k |∇ γ W | H k .
Remark 3.15 This Lemma states that even if ph b Aq is not elliptic (it is essentially I ´µ∇∇ γ ¨with variables coefficients), its inverse allows to recover a full derivative if it is applied to a gradient. The quantity h b B is essentially composed of gradients, except for the term ∇ γK ∇ γK ¨which vanishes in any scalar product with a gradient.

Proof We only give the control of the most difficult terms of the quantity to be controlled, which are:

µpph b Aq ´1p∇ γ ph 3 b ∇ γ ¨V qq, h b ∇ γ W q H k , µpph b Aq ´1p∇ γK ∇ γK ¨V q, h b ∇ γ W q H k (3.83)
(the other terms from h b B are controlled even more easily by similar techniques). For the first term of (3.83), one computes:

µpph b Aq ´1p∇ γ ph 3 b ∇ γ ¨V qq, h b ∇ γ W q H k ď µ|ph b Aq ´1∇ γ ph 3 b ∇ γ ¨V q| H k |h b ∇ γ W | H k ď ? µCp 1 h b , |h b | H N `1 q|h 3 b ∇ γ ¨V | H k |∇ γ W | H k
where we used Proposition (3.13) to derive the last inequality. Finally, one gets:

µpph b Aq ´1p∇ γ ph 3 b ∇ γ ¨V qq, h b ∇ γ W q H k ď Cp 1 h b , |h b | H N `1 q ? µ|∇ γ V | H k |∇ γ W | H k . (3.84)
For the second term of (3.83), one computes, integrating by parts:

µpph b Aq ´1p∇ γK ∇ γK ¨V q, h b ∇ γ W q H k " ´µp∇ γ ¨hb ph b Aq ´1p∇ γK ∇ γK ¨V q, W q H k `µpΛ k´1 ph b Aq ´1p∇ γK ∇ γK ¨V q, Λrh b , Λ k s∇ γ W q 2 `µprh b , Λ k sph b Aq ´1p∇ γK ∇ γK ¨V q, Λ k ∇ γ W q 2 , (3.85)
where we recall that Λ " p1 `|D γ | 2 q 1{2 . One has to notice that for f P H 1 pR d q, u " ph b Aq ´1ph b ∇ γ f q is a term of the form ∇ γ g, since u " ∇ γ p 1 h b ∇ γ ¨ph b uqq `∇γ f , using the definition of A given by (3.42). Therefore, ∇ γK ¨ph b Aq ´1ph b ∇ γ f q " 0 for all f , and by duality ∇ γ ¨ph b ph b Aq ´1∇ γK wq " 0 for all w. The first term of the rhs of term (3.85) is therefore zero.

For the second term of the rhs of (3.85), one easily proves that, for all f P H k pR d q, using the Kato-Ponce estimate of Proposition A.1:

|Λrh b , Λ k sf | 2 ď Cp|h b | H N `1 q|f | H k ,
and thus the second term of the rhs of (3.85) is bounded by

µCp|h b | H N `1 q|ph b Aq ´1p∇ γK ∇ γK ¨V q| H k´1 |∇ γ W | H k
and using Proposition (3.13), one gets the bound

µ|pΛ k´1 ph b Aq ´1p∇ γK ∇ γK ¨V q, Λrh b , Λ k s∇ γ ∇ γ W q 2 | ď µCp 1 h b , |h b | H N `1 q|∇ γ V | H k |∇ γ W | H k . (3.86)
The third term of (3.85) is controlled similarly with the same bound as (3.86). Putting together (3.84) and (3.86), one gets the Lemma. l

