Weakly transverse Boussinesq systems and the Kadomtsev

INSTITUTE OF PHYSICS PUBLISHING Nonlinearity 19 (2006) 2853–2875

NONLINEARITY doi:10.1088/0951-7715/19/12/007

Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation D Lannes1 and Jean-Claude Saut2 1 MAB,

Université Bordeaux I et CNRS UMR 5466, 351, Cours de la Libération, 33405 Talence Cedex, France 2 Universit´ e Paris Sud, UMR 8628, Département de Mathématiques, Bât. 425, 91406 Orsay Cedex, France

Received 11 March 2006, in final form 3 October 2006 Published 2 November 2006 Online at stacks.iop.org/Non/19/2853 Recommended by L Ryzhik Abstract We study here the asymptotic behaviour of weakly transverse water-waves in the long waves regime. It is well-known that the Kadomtsev–Petviashvili (KP) approximation describes formally the dynamics of the exact solutions of the water-waves equations. We provide here a rigorous justification of this approximation, showing that if solutions of the water-waves equations exist over a relevant time scale, then they are well approximated by the KP approximation. A nonphysical zero mass assumption, inherent to the structure of the KP equation, is however needed to obtain this result; this is the reason why we introduce a class of weakly transverse Boussinesq systems. These new systems provide a much more precise approximation than the KP equation and do not require any zero mass assumption. Mathematics Subject Classification: 35B40,76B07,76B15

1. Introduction This paper is in some sense a continuation of [4–6] where new ‘Boussinesq’ systems describing the interaction of long water-waves in two and three spatial dimensions were systematically derived and analysed. In particular, it was proven in [6] that solutions of any of the aforementioned systems yield good approximations to the full Euler equations on the long time scale ε−1 where nonlinear and dispersive effects can have an order one relative effect on the velocity field and the wave profile. Our aim here is to start from the complete free-surface Euler equations with a flat bottom (more precisely, from the free-surface Bernouilli equations, which correspond to the Euler equations written in terms of a velocity potential [22]) to derive new Boussinesq systems in the Kadomtsev–Petviashvili (KP) scaling, that is in the regime of weakly transverse long waves. As is well known (see for instance [3, 13] the KP equation has serious 0951-7715/06/122853+23$30.00 © 2006 IOP Publishing Ltd and London Mathematical Society Printed in the UK

2853

2854

D Lannes and J-C Saut

drawbacks of having a (unphysical) zero mass constraint 3 and a rather poor convergence rate to the full Euler system. This limits severely its applicability as a realistic model for long waterwaves. Adapting the strategy of [4–6] we obtain here a class of Boussinesq systems which have the convergence rate O(ε 2 t) (where ε is the small dimensionless parameter) and which does not suffer from a zero mass constraint. The only incidence of the uni-directionalization implied by√ the KP scaling is that the transverse component of the velocity at the surface may grow as 1/ ε. 1.1. Formulation of the equations As said above, we are interested here in the case of a flat bottom; we denote by h the depth of the fluid at rest. The horizontal variables are denoted by x and y, while z stands for the vertical coordinate (z = 0 being a parameterization of the surface of the fluid at rest). In this setting, it is presumed that the free surface may be described as the graph of a function ζ defined over the bottom. Since we assume that the fluid is incompressible and irrotational, there exists a velocity potential such that the velocity field is given by v = ∇ and the equations of motion are given by = 0, ∂z = 0, ∂t ζ + ∂x ∂x ζ + ∂y ∂y ζ = ∂z , ∂t + 21 |∇|2 + gz = 0,

−h z ζ (t, x, y), z = −h, z = ζ (t, x, y), z = ζ (t, x, y),

(1)

where the gradient operator ∇ and the Laplace operator are taken with respect to the three coordinates x, y and z. The regime of weakly transverse long waves (or KP regime) under study here can be specified in terms of the relevant characteristics of the wave, namely, its typical amplitude a, the mean depth h, the typical wavelength λ along the longitudinal direction (say, the x axis) and µ, the wavelength along the transverse direction (say, the y axis): S1 S2 a λ2 µ2 = = 2, (2) = ε, , 2 2 h h ε h ε where ε 1 is a small dimensionless parameter, while S1 ∼ 1 and S2 ∼ 1 (so that the Stokes number is S1 along the longitudinal direction and S2 /ε along the transverse one). For notational simplicity, we set S1 = S2 = 1 throughout this paper. 1.2. The nondimensionalized water-waves equations The asymptotic study becomes more transparent when working with variables scaled in such a way that the dependent quantities and the initial data which appear in the initial value problem are all of order one. The relations (2) which set the KP regime studied here are connected with small parameters in the nondimensionalized equations of motion. Denoting dimensionless variables with a prime, we set √ gh x y z ζ x = , y = , z = , ζ = , = t, , t = λ µ h a 0 λ √ where 0 = (a/ h) ghλ. 3 This constraint is due to a singular approximation of the dispersion relation of the linear wave equation in the relevant scaling (see, e.g. [16]) which measures the ratio of the typical wave elevation over the mean depth.

