Full justification for the extended Green-Naghdi system for an uneven bottom with surface tension

. This paper is a continuation of a previous work on the extended Green-Naghdi system. We pro-long the system, in arbitrary dimension, with/without surface tension, and for a general bottom topography. Conﬁning the work to the one-dimensional case, well-posedness and consistency with respect to initial data and parameters are proved taken into account the eﬀect of surface tension. The resulting results are local, but long-term in the sense of dependence upon initial data. As a conclusion, our solution remains close to exact solution of the full Euler system with a better (smaller) precision and therefore the full justiﬁcation of the models.


Introduction
1.1 The water-wave problem.Two-dimesional full water-wave problem in arbitrary dimension with general bottom topography is considered.Assume that the fluid is of constant density ρ and denote by Ω t = {(X, z) ∈ R d × R, −h 0 + b(X) < z < ζ(t, X)} the domain of the fluid for each time variable t where the surface of the fluid is a graph parametrized by ζ and its bottom is parametrized by −h 0 +b(X) independent of time with h 0 the depth.Knowing that d = 1, 2 the spatial dimension of the surface of the fluid where X ∈ R d the spatial variable is written as X = (x, y) when d = 2 and X = x when d = 1, while the vertical variable is denoted by z.The motion of an ideal moving fluid is described by the free surface Euler equations for steady flow along a streamline (their well-posedness were recognized after the work of Nalimov [20], Yosihara [6], Craig [21], Wu [13,14] and Lannes [5]) which is a connection between the velocity V , the pressure P , and the density ρ of the fluid that is based on the Newton's second law of motion and can be written under the form (1.1) denoting by −g − → e z the gravitational field which is acting vertically downward with g greater than zero and − → e z is a unit vector in vertical direction.This equation is combined with several physical assumptions.The incompressibility of the fluid is expressed by Another assumption is that the flow is irrotational : Denote by ∇ = ∇ X .These equations are complemented with boundary conditions.At the free surface, a dynamic condition is given by (1.4) where P atm is the (constant) atmospheric pressure, σ > 0 is the surface tension coefficient, and κ(ζ) = −∇ • ∇ζ/ 1 + |∇ζ| 2 is the mean curvature of the surface.Another boundary condition at the top surface is the kinematic condition : (1.5) where the outward unit normal to the upper boundary n + = 1 √ 1+|∇ζ| 2 ∇ζ T , 1 T .This condition states that the fluid particles cannot cross the surface.A similar condition on the velocity at the bottom is given by where the outward unit normal to the lower boundary n − = 1 √ 1+|∇b| 2 ∇b T , −1 T .Another assumption (1.7) states that the fluid is at rest at infinity and is given by (1.7) lim The equations (1.1)-(1.7)are called free surface Euler equations.
1.2 Non-Dimensionalised equations : Bernoulli and Zakharov/Craig-Sulem.The complexity of Euler equations embodied in the many unknowns on a moving with time domain Ω t .Thus we will follow the Eulerian approach that focuses on specific locations in the space through which the fluid flows as time passes.
More precisely, we derive simpler asymptotic models in some geophysical regimes that requires identification of small parameters.Neglecting rotational effects is the starting point towards this end.In other words, assumption (1.3) ensures the existence of the potential velocity of the fluid ϕ(t, X, z).Consequently, the Euler system can be rewritten under Bernoulli's formulation : (1.8) The Laplacian equation is obtained by taking the divergence of the potential velocity combined by (1.2).