We can now apply Lemma 3.84 to get immediately (note that |∇ γ ¨∇ζ| 2 " |∇ γ ∇ γ ζ| 2 ):

|pph b Aq ´1ph b BqV k`1 , h b pI ´µ∇ γ ∇ γ ¨q∇ γ ζ k q H N ´k´1 | ď Cp 1 h b , |h b | H N `1 qp ? µ|∇ γ V k`1 | H N ´k´1 |∇ γ ζ k | H N ´k´1 `?µ|∇ γ V k`1 | H N ´k´1 ? µ|∇ γ ¨∇γ ζ k | H N ´k´1 q ď Cp 1 h b , |h b | H N `1 qp CpKqpt `εq `C0 q|∇ γ ζ k | X N ´k´1 (3.87)
using the notations of Proposition (3.13), and using the induction hypothesis. Putting (3.82) and (3.87) into (3.80), one gets:

|∇ γ ζ k | 2 X N ´k´1 ď CpKqε `pCpKqpt `εq `C0 q|∇ γ ζ k | X N ´k´1 .
By noticing that, for all u smooth enough: ∇ γ ¨ph b V k q " ´hb pq k`1 `εV ¨∇γ q k q and using induction hypothesis to control q k`1 , one easily gets

|u| X N ´k ď |u| X N ´k´1 `|∇ γ u| X N
|∇ γ ¨ph b V k q| H N ´k´1 `?µ|∇ γ ∇ γ ¨ph b V k q| H N ´k´1 ď CpKqpt `εq `C0 .
If d " 1, then we controlled a full derivative of V k , and the induction hypothesis is true for k. l Remark 3.16 If d " 2, of course, it is not sufficient to control only ∇ γ ¨V k to recover a good control for V k in norm H N ´k. One should look after a good control for ∇ γK ¨V k . This is obtained by taking ∇ γK ¨of the second equation of (3.78):

B t p∇ γK ¨V k q`B t µ∇ γK ¨∇γK p∇ γK ¨V k q`pV ¨∇γ q∇ γK ¨V k " ∇ γK ¨pR k 2 q`r∇ γK ¨, V sV k ´∇γK ¨µT b B t V k (3.88)

However, it is difficult to control ∇ γK ¨µT b B t V k . Indeed, ∇ γK ¨Tb is no longer symmetric, which means that multiplying the equation (3.88) by ∇ γK ¨V creates a term of the form p∇ γK ¨Tb B t V , ∇ γK ¨V q 2 which is not the time derivative of a positive quantity.

The key Lemma 3.10 is this latter result with k " 0. We now end the proof of Theorem 3.7. We set ε 0 " C 0 2Cp2C 0 q , T 0 " C 0 2Cp2C 0 q .

Let fix an ε ą 0 such that ε ă ε 0 . There exists T ε and a unique solution pζ ε , V ε q P Cpr0; T ε r; E N q to the equation (3.77). We set T ε ˚" sup tPr0;T ε r tt, pζ ε , V ε q, exists on r0; ts with :@s ď t, E N pζ ε , V ε qpsq ď 2C 0 u

Then, one has T ε ˚ě T 0 . Indeed, suppose it is not true. One has for all t ă T ε , using Lemma 3.10:

E N pζ ε , V ε qptq ď CpKqpt `εq `C0 , with K " sup tPr0;T ε ˚rE N pζ ε , V ε q.
Notice that K ď 2C 0 by definition of T ε ˚. Since C is non decreasing, one has, for all t ď T ε ˚:

E N pζ ε , V ε qptq ď Cp2C 0 qpt `εq `C0 ă Cp2C 0 qpT 0 `ε0 q ă 2C 0 and therefore, by continuity, there exists T ε ą T ε ˚such that pζ ε , V ε q exists on r0; T ε s with E N ptq ď 2C 0 for all t ď T ε . It is absurd, by definition of T ε ˚. Therefore, the solution exists on r0; T 0 s which is the result of Theorem 3.7. l

A Classical results on Sobolev spaces

We recall here some classical results on Sobolev spaces. Proofs can be found in [START_REF] Michael | Partial differential equations[END_REF]. The first result is the Kato-Ponce estimate on commutators:

Proposition A.1 (Kato-Ponce) For all s ě 0 and f P H s X W where C is a positive constant independent of f and u.