Weakly transverse Boussinesq systems and the Kadomtsev–Petviashvili approximation

2855

The equations of motion (1) then become h2 2 h2 2 ∂ + 2 ∂y + ∂z2 = 0, −1 z ha ζ , λ2 x µ z = −1, ∂z = 0, 2 2 a aλ λ ∂t ζ + ∂x ∂x ζ + ∂y ∂ y ζ = 2 ∂z , z = ha ζ , 2 h hµ h h 1 a aλ2 aλ2 2 2 ) + ) + z = 0, ∂t + (∂x )2 + (∂ (∂ z = ha ζ . y z 2 3 2 h hµ h a Using the relations (2) and omitting the primes for dimensionless quantities, the above system yields ε∂x2 + ε 2 ∂y2 + ∂z2 = 0, ∂z = 0,

−1 z εζ (t, x, y), z = −1,

1 ∂t ζ + ε∂x ∂x ζ + ε 2 ∂y ∂y ζ = ∂z , ε ∂t + 21 ε(∂x )2 + ε 2 (∂y )2 + (∂z )2 + ζ = 0,

z = εζ (t, x, y),

(3)

z = εζ (t, x, y).

1.3. Reduction to a system of two scalar evolution equations It is well known (see for instance [8, 9, 23] that the water-waves equations reduce to a system of two evolution equations coupling the parameterization of the free surface ζ to the value of the velocity potential at the surface, which we denote ψ. Such a reduction involves usually a Dirichlet–Neumann operator (which maps ψ to the normal derivative of the velocity potential at the surface); as in [6], we rather use the operator Z ε (ζ )· defined for all ζ smooth enough and all s > 1/2 as H s (R2 ) → H s−1 (R2 ) Z ε (ζ ) : , (4) ψ → ∂z |z=εζ where solves the boundary-value-problem ε∂x2 + ε 2 ∂y2 + ∂z2 = 0, −1 z εζ (x, y), ∂z = 0, z = −1, = ψ, z = εζ (x, y). Remark 1. (i) The 2D-surface waves studied in [6] were assumed to have wavelength of the same order of magnitude in all directions. The elliptic equation determining was therefore different, namely, ε∂x2 + ε∂y2 + ∂z2 = 0. (ii) The dimensionless version of the Dirichlet–Neumann operator Gε (ζ ) used in [8, 9] does not coincide with the operator Z ε (ζ ); the link between both operators is given by the relation 1 Z ε (ζ )ψ = (Gε (ζ )ψ + ε 2 ∂x ζ ∂x ψ + ε 3 ∂y ζ ∂y ψ); 3 1 + ε (∂x ζ )2 + ε 4 (∂y ζ )2 one can of course work interchangeably with one or the other operator. Remarking that ∂i |z=εζ = ∂i ψ − ε∂i ζ Z ε (ζ )ψ, with i = x, y or t, and plugging these relations into (3) yields the following dimensionless version of the equations derived in [8,9,23]: ε3 1 ε ∂t ψ + ζ − |Z ε (ζ )ψ|2 + (∂x ψ)2 + ε(∂y ψ)2 − |Z ε (ζ )ψ|2 (∂x ζ )2 + ε(∂y ζ )2 = 0, 2 2 2 (5) 1 ∂t ζ + ε ∂x ψ∂x ζ + ε∂y ψ∂y ζ − ε 2 Z ε (ζ )ψ (∂x ζ )2 + ε(∂y ζ )2 = Z ε (ζ )ψ, ε which is the formulation of the water-waves problem we work on in this paper.