Second and third equations is written using the boundary conditions (1.6)-(1.5)respectively.While the last equation is established by commuting V = ∇ X,z ϕ in (1.1).Although the present system has less unknowns, but still in order to solve the Laplacian equation we need information from the boundaries that move with time and its location is determined by two coupled nonlinear partial differential equations which is a basic difficulty.This leads us to the identification of small parameters that it is often possible to deduce from their values some insight on the behavior of the flow.More precisely, let us introduce some dimensionless parameters that encodes the various regimes of interest : • Nonlinearity or the amplitude parameter : ε = a h0 ∈ (0, 1), • Dispersion or the shallowness parameter : µ = ( h0 λ ) 2 ∈ (0, 1), • Bottom topography parameter : β = b0 h0 ∈ (0, 1), • Classical Bond number which measures the ratio of gravity forces over capillary forces : Bo = ρgλ 2 σ > 1, where a is the amplitude of the wave, λ the wave-length of the wave, b 0 is the size of the bottom topography variations, h 0 the reference depth, ρ the density of the fluid, and σ is the surface tension coefficient.We now execute the classical shallow-water (µ 1) non-dimensionalization using the following variables : Therefore, the equations of motion (1.1)-(1.7) is then rewritten under the dimensionless Bernoulli's formulation (we eliminate the primes for sake of clarity) (1.9) To reduce (1.9) into a system where all functions are evaluated at the free surface (i.e. in R + × R d ) that is known as the dimensionless version of Zakharov/Craig-Sulem [19] formulation of the water-waves equations.The demonstration is commenced by introducing ψ : R + × R d −→ R the trace of the velocity potential at the free surface (1.10) ψ(t, X) = ϕ t, X, εζ(t, X) = ϕ | z=εζ , and the Dirichlet-Neumann operator G µ [εζ, βb]• is defined by with ϕ solving ( see [4] for accurate analysis) the Laplace equation with Neumann (at the bottom) and Dirichlet (at the surface) boundary conditions (1.12) where . The solutions of such systems (1.13) are very hard to be explained and appear more complex than necessary for the modeling situation.At this stage, a traditional technique is to pick an asymptotic regime, in which we search for an approximate models and hence an approximate solutions.In the sequel, we seek solutions up to third-order error in the dispersion parameter.Formally, under this condition an approach is based on a perturbation method with respect to a small parameter µ 1 when no assumptions are made on ε ∼ 1 and β ∼ 1.The Saint-Venant equations who first derived them in 1871 [1,2], are order O(µ) corrections of the shallow water equations.These equations couple the evolution of the free surface ζ to the evolution of the vertically averaged horizontal component of the velocity v, they are expressed such as where the non-dimensional height of the liquid.The first justification of such models goes back to Ovsjannikov [9], and Kano and Nishida [34] who proves that the solution of the shallow-water equations (1.14) converges to the solution of the water-waves equations as µ → 0 in the one-dimensional case, and under some restrictive assumptions.However, Y. A. Li in [22] removes these assumptions and gives rigorous justification.Note that, for such a symmetrizable hyperbolic system (multiply the second equation by h) a classical local existence can be established see [15,31].Up to a second-order error in µ the classical Green-Naghdi system takes into account dispersive effects ignored by (1.14).This system is derived in [38] through Hamilton's principle and in [27] through a general method of derivation (see section 2) is exactly the same as the one found in [23], that is where we recall that h = 1 + εζ − βb and T v = T [h, βb]v and the quadratic form Q[h]v is defined as and For this approximation we refer to several works that give a rigorous justification on the well-posedness of this system using various arguments such as Y. A. Li in [22], Alvarez-Samaniego and Lannes in [25], T.
Iguchi in [16], Israwi in [7,8,22], Khorbatly in [3], and H. Fujiwara and T. Iguchi in [32].We also refer to Lannes and Marche [26] for recent progress made to this system in the derivation of an alternative new class of equations having better mathematical structure which makes them much more suitable for the numerical simulations.

Main results
: comments and organization.The aim of this paper is to derive and full justify (i.e.local existence+consistency+convergence) the extended Green-Naghdi (modified) system to one order higher in µ in the presence of a nontrivial bottom topography, thereby introducing quite a few new and nontrivial higher-order terms.In other words, this system is much higher order approximation to the full Euler system with less error up to order O(µ 3 ), i.e. combines much better dispersive properties, and thus wider range of application on oceanography.In the spirit of the work done in [11,12], Y. Matsuno derived the extended equations for flat bottom by a slightly different method with respect to the one used here [27,40].He showed that they permit Hamiltonian structure and stated that the linear dispersion relation of his model does not have good structures, i.e. we cannot expect the well-posedness of the initial value problem so that it is hopeless to obtain an error estimate of the solutions of order O(µ 3 ).In fact, this is not the case in presence of surface tension, as the linear part of the µ 3 modified model has better structure and the solution might approximate the solution to the full water wave equations up to order O(µ 3 ) (see remark 2).