The following result stands that one can compose any H s X L 8 function with a smooth function.

Proposition A.2 (Moser) Let F : R Ñ R be a smooth function, null at zero. Then, for all s ě 0, and all u P H s pR d q, F puq P H s pR d q and |F puq| H s ď cp|u| 8 q|u| H s where c is a smooth non decreasing function.

B Results on the operator A

We prove in this section the regularity of the inverse of h b A stated by Proposition 3.13.

Proposition B.1 Let N P N and let h b P H N pR d q be such that there exists h min ą 0 such that @X P R d , h b pXq ě h min .

We set, for all f P L 2 pR d q : |f | 2 X N " |f | 2 H N `µ|∇ γ ¨f | 2 H N . The operator h b A is invertible on H N pR d q d . Moreover, the following estimates stand.

( 1) For all f P H N pR d q d ,

|ph b Aq ´1f | X N ď Cp 1 h b , |h b | H N `1 q|f | H N .
( 2) For all g P H N pR d q,

? µ|ph b Aq ´1∇ γ g| X N ď Cp

1 h b , |h b | H N `1 q|g| H N .
Proof Let N P N. We define X N " tV P L 2 pR d q, ∇ γ ¨V P L 2 pR d qu.

Endowed with the scalar product p¨, ¨qX N " p¨, ¨qH N `µp∇ γ ¨, ∇ γ ¨qH N , X N is an Hilbert space with norm

| ¨|2 X N " | ¨|2 H N `µ|∇ γ ¨|2 H N .
We start to prove that h b A is invertible, by using a Lax Milgram's Theorem with the bilinear form

T : pV 1 , V 2 q P X 0 ˆX0 Þ ÝÑ ph b AV 1 , V 2 q 2 .
i) The bilinear form T is continuous:

Indeed, one has, for all V 1 , V 2 P X N :

ph b AV 1 , V 2 q 2 " ph b V 1 , V 2 q 2 `µp 1 h b ∇ γ ¨ph b V 1 q, ∇ γ ¨ph b V 2 qq 2
and therefore one has

ph b AV 1 , V 2 q 2 ď C 1 ph min , |h b | W 1,8 q|V 1 | X 0 |V 2 | X 0
with C 1 a non decreasing function of its arguments.

ii) The bilinear form T is coercive:

Let us write for all V P X 0 :

ph b AV, V q 2 " ph b V, V q 2 `µp 1 h b ∇ γ ¨ph b V q, ∇ γ ¨ph b V qq 2
which already gives

|V | 2 2 ď 1 h min ph b V, V q 2 . (B.89)
Moreover, one has

ph b V, V q 2 `p 1 h b ∇ γ ¨V, ∇ γ ¨V q 2 " ph b AV, V q 2 ´2µp∇ γ ¨V, V ¨∇γ h b q 2 ´µp 1 h b ∇ γ ph b q ¨V, ∇ γ ph b q ¨V q 2 ď ph b AV, V q 2 `2µ|∇ γ ¨V | 2 |V | 2 |h b | W 1,8 `1 h min µ|V | 2 2 |h b | W 1,8
and one can conclude using (B.89) and Young's inequality that

|V | 2 X 0 ď C 2 p 1 h min , |h b | W 1,8 qph b AV, V q 2 . (B.90)
Using Lax Milgram's Theorem, for all f P L 2 pR d q, there exists a unique V f P X 0 be such that h b AV f " f . We now prove the first estimate on V f stated by the Proposition by induction on N . Taking V " V f in (B.90), one has this estimate for N " 0. Let us suppose that the result is true for N ´1 with N ě 1, and let us prove it for N . One has, differentiating N times the relation h b AV f " f (we denote by B N any derivative of order N below):