2856


1.4. Description of the results Proceeding as in [4, 5] we first derive a class of formally equivalent Boussinesq systems for weakly transverse waves, by essentially performing various transformations on the dispersive (linear) part of a reference Boussinesq system in the KP scaling. Next following [6], we perform a nonlinear transform which symmetrizes the nonlinear part, keeping the same order of approximation. We then restrict somehow this (formally equivalent) class of fully symmetric systems by eliminating those which are not linearly well-posed and (using Padé approximants) by choosing those whose dispersions have ‘good’ approximation properties. In the next section, we derive rigorously the uncoupled KP approximation from the weakly transverse Boussinesq systems obtained in the previous sections. We also prove a convergence theorem for the aforementioned completely symmetric systems. The last section is devoted to convergence results for the water-waves system. As advocated above we obtain a better approximation for the weakly transverse Boussinesq systems (convergence rate of order O(ε 2 (t)) than for the uncoupled KP system (convergence rate o(1)) confirming the advantage of the former system. Notations – We use the generic notation C for any numerical constant; when we want to stress out the dependence of a constant on some parameters λ1 , λ2 , . . ., we write C(λ1 , λ2 , . . .). – We denote by S the strip R2x,y × (−1, 0)z . – For all 1 p ∞, the usual Lp (R2 )-norm is denoted by | · |p , and the usual Lp (S )-norm by · p . – For all f ∈ L2 (R2 ), we denote by F f or fits Fourier transform and by F −1 f its inverse Fourier transform. – We use the classical notation H s (R2 ) for Sobolev spaces over R2 . – For all s ∈ R, we define the space H s,0 = H s,0 (S ) by 0 1/2 s 2 s Htg (S ) := {ϕ ∈ D (S ), ϕ Htg := |ϕ(·, ·, z)|H s (R2 ) dz < ∞}. −1

– The unit vertical vector is written ez . 2. Boussinesq systems for weakly transverse water-waves 2.1. Derivation of a weakly transverse Boussinesq system We derive in this section a Boussinesq system which formally describes the asymptotic behaviour of the solutions to (5). This is achieved via the asymptotic expansion of the operator Z ε (ζ )· in terms of ε provided by the following proposition (see also, for instance, [6–9] for rigorous expansions of Dirichlet–Neumann operators). Proposition 1. Let k ∈ N and ζ ∈ W k+4,∞ (R2 ). Then for all ψ such that ∇x,y ψ ∈ H k+8 (R2 ), one has for all ε > 0 small enough: ε Z (ζ )ψ − εZ1 + ε 2 Z2 + ε 3 Z3 k+1/2 ε4 C(|ζ |W k+4,∞ )|∇x,y ψ|H k+8 , H

with Z1 := −∂x2 ψ, Z2 := − 13 ∂x4 ψ + ζ ∂x2 ψ + ∂y2 ψ , Z3 := − ζ ∂y2 ψ + 23 ∂x2 ∂y2 ψ + ζ ∂x4 ψ + 2∂x ζ ∂x3 ψ + ∂x2 ψ∂x2 ζ +

2 6 ∂ ψ 15 x

.


2857

Remark 2. For small amplitude waves, it is possible to make an asymptotic expansion of the operator Z ε (ζ )· in terms of the corresponding operator for unperturbed surfaces Z ε (0)· (see [8, 9]); expanding the operator Z ε (0)· in terms of ε would lead in the end to the same result as proposition 1. One could also keep the full operator Z ε (0)· instead of expanding it: this is the approach used in [1,15] and which leads to models with full dispersion; this leads to a higher computational cost (the operator Z ε (0)· being non-local) and, more specifically, this method is not adapted to our present purpose of justifying rigorously the KP approximation. Proof. First recall that if solves ε∂x2 + ε 2 ∂y2 + ∂z2 = f on the fluid domain, then we y, z) = (x, y, (1 + εζ )z + εζ ) solves the following know by lemma 2.5 of [12] that (x, elliptic equation on the flat strip S = R2 × (−1, 0): = 0, ∇ · P ε (ζ )∇ with

  P ε (ζ ) := 

(6)

ε(1 + εζ ) 0

0 2 ε (1 + εζ )

−ε2 (1 + z)∂x ζ

−ε 3 (1 + z)∂y ζ

 −ε 2 (1 + z)∂x ζ −ε 3 (1 + z)∂y ζ  . 1+ε 3 (1+z)2 (∂x ζ )2 1+εζ

One checks easily that P ε (ζ ) is coercive and that for all X = (X1 , X2 , X3 )T ∈ R3 , one has X · P ε (ζ )X c0 (|ζ |W 1,∞ ) εX12 + ε 2 X22 + X32 , (7) for some c0 (|ζ |W 1,∞ ) > 0 independent of ε. We have the following elliptic estimates. app satisfy Lemma 1. Let p ∈ N∗ , k ∈ N and ζ ∈ W k+2,∞ (R2 ). Let app = εp R ε −∇ · P ε (ζ )∇ app |z=0 = ψ, app |z=−1 = 0, ∂z with (R ε )ε bounded in Htgk+1 (S ). Then one has, for all ε > 0 small enough, ε Z (ζ )ψ −