Here it is wroth noticing that in recent papers T. Iguchi [17,18] obtained and mathematically justified a model in a similar asymptotic regime than this work.Also it was shown that the so-called Isobe-Kakinuma model permits Hamiltonian structure [36].Remark that one of its strong advantages is that it does not contain any higher order derivatives and thus less troublesome terms in numerical computation.However, the only drawback of such hierarchy is that the Green-Naghdi/Boussinesq systems are not one of the models.Regarding this, a number of challenging/interesting problems associated with the extended system that are worthy of additional study.We list some of them below : • Identification of physically relevant models among various extended models, i.e. having asymptotic models of order O(µ 3 , ε 2 µ 2 , εβµ 2 , β 2 ) (medium amplitude and bathymetry models) and also one can derive higher order version of various shallow-water models such as the well-known Boussinesq model in multiple surface and topography variations etc... • Numerical computations of the initial value problems as well as solitary and periodic wave solutions for less pain models previous mentioned above .• The effect of higher-order dispersion on the wave characteristics and the affect of the presence, or lack thereof, of surface tension on the justification of the asymptotic models by means of the rigorous mathematical analysis .
Following the general method for the derivation of asymptotic nonlinear models in shallow and deep water first introduced in [27], we derive the new system generalizing (in presence of an arbitrary topography) the investigations of [40] done by the authors.What is different for this system is the pure intricacy and number of terms that has not been yet derived or analyzed especially in the case when the bottom is not flat.For the sake of constructing the solution by an energy estimate method, an impressive set of estimates and calculations one must go through to obtain the correct energy estimate.Due to this, the work is to be confined to one space dimension.The strategy of the proof is to write the system as a quasilinear form, i.e. to be treated as hyperbolic system, then use symmetrizer to derive good energy estimate on the solution.
The existence is then obtained through a fixed point argument [15,31].On behalf of the inadequacy linear dispersion structure of the original extended system (2.19), the study is to be done on a modified asymptotic variant: (1.17) where the non-topographical terms are represented by while the purely-topographical terms are introduced by This new variant shares the same order of precision as the original one but having a mathematical structure that is more suitable for well-posedness, in particular, its linear dispersion relation (see for instance remark 2).Consequently, a problematic term appears under the form of 5th order derivatives on the surface elevation which requires special treatment.Several different techniques are used throughout the proofs towards generalizing the flat bottom case [40] such as invertibility of the = h + µT [h, βb] − µ 2 T[h, βb] operator, and re-expression of terms.The main point regarding the justification of such higher order asymptotic model is the following well-posedness result : • Long-time existence with surface tension.Theorem 1 states that (1.17) admits a unique solution in U ∈ X s = H s+2 (R) × H s+2 (R) on existence time scales T max up to order 1 max(ε,β) such that 1/T max depends on initial data and the rescaled bond number (2.17) bo −1 as the energy estimate constant (4.8) depends on same bo −1 ∈ [0, 1).In the sequel, section 2 is devoted to the derivation of the extended two-dimensional Green-Naghdi system for non-flat bottom topography and the modification process held for the one-dimensional system to be studied.In section 3, some properties on the operator and its inverse are given.In section 4, a suitable energy norm and symmetrizer are sought, and the linearized system with surface tension is studied for an appropriate energy preservation.The main results (existence theorem 1, stability property 2, consistency proposition 2, and convergence theorem 3) are deduced and proved in section 5.