h b AB N V f " B N f `rh b A, B N sV f (B.91)
and rh b A, B N sV f is sum of terms of the form R N 1 " µB k1 ph b q∇ γ pB k2 p 1 h b q∇ γ ¨pB k3 ph b qB k4 V f qq and R N 2 " B l1 ph b qB l2 V f with k 1 `k2 `k3 `k4 " N and k 4 ă N , and with l 1 `l2 " N and l 2 ă N . Taking the L 2 scalar product of (B.91) with B N V f , and noticing that pR N 1 , B N V f q " µpB k2 p 1 h b q∇ γ ¨pB k3 ph b qB k4 V f q, ∇ γ p¨B k1 ph b qB N V f qq 2 , one gets:

ph b AB N V f , B N V f q 2 ď Cp 1 h b , |h b | H N `1 qp|B N V f | 2 |B l2 V f | 2 `µ|∇ γ ¨BN V f |p|∇ γ ¨Bk4 V f | 2 `|B k4 V f | 2 qq.
Using the induction hypothesis, the terms ∇ γ ¨Bk4 V f and B l2 V f are already controlled by

Cp 1 h b , |h b | H N `1 q|f | H N ,
since k 4 ă N and l 2 ă N . One finally gets with a Young's inequality that To prove the second point of the Proposition, one has to notice that for all f " ? µ∇ γ g with g P H 1 pR d q and all V P X 0 : pf, V q 2 " ´pg, ? µ∇ γ ¨pV qq 2 and one can adapt all the proof of the first point to get the desired result, since ? µ∇ γ ¨pV q P L 2 pR d q. l The author has been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01.

ph b AB N V f , B N V f q 2 ď Cp 1 h min , |h b | H N `1 q|f | H N (B.

Theorem 1 . 2 ( 1 )

 121 The equation (1.10) is locally well-posed on a time interval r0; T s where T only depends on the initial data (and not on µ), in dimension d " 1, 2.

26 )

 26 One has to control this term uniformly with respect to δ, and deals with the fact that ph b pI `µT b qq ´1 is not optimally estimated. More precisely, that is absolutely not clear that 4 |h b pI `µT b qf ph b pI `µT b qq ´1u| H s ď |u| H s

  .40) Under the change of unknown (3.40), the equation (3.36) takes the form:

  .47) One differentiates the equation (3.46) to find the following equation in terms of the unknowns ζ δ pαq , V δ pαq (recall the notation (3.47)):

  .74) Using successively the identities (3.72), (3.74), (3.71) and (3.73), the first term of the right hand side of (3.1) is bounded by: |pI ´δ∆qrh b µT b , B α sph b Bq ´1pI ´δ∆q ´1`ε

  92) and combining (B.92) with (B.90), one gets the estimate of the Theorem by a duality argument.

  1,8 " `8. If one could prove an estimate of the form |pV , ζqptq| H N ď gp|pV , ζq| H N qtε (1.7)

  ´k´1 , one finally recovers the "good estimate" k for ζ k :|ζ k | H N ´k `?µ|∇ γ ζ k | H N ´k ď CpKqpt `εq `C0 .Now, we get the "good estimate" for V k . The equation (3.78) gives:

  1,8 and u P H s´1 X L 8 , one has the following inequality:|rΛ s , f su| 2 ď Cp|∇f | H s´1 |u| 8 `|∇ γ f | 8 |u| H s´1 q

The pressure is hydrostatic if, in dimensional form, P pX, zq " Patm ´ρgpz ´ζq. This is always true at the leading order in µ. See for instance[START_REF] Lannes | The water waves problem[END_REF] Section 5.5

Note however that a variant of the Green-Naghdi equation has where the operator to invert in time is time independent has been recently derived recently in[START_REF] Lannes | A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations[END_REF]

The operator h b pI `µT b q is not technically elliptic of order 1, since its inverse only controls the divergence (and not a full derivative). This is actually a big issue for all the local existence results for the Boussinesq-Peregrine equation(2.11). This is also the reason why a Nash-Moser scheme must be used to solve the Green-Naghdi equations in 2d (see[START_REF] Alvarez-Samaniego | A Nash-Moser theorem for singular evolution equations. Application to the Serre and Green-Naghdi equations[END_REF]).