1 app )|z=0 k+1/2 εp C(|ζ |W k+2,∞ ) R ε H k+1 . (∂z H tg 1 + εζ

|z=0 . It follows that Proof. By definition, one has Z ε (ζ )ψ = (∂z )|z=εζ = (1/(1 + εζ ))∂z ε app )|z=0 = (1/(1 + εζ )∂z ( − app )|z=0 . Writing := app − , Z (ζ )ψ − (1/(1 + εζ ))(∂z it follows from the trace theorem that ε Z (ζ )ψ − 1 (∂z app )|z=0 k+1/2 C(|ζ |W k+1,∞ ) ∂z H k+1 + ∂z2 H k , tg H tg 1 + εζ and we are thus reduced to bound ∂z Htgk+1 and ∂z2 Htgk from above by εp C(|ζ |W k+2,∞ ) R ε Htgk+1 . Remark that for all k ∈ N and 0 l k, the quantity ∂xk−l ∂yl solves the boundary value problem −∇ · P ε (ζ )∇∂xk−l ∂yl = εp ∂xk−l ∂yl R ε + ∇ · [∂xk−l ∂yl , P ε (ζ )]∇ , ∂z (∂xk−l ∂yl ) |z=−1 = 0. ∂xk−l ∂yl |z=0 = 0,

2858


Introducing the notation ϕ 2H˙ 1 := ε ∂x ϕ 22 + ε2 ∂y ϕ 22 + ∂z ϕ 22 , for all functions ϕ defined ε

over the strip R2 × (−1, 0), we now prove that ∀k ∈ N,

sup ∂xk−l ∂yl H˙ ε1 εp C(|ζ |W k+1,∞ ) R ε Htgk ,

(8)

0lk

from which the Htgk+1 -estimate on ∂z follows directly. Multiplying the equation by ∂xk−l ∂yl , integrating by part and using (7) and Poincaré’s inequality yields classically c0 (|ζ |W 1,∞ ) ∂xk−l ∂yl 2H˙ 1 εp ∂xk−l ∂yl R ε 2 ∂xk−l ∂yl H˙ ε1 ε + [∂xk−l ∂yl , P ε (ζ )]∇ , ∇∂xk−l ∂yl L2 ;

(9)

(note that the boundary terms arising in the integration by parts of the commutator term vanish since ∂xk−l ∂yl ez · [∂xk−l ∂yl , P ε (ζ )]∇ = 0 on the upper and lower boundaries of the strip R2 × (−1, 0)). When k = l = 0, the second term of the rhs of (9) vanishes and the result follows easily. When k 1, remark that from the explicit expression of P ε (ζ ) and Poincaré’s inequality, one also gets k−l l ε [∂ ∂ , P (ζ )]∇ , ∇∂ k−l ∂ l 2 x

y

εC(|ζ |W k+1,∞ ) sup

0j k

x y L k−j j ∂x ∂y 2H˙ 1 ε

and one deduces easily from (9) that, for all 0 < ε < c0 (|ζ |W 1,∞ )/C(|ζ |W k+1,∞ ), sup ∂xk−l ∂yl H˙ ε1 εp C(|ζ |W k+1,∞ ) R ε Htgk ,

0lk

which concludes the proof of (8). To find the required estimate on ∂z2 , we use the equation satisfied by and (8), to get that, for all k ∈ N, ∂z2 Htgk εp C(|ζ |W k+2,∞ ) R ε Htgk+1 , which concludes the proof of the lemma. app to The end of the proof of the proposition consists of constructing an approximate solution (6). This is done via a standard BKW procedure, as in [6]. To avoid too lengthy computations, we only construct an approximation of order O(ε3 ) (that is, p = 3 in lemma 1); higher order app = 2 such 0 + ε 1 + ε2 terms are obtained exactly in the same way. We seek therefore that 0 |z=0 = ψ, j |z=0 = 0 (j = 1, 2) and (∂z 0 ) |z=−1 = 0 (j = 0, 1, 2). Plugging this expression into (6), one obtains app = R0 + εR1 + ε 2 R2 + ε 3 R ε , ∇ · P ε (ζ )∇