1.5 Notation.We denote by C(λ 1 , λ 2 , ...) a constant depending on the parameters λ 1 , λ 2 , ... and whose dependence on the λ j is always assumed to be nondecreasing.The notation a b means that a ≤ Cb, for some non-negative constant C whose exact expression is of no importance (in particular, it is independent of the small parameters involved).Also, the notation a ∨ b stands for the maximum between a and b.We denote the norm | • | L 2 simply by | • | 2 .The inner product of any functions f 1 and f 2 in the Hilbert space For any real constant s, H s = H s (R d ) denotes the Sobolev space of all tempered distributions f with the x ) s/2 .For any functions u = u(t, X) and v(t, X) defined on [0, T ) × R d with T > 0, we denote the inner product, the L p -norm and especially the L 2 -norm, as well as the Sobolev norm, with respect to the spatial variable, by (u, v) ) denote the space of infinitely differentiable functions, with compact support in R d .We denote by C ∞ b (R) the space of infinitely differentiable functions that are bounded together with all their derivatives.For any closed operator T defined on a Banach space Y of functions, the commutator [T, f ] is defined by [T, f ]g = T (f g) − f T (g) with f , g and f g belonging to the domain of T .
2 Derivation of the extended GN model with general bottom topography 2.1 Derivation of the system in two space dimensions.To derive the Green-Naghdi equations (2D case), the starting point is introducing the depth averaged horizontal velocity where h(t, X) = 1 + εζ(t, X) − βb(X) the non-dimensionalised height of the liquid.The Exact expression for − 1 µ G µ [εζ, βb]ψ = ∇ • (hv) stemming from a clear outcome of Green's identity or by a straightforward calculation and rearranging terms using (1.9), the first evolution equation on ζ in terms of (ζ, v) is which exactly coincides with the first equation of (1.13).Proceeding as in [27], for the second evolution equation on v, ∇ψ do not have an exact expression in terms (ζ, v).Therefore, since µ 1, we look for an asymptotic expansion with respect to µ on ∇ψ in terms of (ζ, v) and this is obtained through an asymptotic description of ϕ in the fluid by constructing Plugging expression (2.2) into the boundary value problem (1.12), and dropping all terms of order O(µ N +1 ), one gets with the convention ϕ −1 = 0 by definition and the boundary condition where δ 0,j = 1 if j = 0 and zero otherwise.Solving the ODE (2.3)-(2.4)yields to the three solutions that are polynomials of order 0, 2, 4 in z such that So the horizontal component of the velocity in the fluid domain is given by The averaged velocity is thus given by As in [27], the second order approximation O(µ 2 ) of ∇ψ in terms of ζ and v is as follows (2.10) The new ingredient is writing a third order approximation O(µ 3 ) of ∇ψ in terms of (ζ, v).Thus we shall need to compute the integral ∇ϕ 2 dz.As the terms are many we refer to appendix A.1 for detailed calculations and clear presentation of J 2 .

Remark 1. Here and throughout the rest of this paper we shall introduce two new notions 'non-topograpical' and 'purely-topographical' to differentiate many terms. Apart from the expression of the height of the fluid
), the non-topographical expressions are the terms that do not include a bottom parameter β k∈N in front of the term.Otherwise, the terms are purely-topographical.
The averaged velocity expression becomes Hence, (see the details of T (h −1 T ) in appendix A.2 and appendix A.
It follows the lines of derivation using (multiple) scales in general, and the derivation of the Green-Naghdi system itself in particular.As such, it is formal and essentially algebraic (there is no functional analytic framework), but it is also lengthy and a feat in thoroughness and endurance, as the terms of the extended topography order are many, involved and require clever tricks to gather in a suitable form.The main steps are (1) Take the gradient of the second equation of (1.13) then multiply it by h .
(5) Expand then reduce terms of same size .( 6) Take advantage of the the following vector triple products and the vector identities where G is a differentiable scalar function and u,ν,ω, and F are differentiable vector fields.
Finally, after capturing the information above we obtain the extended 2D Green-Naghdi system for an uneven bottom topography (β = 0) without surface tension (σ = 0) with an error of order µ 3 represented by (2.16) where the non-topographical terms represented by while the purely-topographical terms are introduced by where the expression of Q 2 introduces the Laplacian operator

The capillary components.
In the presence of surface tension (σ = 0), different strategies exist to deal with it in the water-wave problem such as [19,[28][29][30], the main contrast in our work is that the gradient of the capillary term − 1 Bo multiplied by h must be added to the right-hand side of the second equation of (2.16).From a physical point of view, the effect of surface tension on the water surface is negligible, so the only condition we need is that there is a small amount of surface tension.More precisely, since the Bond number Bo is generally large, we assume that the capillary parameter Bo −1 is at the same order as the shallowness parameter µ 1.And therefore we define the rescaled Bond number bo instead of the classical Bond number Bo, as follows (2.17) 0 where h 0 the reference depth, ρ the positive constant density of the fluid, g the acceleration of gravity, and σ the surface tension coefficient.Regarding this, bo is not assumed too small so that Bo −1 = µbo −1 = O(µ), and the 2D capillary terms that should be added stands for

The one-dimensional case.
Here and throughout the rest of this paper we will confine the work to one space dimension.The extended Green-Naghdi system (2.16) with surface tension is rearranged after few calculations taking into account the capillary terms (2.18), as follows where U = (ζ, v) T and denoting by h = h(t, x) = 1 + εζ(t, x) − βb(x) the total non-dimensional height of the liquid, with The non-topographical terms are represented by while the purely-topographical terms are introduced by

New invariant of (2.19).
The main interest of this reformulation is to gather all terms of fifth-order derivatives in the left-most term, that is h ), where U = (ζ, v) T and h(t, x) = 1 + εζ(t, x) − βb(x), and one may write the above expressions as follows where the reformulated non-topographical terms are represented as while the reformulated purely-topographical terms are introduced as x , and 2.5 The modified system to be studied (2.23) βb] with an arbitrary real parameter α > 0. The special case α = 1 recovers the present definition.In this general setting, also the linear dispersion relation would not give rise to any singularity.In this case, the modified linear dispersion relation of the new system (2.23) (with bo −1 = 0) exhibits no singularity and reads On the other hand, in presence of surface tension (with bo −1 = 0) the modified linear dispersion relation of the new system (2.23) reads In view of the above remark, we introduce the new operator . This has to be followed by some necessary rearrangements due to a suitable specification of an appropriate symmetrizer (4.5).The modified 1D extended Green-Naghdi system with surface tension then reads (2.23) where where the non-topographical terms are represented by while the purely-topographical terms are introduced by 3 Preliminary results.
From a physical point of view we will assume that the fluid depth is constantly limited.This assumption is essential for the mathematical analysis.Therefore, the analysis is to be done under the non-zero depth condition : We shall use intensively two formulations of the left-most operator 25) the definition of T and T) where at some points provides more conveniency in the energy estimate derivation and the analysis of the operator itself.These two expressions corresponds to two formulations of operator T[h, βb] defined by (2.25) and the other by : The following two lemmas provides important invertibility results on and specify some properties on its inverse −1 in the following lemmas .

is a differentiable scalar function under the condition (3.1). Then, the operator
is well defined, one-to-one and onto .
Proof.The proof of the invertibility of is a direct application on the Lax-Milgram theorem.We define by but not uniformly with respect to µ ∈ (0, 1).Let f ∈ L 2 (R).Consider the weak problem with L(u) = (f, u) and the bilinear form a(v, u) = ( v, u) which can be written as follows It is easy to see that a and L are continuous on H 2 µ (R) × H 2 µ (R) and H 2 µ (R) respectively.In addition, using (3.1) we have Therefore by Lax-Milgram theorem, for every f ∈ L 2 (R), there exists a unique v ∈ H 2 µ (R) such that for all u ∈ H 2 µ (R) we have : a(v, u) = ( v, u) = L(u) = (f, u).Equivalently, there is a unique variational solution to the equation

Therefore
It remains to prove that v ∈ H 4 (R).Indeed, fix µ ∈ (0, 1) and let us introduce the well-defined invertible operator J : From the definition of operators and J combined with (3.5), it holds that v xx = J −1 ψ such that R) and using (3.6) one may deduce that J −1 ψ = v xx ∈ H 1 (R).Thus by (3.5) and (3.1) we have Hence the proof is complete.
The following lemma gives functional properties to the operator −1 .