(10)

with 0 , R0 = ∂z2 2 0 , R1 = ∂z 1 + (∂x2 − ζ ∂z2 ) 1 + ∂x (ζ ∂x 2 + (∂x2 − ζ ∂z2 ) 0 ) − ∂x ((1 + z)∂x ζ ∂z 0 ) + ∂y2 0 R2 = ∂z2 0 ) + ζ 2 ∂z2 0 , − ∂z ((1 + z)∂x ζ ∂x 1 and 2 in order to cancel R0 , R1 and R2 gives an approximate 0 , so that choosing solution app satisfying the assumptions of lemma 1 with p = 3. One can easily check


2859

0 , 1 and 2 cancel R0 , R1 and R2 and satisfy the boundary that the following values of conditions stated above: 0 = ψ, 2 1 = − z + z ∂x2 ψ, 2 4 2 z3 z z z 4 2 = + − ∂ ψ− + z (2ζ ∂x2 ψ + ∂y2 ψ). 24 6 3 x 2 With such a choice, the rhs of (10) reduces to ε 3 R ε , and it is easy to check that R ε Htgk+1 C(|ζ |W k+3,∞ )|∇x,y ψ|H k+6 . app )|z=0 and to use lemma 1 to obtain the Therefore, one just has to compute explicitly (∂z proposition (as said above, we did not give the details of the computations for the O(ε 3 )-order term of the expansion of Z ε (ζ )ψ; note also that the whole procedure has been implemented and checked with the symbolic computational software MuPad). Replacing Z ε (ζ )ψ by its asymptotic expansion √provided by proposition 1 in the waterwaves equations (5) and setting v = ∂x ψ and w = ε∂y ψ yields ε ∂t ψ + ζ + (v 2 + w 2 ) = O(ε2 ), 2 √ (11) ∂t ζ + ∂x v + ε∂y w + ε v∂x ζ + ζ ∂x v + 13 ∂x3 v +ε3/2 (w∂y ζ + ζ ∂y w + 23 ∂x2 ∂y w) = O(ε 2 ). √ √ √ Remark now that √ ε∂y v 2 = 2v∂x w = √ 2 εv∂y v = v∂x w + εv∂y v and that, similarly, ∂x w 2 = 2w∂x w = 2 εw∂y v = w∂x w + εw∂y v. Differentiating the evolution equation on ψ in the above system with respect to x and y gives therefore the following weakly transverse Boussinesq system 1 ∂t v + ∂x ζ + ε(v∂x v + w∂x w) + ε 3/2 21 w∂y v = O(ε2 ), 2 √ ∂t w + ε∂y ζ + ε(w∂y w + 21 v∂x w) + ε 3/2 21 v∂y v = O(ε2 ), √ ∂t ζ + ∂x v + ε∂y w + ε v∂x ζ + ζ ∂x v + 13 ∂x3 v

(12)

ε3/2 (w∂y ζ + ζ ∂y w + 23 ∂x2 ∂y w) = O(ε2 ). Remark 3. We treat √ w as a O(1) quantity with respect to√ε, which can seem surprising since by definition w = ε∂y ψ, which is formally of order O( ε). Our choice is motivated by the structure of the equations. For instance, for initial data of the form ψ|t=0 = 0, ζ|t=0 = f (y), an explicit solution to the linearization of (11) around the rest state is given by √ √ 1 ψ(t, x, y) = √ (f (y − εt) − f (y + εt)), 2 ε √ √ ζ (t, x, y) = 21 (f (y − εt) + f (y + εt)). √ √ For this solution, ε∂y ψ is of size O(1) and not O( ε). 2.2. A first class of formally equivalent systems In (12), v and w denote approximations of the horizontal components of the velocity field at the free surface. Inspired by [4, 6, 17, 20], we define vθ and wσ as −1 −1 ε ε v, wσ = 1 − (1 − σ 2 )∂x2 w, (13) vθ = 1 − (1 − θ 2 )∂x2 2 2

2860


where θ, σ ∈ [0, 1]. The quantities vθ and wσ are therefore approximations of the horizontal components of the velocity field at height z = −1 + θ (x-axis) and z = −1 + σ (y-axis). As shown below, using these quantities will provide formally equivalent weakly transverse Boussinesq systems, with improved dispersion relation for some of them. Rewriting (12) in terms of vθ and wσ yields therefore the following two parameters (namely, θ and σ ) family of formally equivalent systems 1 1 ∂t vθ + ∂x ζ + ε − (1 − θ 2 )∂x2 ∂t vθ + vθ ∂x vθ + wσ ∂x wσ 2 2 1 +ε3/2 wσ ∂y vθ = O(ε 2 ), 2 1 √ 1 ∂t wσ + ε∂y ζ + ε − (1 − σ 2 )∂x2 ∂t wσ + wσ ∂y wσ + vθ ∂x wσ 2 2 1 +ε3/2 vθ ∂y vθ = O(ε 2 ), 2 2 √ θ 1 3 ∂t ζ + ∂x vθ + ε∂y wσ + ε vθ ∂x ζ + ζ ∂x vθ + − ∂ vθ 2 6 x 2 σ 1 2 +ε3/2 wσ ∂y ζ + ζ ∂y wσ + + ∂x ∂y wσ = O(ε2 ). 2 6