Lemma 2. Let ζ ∈ H t0+1 (R) and b ∈ H t0+3 (R) be such that (3.1) is satisfied.Then, we have the following : Proof.The proof is a generalization of the proof when β = 0 of operator in [40].Remark that here it is more convenient to use the second formulation (3.2) of the operator .Assume that f ∈ H s (R) and u = −1 f , then u = f .Apply Λ s to both sides, then multiply by Λ s u which yields the following equality (note that Λ such that f , g, and p reads Integrating by parts and using (3.4), we get Now, using the necessary Kato-Ponce commutator estimates below (see Lemma 4.6 of [25]) Hence, the inequality (i) holds after a continuous induction on s.For the proof of (ii), one has to replace u = √ µ −1 ∂ x f and u = µ −1 ∂ 2 x f for a second time.The general strategy is the same as in (i) noticing that Λ s commutes with ∂ x , ∂ 2 x .The only difference is in the expression of f , g, p when setting u = √ µ −1 ∂ x f and similarly when u = µ −1 ∂ 2 x f .The rest of the proof is as in [40].

The linearized system
In order to rewrite the extended Green-Naghdi system for an uneven bottom with surface tension in a condensed form, we introduce a new operator symmetric J bo as follows The first equation in (2.23) can be written as For the second equation in (2.23) apply −1 to both sides, we get Hence the extended Green-Naghdi system (β = 0) with surface tension can be written under the form (4.2) where and .
We consider now the linearized system around some reference state U = (ζ, v) T (4.4) The proof of the energy estimate which permits the convergence of an iterative scheme to construct a solution to the extended system (2.23) for the initial value problem (4.4) requires to define the X s spaces, which are the energy spaces for this problem .
Definition 1.For all s ≥ 0 and T > 0, we denote by X s the vector space First, recall that a pseudo-symmetrizer for A[U ] is given by where h = 1 + εζ − βb.A natural energy for the initial value problem (4.4) is suggested to be The connection between E s (U ) and the X s -norm is examined using the lemma below .
Using the expressions of , J bo , the proof of Lemma 1 and by integrating by parts, it holds that From the assumption of the lemma we know that b , and the fact that the water depth is constantly limited (4.7), then by Cauchy-Schwartz inequality and the help of the proof of Lemma 1, we get the two inequalities of the desired lemma.
A derivation of the prior energy estimate is given in the following proposition .
Remark 3. In the following proof and for sake of simplicity, we will not attempt to show the dependence on the bottom parametrization b ∈ H s+3 (R) in all the verifications.The reason why s + 3 can be easily seen when controlling Proof.The existence and uniqueness of the solution is a direct adaptation of the proof in Appendix A of [7] (one may see also [15,31] for general details).Our attention targets mainly the demonstration of the energy estimate.Consider any λ ∈ R, the key point is to bound from above in terms of E s (U ) the below component Using the fact that and J bo are symmetric, in addition to the following identities (4.9) one gets after using (4.4) the following Therefore, we obtain We will focus now on bounding from above the purely-topographical components of the r.h.s of (4.10), knowing that the non-topographical expressions have been controlled in [40].Remark that by Parseval's identity, Cauchy-Schwarz inequality, then Young's inequality we shall use the below inequality (4.11) . By definition we have Then, it holds that From [40] and with inequality (4.11) in hands, one may deduce that the non-topographical terms A 1 , A 2 , A 3 , A 5 , A 7 are controlled as follows (4.12) To control A 4 , an integration by parts yields to write = A 41 + A 42 + ... + A 4 (10) .