(14)

The same kind of observation made in the derivation of the BBM equation from the KdV equation [2] can be used here to introduce three more parameters. More precisely, one has formally √ ∂x3 vθ = λ∂x3 vθ − (1 − λ)∂x2 ∂t ζ − ε(1 − λ)∂x2 ∂y wσ + O(ε), ∂x2 ∂t vθ = (1 − µ)∂x2 ∂t vθ − µ∂x3 ζ + O(ε), √ ∂x2 ∂t wσ = (1 − ν)∂x2 ∂t wα − εν∂x2 ∂y ζ + O(ε), with λ, µ, ν ∈ R (the introduction of these three parameters gives rise to new Boussinesq systems, some of them having very good dispersive properties as shown in section 2.5). Plugging these relations into (14) gives therefore ∂t vθ + ∂x ζ + ε a∂x3 ζ − b∂x2 ∂t vθ + vθ ∂x vθ + 21 wσ ∂x wσ 1 +ε3/2 wσ ∂y vθ = O(ε 2 ), 2 √ ∂t wσ + ε∂y ζ + ε − e∂x2 ∂t wσ + wσ ∂y wσ + 21 vθ ∂x wσ (15) +ε3/2 f ∂x2 ∂y ζ + 21 vθ ∂y vθ = O(ε2 ), √ ∂t ζ + ∂x vθ + ε∂y wσ + ε vθ ∂x ζ + ζ ∂x vθ + c∂x3 vθ − d∂x2 ∂t ζ +ε3/2 wσ ∂y ζ + ζ ∂y wσ + g∂x2 ∂y wσ = O(ε2 ), with 1 − θ2 1 − θ2 µ, b= (1 − µ), 2 2 2 2 1 θ 1 θ − λ, d= − (1 − λ), c= 2 6 2 6 1 1 e = (1 − σ 2 )(1 − ν), f = (1 − σ 2 )ν, 2 2 2 2 σ θ 1 1 g= − − (1 − λ). + 6 2 6 2 a=

(16) (17) (18) (19)


2861

Notation. Writing p := (θ, σ, λ, µ, ν) ∈ [0, 1]2 ×R3 , we denote by Sp (∂) the operator formed by the lhs of (15) and corresponding to this set of parameters. We denote by S the class of all the operators Sp (∂) obtained for all p ∈ [0, 1]2 × R3 . As in [6], we introduce the following notion of consistency. Definition 1. Let T > 0, ε0 > 0, s > 2 and p := (θ, σ, λ, µ, ν) ∈ [0, 1]2 × R3 . A family (U ε )0 0, (i) There exists a unique solution u± ∈ C([0, T ]; H s (R2 ))∩Lip([0, T ]; H s−3 (R2 )) to the KP equation (49) with initial condition u0± . Moreover, one has u± ∈ L∞ ([0, T ], ∂x H s (R2 )). (ii) The function u0 given by (46) satisfies u0 ∈ L∞ ([0, T ] × R+t ; H s−1 (R2x,y )). ε (iii) One can construct the approximate solution Uapp defined in (33), with U0 (τ, t, x, y) = u+ (τ, x − t, y)e+ + u− (τ, x + t, y)e− , U1 (τ, t, x, y) = u0 (τ, t, x, y)e0 and U2 solving the profile equations (51) and (50). Moreover, e± · Uapp (t, x, y) − u± (εt, x ∓ t, y) = o(1), √ e0 · Uapp (t, x, y) − εu0 (εt, t, x, y) = O(ε) in L∞ ([0, T /ε] × R2x,y ). (iv) If moreover ∂y2 u0± ∈ ∂x2 H s−3 (R2 ) and s > 4, then the o(1) error estimate given in (iii) can √ be improved to o( ε). Proof (i) Existence and uniqueness of the functions u± is provided by classical results on the KP equations (e.g. [21]). The fact that u± ∈ ∂x H s (R2 ) is proved in [19]. (ii) From (46) one deduces easily |u0 (τ, t, ·)|H s−1 (Rd ) C(|u00 |H s−1 + |u+ (τ, ·)|∂x H s + |u− (τ, ·)|∂x H s ), for all τ ∈ [0, τ ] and t 0,√so that the result follows from the first point. (iii) First remark that e0 ·Uapp − εu0 = εe0 ·U2 (εt, t, x, y).From (51), one obtains easily that |e0 · U2 (τ, t, ·)|H s−2 (R2 ) C |u+ (τ, ·)|H s + |u− (τ, ·)|H s , so that the estimate on e0 · Uapp follows from the first point and the Sobolev embedding H s−2 (R2 ) ⊂ L∞ (R2 ). Since e± · Uapp − u± = εe± · U2 (εt, t, x, y), we now seek an estimate on e± · U2 , which is given by (50). We need the following lemma. Lemma 2. Let c, c1 ∈ R, c = c1 and f ∈ H r (R2 ) for some r > 1, and define F (t, x, y) = f (x − c1 t, y). Then (i) If u(t, x, y) solves (∂t + c∂x )u = F , with u(t = 0) = 0, then 1 lim |u(t, ·)|H r (R2 ) = 0. t→∞ t If moreover f ∈ ∂x H r (R2 ), then for all t 0, |u(t, ·)|H r C|f |∂x H r .