For controlling A 41 and A 42 , by integration by parts and (4.11) it holds that Again, using integration by parts one can write The rest of the components are controlled similarly so that it holds To control A 6 one should notice that the non-topographical terms are bounded from above in [40] by while the purely-topographical terms can be written as follows Again, by integration by parts one has Hence, we get Similarly, using the expressions of and integrations by parts one has Therefore, we get To control D 1 , D 2 we use the expression of J bo , the commutator estimate (3.8), and the fact that Then, with (4.11) in hands, it holds that (4.14) To control By using the explicit expression of , integration by parts and the fact that The above equality the implies In order to facilitate our way in controlling D 3 + D 4 , we recall the re-expression (3.2) of such that Using this, the commutator estimate (3.8), the identities (4.15) and the help of lemma 2, it holds that For controlling D 345 , one should use the explicit expression of J bo and the fact that to write D 345 as follows For controlling D 3454 , one should notice the following commutator identity x N .Then, with (4.11) in hands, it holds that Therefore, we obtain For controlling D 346 , by integration by parts and using the explicit expression of , the commutator estimate (3.8), the identities (4.15) and the help of Lemma 2, we get For controlling D 347 , by integration by parts, the commutator (3.8) with the help of (4.16)-(4.17)-(4.11)and the commutator identity below The non-topographical terms are bounded from above by On the other side, for the purely-topographical terms in D 347 we use ]N, and by integration by parts and (3.8) to write Thus, after collecting the information above, we obtain To control D 5 , let us notice the identities (4.19) and (4.20) [Λ s , M ∂ (ı)  x (N x [Λ s , N ]P where ı = 1, 2 .Now as in D 3 , using (4.15), by integration by parts, lemma 2, and the Kato-Pance commutator (3.8) estimate, it holds that To control D 6 , one can write after checking the expression of and using (4.13) with integration by parts and the fact that To control D 7 , one can realize D 3 and integrate by parts with the help of (4.15) to write Furthermore, as above using the expressions of , with the help of lemma 2, the commutator (3.8), in addition to (4.15) and (4.19) one gets To control D 8 , one can write by integration by parts Now, as above using the expressions of , Eventually, as a conclusion, one gets • Estimation of Λ s B(U ), SΛ s U .We recall that where we recall the expressions of the operators : Now, as in D 3 , one may write Using the expressions of , R[µ, εh, βb](U ), J bo with the help of lemma 2, the commutator estimate (3.8), in addition to (4.15) and (4.19) one gets One can write after checking the expression of J bo and doing an integration by parts, that , so one gets using integration by parts Gathering the information provided by the above estimates and using the fact that Taking λ = λ T large enough (how large depending on sup t∈[0, T ε∨β ] C E s (U ), |∂ t h| L ∞ , bo −1 to have the first term of the right hand side negative for all t ∈ [0, T ε∨β ], one deduces Integrating this differential inequality with the help of Grönwall's inequality yields therefore which is the desired estimate.
5 Full justification of the asymptotic model (2.23) The following theorem proves the main result of this paper, that is a large-time existence of solution to the extended system (2.23) in some Sobolev spaces X s = H s+2 (R) × H s+2 (R) as soon as s > 3/2 of time order t = O( Proof.The proof of the well-posedness is a straightforward readjustment of the proof of Theorem 7.3 in [37] or Theorem 1 in [40] using the energy estimate from the linear analysis proved in Proposition 1.The techniques here are those used for hyperbolic systems (see [15,31] for general details) with additional standard arguments, where no smallness assumption on the parameters ε, µ, β is required in the theorem.Remark that, when a sequence of nonlinear problems are devised, the difference with respect to flat bottom case (see Theorem 1 of [40] for details of the proof) occurs in the convergence of solutions that is established using the energy estimate.So, one has to deal with topography terms such as (B(U where C 0 is a constant depending on initial data U 0 .It is worth noticing that the constant λ T appearing in Proposition 1 is independent of ∂ t ζ n and depends only on |U 0 | X s .Indeed, by induction on n and using the mass conserved equation it holds that Theorem 1 is complemented by the following result that shows the stability of the solution with respect to perturbations, which is very useful for the justification of asymptotic approximations of the exact solution.(The solution U = (ζ, v) T and time T max that appear in the statement below are those furnished by Theorem 1).

Theorem 2. (A stability property) Let the assumption of Theorem 1 be satisfied and moreover assume that there exists
T with respect to U given by Theorem 1 satisfies for all 0 ≤ (ε ∨ β)t ≤ T max the following inequality Proof.The proof is a direct and classical consequence of the same energy estimate evaluated in Theorem 1 wich is itself similar to the energy estimate proved in Proposition 1. Subtracting the equations satisfied by U = (ζ, v) T and U = ( ζ, v) T we get the following system Therefore, in view of the proof of Proposition 1 one may deduce similarly the following estimate below 1 2 X s (R) .Integrating the differential inequality (5.1) by applying Grönwall's inequality yields therefore to the desired result.