(ii) Let also g ∈ H r (R2 ) and G(t, x, y) = g(x − ct, y). (∂t + c∂x )v = F G, with v(t = 0) = 0, then 1 lim √ |v(t, ·)|H r (R2 ) = 0. t→∞ t

2871

Then if v(t, x, y) solves

Proof. The first estimate of (i) is classical (see, e.g. proposition 2.2. of [14]). Remark now that under the assumption f ∈ ∂x H r , one can write f = ∂x f, with f ∈ H r (R2 ); the solution u of the i.v.p. under consideration is then given by 1 u(t, x, y) = f (x − c1 t, y) − f(x − ct, y) , c − c1 so that the second estimate follows. For the proof of (ii), we refer to proposition 3.6 of [14]. From the lemma and (50), it is obvious that for all 0 τ T , |e± · U2 (τ, t, ·)|H s−2 C(|u+ |HTs , |u− |HTs ) + C(|u+ |∂x HTs , |u− |∂x HTs )(t)t √ +C(|u+ |HTs , |u− |HTs )(t) t,

(52)

where (t) denotes a function such that limt→∞ (t) = 0. It is then easy to conclude that εe± · U2 (εt, t, ·) = o(1) in L∞ ([0, T /ε] × R2 ), which concludes the proof of (iii). Before proving (iv), let us prove the following lemma. Lemma 3. If u0± ∈ ∂x H s+1 (R2 ), with s > 3, and if moreover one has ∂y2 u0± ∈ ∂x2 H s−3 (R2 ), then ∂y2 u± ∈ L∞ ([0, T ], ∂x2 H s−3 (R2 )) and |∂y2 u± |∂x2 HTs−3 C(|u± |HTs ). Proof. We must prove that ∂x−2 ∂y2 u± ∈ L∞ ([0, T ], H s−3 (R2 )). Remarking that 1 −1 γ± 2 ∂x−2 ∂y2 u± = ∓ ∂x ∂τ u± + β± ∂x2 u± + u , α± 2 ± it suffices to prove that ∂τ u± ∈ L∞ ([0, T ], ∂x H s−3 ). This can be deduced from the identities (3.9) and (3.10) of [18] under the assumption that ∂y2 u0± ∈ ∂x2 H s−3 (R2 ). Owing to this lemma, we can use the second estimate of point (i) of lemma 2 to control the first terms of the rhs of (50). It follows that one can replace (52) by |e± · U2 (τ, t, ·)|H s−3 C(|u+ |HTs , |u− |HTs ) √ +C(|u+ |HTs , |u− |HTs )(t) t, so that the proof of (iv) follows easily. ε 3.2.4. Proof of theorem 1. We first prove that the approximate solution Uapp (εt, t, x, y) solves the i.v.p. (30) up to a small residual term.