We state here that the solutions to the water wave equations (1.13) are consistent at order O(µ 3 ) with the extended Green-Naghdi equations (2.19).
Proposition 2 (Consistency).Let U euler = (ζ, ψ) T be a family of solutions to the full Euler system (1.13) such that there exists T > 0, s > 3/2 for which (ζ, ψ ) T is bounded in L ∞ ([0; T ); H s+N ) 2 with N sufficiently large, uniformly with respect to ε, µ, β, bo −1 ∈ (0, 1) 4 .Moreover, assume that b ∈ H s+N and that ζ satisfies ) .Proof.The proof is following the same lines as for the proof of Theorem 6.10 in [4] with N sufficiently large.In fact, it is sufficient to show that the second equation of (2.23) is satisfied up to a term µ 3 R with R as in the statement of the Proposition.Indeed, taking the derivative of the second equation of (1.13) and replacing G[εζ, βb]ψ by −µ∂ x (hv) and in view of (2.13) replace ψ by v + µ h T [h, βb]v + µ 2 h T[h, βb]v + µ 3 R 3 .At this stage, denote by µ 3 R all the terms of order µ 3 , then taking advantage of similar estimates as those of Lemma 5.11 in [4]  Finally, the following convergence result states that the solutions of the full Euler system, remain close to the ones of our system (2.23) with a little more precision as µ 3 is smaller.Proof.The existence of U ex is given by our Theorem 1 (we choose T as the minimum of the existence time of both solutions; it is bounded from below, independently of ε, µ, β, bo −1 ∈ (0, 1) 4 ).Assuming that U euler satisfies the assumptions of our consistency result, Proposition 2, therefore (ζ, v) T solves (2.23) up to a residual R bounded from above by µ 3 .The result then follows from the stability property (Theorem 2).

A Appendix
In this section we denote by w = ∇ψ (note that w is independent of z).The strategy is to expand then reduce all terms of same size.First, find ∇ϕ 2 , and note that using the expression of ∇ϕ Rearranging the many expressions of ∇ϕ 2 , then integrating between on ] − 1 + βb, εζ[ and taking advantage of the integrals above one may simplify J 2 as follows.

A.1 Computation of the integral J
The non-topographical expressions (see remark z ϕ refers to the upward normal derivative at the bottom.A set of two equations on the free surface parametrization ζ and the trace of the velocity potential at the surface ψ = ϕ | z=εζ involving the Dirichlet-Neumann operator is introduced as (1.13)
. In this subsection, we will state in remark 2 the reason behind the use of the BBM trick and consequently the corresponding modified model(2.23).Actually, in high frequency regime |ξ|1 an instability appears.As a result, the Cauchy problem for the µ 3 model is ill-posed.Therefore, a modification is required on the structure of the model so that the dispersion relation do not give rise to any singularity.In other words, the positive sign of T[h, βb] in the µ 2 -fifth order factorized term is problematic and prevents the invertibility proof of the operator h+ µT [h, βb] + µ 2 T[h,βb] by a Lax-Milgram theorem.This property is of highest interest for the well-posedness demonstration.To overcome this difficulty we replace the positive sign by a negative sign.A remainder term under the expression 2µ 2 T[h, βb](∂ t v + εvv x ) appears.At this stage, in order to trade the time-dependent derivative on v by a spacial one, a BBM trick is used and is represented by the following approximate equation ∂ t v + εvv x = −ζ x + O(µ).It is noteworthy that one may replace the relation h + µT [h, βb] + µ 2 T[h, βb] = + 2µ 2 T[h, βb] which defines the new operator by an alternative one 1 ε∨β ), realizing that if some smallness assumption is made on ε ∨ β, at that point the presence time t = O( 1 ε∨β ) ends up bigger.