Proposition 6. Let s > 1. Suppose that assumptions 1 and 2 are satisfied and assume moreover that u0± ∈ ∂x H s+7 (R2 ) and ∂y2 u0± ∈ ∂x2 H s+3 (R2 ); assume also that u00 ∈ H s+4 (R2 ). ε Then, for all T > 0, the approximate solution Uapp given in (33) is defined on [0, T /ε]×R2 and one has ε ε P ε (∂, Uapp )Uapp = o(ε) ε and Uapp (t = 0) = U 0,ε .

in

L∞ ([0, T /ε]; H s (R2 )),

2872


Proof. Recall first that U0 = u+ e+ √ + u− e− , U1 = u0 e0 and that we showed in the proof of proposition 5 that U2 (εt, t, ·) = o(1/ ε) in L∞ ([0, T /ε]; H r (R2 )) provided that u0± ∈ ∂x H r+4 and ∂y2 u0± ∈ ∂x2 H r , for any r > 1. Except for the terms ∂τ U1 and ∂τ U2 , it is thus easy to check √ that all the terms appearing in the expansion (39)–(43) and evaluated at τ = εt are o(1/ ε) (or better) in L∞ ([0, T /ε]; H s (R2 )) under the assumptions on the initial data made in the statement of the proposition. To give an estimate on ∂τ U1 (εt, t, ·) and ∂τ U2 (εt, t, ·), we need the following lemma. Lemma 4. One has, for all (τ, t) ∈ [0, T ] × R+ , |∂τ U1 (τ, t, ·)|H s C(|u+ |HTs+4 , |u− |HTs+4 ) and |∂τ U2 (τ, t, ·)|H s C(|u+ |HTs+6 , |u− |HTs+6 )(1 + (t)t), with (t) → 0 when t → ∞. √ Proof. One has ∂τ U1 = ε∂τ u0 e0 , and we recall that u0 is given by (46). Differentiating (46) with respect to τ and using (49) to express ∂τ u± in terms of spatial derivatives of u± , one can express ∂τ u0 in terms of ∂x−2 ∂y3 u± , ∂x2 ∂y u± and ∂y (u2± ), and the first estimate of the lemma follows easily from lemma 3. Similarly, differentiating (51) and (50) with respect to τ and using lemma 2 yields the second estimate. Using the lemma, one gets that ε 3/2 ∂τ U1 (εt, t, ·) = O(ε 3/2 ) and ε2 ∂τ U2 (εt, t, ·) = o(ε) in L∞ ([0, T /ε]; H s (R2 )) and it is thus easy to conclude the proof. Remarking that ε ε Uapp − UKP = (e+ · Uapp (t, x, y) − u+ (εt, x − t, y))e+ ε +(e− · Uapp (t, x, y) − u− (εt, x + t, y))e− √ ε +(e0 · Uapp (t, x, y) − εu0 (εt, x, y))e0 √ + εu0 e0 (t, x, y),

the first assertion of theorem 1(i) follows from proposition 5, while the second assertion is a consequence of proposition 6. Moreover, under assumption 3, the system (30) is completely symmetric and thus wellposed in Sobolev spaces. Denoting Uexact the exact solution of the i.v.p. (30) and using classical arguments as in [3, 10], one can prove that its existence time is at least the same as for the ε ε approximate solution Uapp constructed above (that is, [0, T /ε])) and that Uexact − Uapp = o(1) ∞ 2 ε in L ([0, T /ε] × R ). From the first part of the theorem, one also has Uapp − UKP = o(1) in L∞ ([0, T /ε] × R2 ), so that the second part of the theorem follows easily. 4. Convergence results for the water-waves equations We describe in this section some asymptotic properties of the solutions of the water-waves equations. The first theorem shows that one can describe the solution of the water-waves


2873

equations using any of the systems of the class S introduced in this paper. More precisely, if p = (θ, σ, λ, µ, ν) ∈ [0, 1]2 × R3 and Upε := (vpε , wpε , ζpε ) solves the i.v.p Sp (∂)Upε = 0, vpε | := (1 − 2ε (1 − θ 2 )∂x2 )−1 v 0 , t=0

wpε | ζpε |

t=0

t=0

:= (1 − 2ε (1 − σ 2 )∂x2 )−1 w 0 ,

(53)

:= ζ 0 ,

one can approximate the solution of the water-waves equations in terms of Upε . Theorem 2. Let 0 < ε0 < 1, T > 0, s large enough, and (ψ 0 , ζ 0 ) be such that (∇ψ 0 , ζ 0 ) ∈ H s (R2 )2+1 . Let√also (ψ ε , ζ ε )0 0, s large enough, and (ψ 0 , ζ 0 ) be such that (∇ψ 0 , ζ 0 ) ∈ H s (R2 )2+1 . Let√also (ψ ε , ζ ε )0