## Abstract

For the Camassa–Holm equation with *κ*≥0, we first prove that any global solution that is *H*^{1}-localized and moves fast enough to the right decays exponentially in space uniformly with respect to time. We also prove that for *κ*>0, a train of *N* solitary waves, which are sufficiently decoupled, is orbitally stable in *H*^{1}().

## 1. Introduction

We consider the Camassa–Holm (CH) equation(1.1)where *κ*≥0. This equation can be derived as a model for the propagation of unidirectional shallow water waves over a flat bottom by writing the Green–Naghdi equations in Lie–Poisson Hamiltonian form and then making an asymptotic expansion that keeps the Hamiltonian structure (Camassa & Holm 1993; Johnson 2002). Note that the CH equation was also found independently by Dai (1998) as a model for nonlinear waves in cylindrical hyperelastic rods and was, in fact, first discovered by the method of recursive operator by Fokas & Fuchssteiner (1981), as an example of a bi-Hamiltonian equation.

The CH equation is completely integrable (Beals *et al*. 1998; Constantin & McKean 1999; Constantin 2001; Lenells 2002) for *κ*≥0. It possesses, among others, the following invariants:(1.2)and can be written in Hamiltonian form as(1.3)For *κ*>0, solitary waves propagating to the right exist only with speed *c*>2*κ* and, conversely, each such speed determines uniquely the profiles *φ*_{c} of the solitary waves up to translations. In Constantin & Strauss (2002), properties of the solitary waves profile *φ*_{c} are derived as follows.

*φ*_{c}is smooth and positive with an even profile decreasing from its peak height*c*−2*κ*.*φ*_{c}is concave for values in and convex elsewhere.on and(1.4)

Requiring that the profile

*φ*_{c}reaches its maximum at*x*=0,*φ*_{c}converge uniformly on every compact subset of as*κ*tends to zero to the peakon profile*x*↦*c*e^{−|x|}.

The orbital stability of these solitary waves is proved by Constantin & Strauss (2002) by applying the general spectral method developed by Benjamin (1972) and Grillakis *et al*. (1987)

In Martel *et al*. (2002), the stability of the train of *N* solitary waves for the generalized Korteweg–de Vries (gKdV) equation was proved. This was the first result of the stability of *N* solitary waves in the energy space *H*^{1}(). In El Dika & Martel (2004), these arguments were generalized to prove a similar result for the generalized Benjamin–Bona–Mahony (BBM) equation. As one can see in El Dika & Martel (2004), the approach of Martel *et al*. (2002) does not depend on specific calculations for the gKdV equations. It is a general method to prove the orbital stability of the train of *N* solitary waves for nonlinear Hamiltonian dispersive equations. This mainly requires two ingredients that are as follows.

A property of almost monotonicity which says that for a solution close to

*φ*_{c}, the part of the energy travelling at the right of*φ*_{c}(.−*ct*) is almost decreasing in time.A dynamical proof of the stability of the solitary wave.

In this paper, we first show that the CH equation possesses, for all *κ*≥0, this property of almost monotonicity of the energy at the right of an *H*^{1}-localized solution. As a consequence, we establish that all *H*^{1}-localized solutions of the CH equation decay exponentially in space. In the second part, we use the framework developed by Martel *et al*. (2002) and El Dika & Martel (2004) to show the stability of the train of *N* solitary waves of the CH equation for any *κ*>0. Note that even if the limit case *κ*=0 does not enter in this general framework (see Constantin & Strauss (2000) and also Constantin & Molinet (2001)), we will treat this case in a forthcoming paper (El Dika & Molinet submitted).

## 2. Well-posedness result

We will work with solutions that are principally in *H*^{1}(). To give sense to these solutions, equation (1.1) has to be rewritten as(2.1)In Constantin & Molinet (2000) and Danchin (2001)—see also Molinet (2004)—the following existence and uniqueness result is derived.

*Let u*_{0}∈*H*^{1}() *with y*_{0}≔*u*_{0}−*u*_{0,xx}∈(), *then there exists T*=*T*(‖*y*_{0}‖)>0 *and a unique solution*1 *to* *(2.1)**,*(2.2)*with initial data u*_{0}. *The functionals E*(.) *and F*(.) *are constant along the trajectory and if y*_{0} *has a definite sign, then u is global in time*.

However, if the potential changes sign, singularities may appear in the solution in finite time in the form of wave breaking (the wave stays bounded but its slope becomes unbounded; see Constantin & Escher (1998*a*,*b*) and Constantin (2000)). In the sequel, we denote by *Y*([−*T*, *T*]) the function space defined by (2.2).

## 3. Exponential decay of *H*^{1}-localized solutions

We closely follow the proof of the exponential decay of *H*^{1} solutions to the BBM equation derived by El Dika (2005*a*,*b*). Let us first define the notion of *H*^{1}-localized solutions (moving to the right).

We say that an *H*^{1} solution of the CH equation is *H*^{1}-localized if there exists *c*_{1}>0 and a *C*^{1} function *x*(.), satisfying , such that for any *ϵ*>0, there exists *R*_{ϵ}>0 such that for all real *t*,(3.1)

The aim of this section is to prove the following theorem.

*Consider u*∈*Y*(]−∞, ∞[) *an H*^{1}-*localized solution to the CH equation, such that c*_{1}>2*κ*, *then there exists C*>0 *depending only on ϵ*↦*R*_{ϵ}, , *κ*≥0 *and* , *such that for all t*∈ *and x*∈,(3.2)

Note that for the limit case *κ*=0, we obtain that all *H*^{1}-localized solution decays faster that e^{−|x|/K} for any *K*>1. Moreover, since for *κ*=0 the equation is invariant by the change of unknown *u*(*t*, *x*)↦−*u*(−*t*, *x*), theorem 3.1 also holds for *H*^{1}-localized solutions that are moving to the left, i.e. *x*_{t}<−*c*_{1}<0 in (3.1).

Theorem 3.1 mainly follows from the crucial almost monotonicity of the energy at the right of an *H*^{1}-localized solution (see lemma 3.1). First, let us define our localized energy. For a non-negative even function , such that , we setWe easily checked that andAlso,(3.3)For 0<*α*<1, such that (1−*α*)^{2}*c*_{1}>2*κ*, we set(3.4)Note that is close to , where *X*_{0}=*x*_{0}+*αx*(*t*_{0}). Moreover, since *ϕ* is even, one can easily check that(3.5)

*Consider u*∈*Y*(]−∞, ∞[) *an H*^{1}-*localized solution to the CH equation, such that c*_{1}>2*κ*, *then there exists C*>0 *and B*>0 *depending only on ϵ*↦*R*_{ϵ}, , *κ*≥0 *and* 0<*α*<1, *such that for all t*≤*t*_{0}, *x*_{0}>*B and* (3.6)

We assume that *u*∈*C*^{1}(; *H*^{∞}). We indicate how to modify the arguments for *u*∈*Y*(]−∞, ∞[) in remark 3.2. Differentiating with respect to time and setting *y*(*t*)=*x*(*t*)+*x*_{0}+*α*(*x*(*t*_{0})−*x*(*t*)), we get(3.7)In the sequel, to simplify the notation, we write *Ψ*_{K} for *Ψ*_{K}(.−*y*(*t*)). Integrating by parts and using the equation, we infer that *J*(*t*) can also be written as(3.8)According to equation (1.2), separating the terms coming from the linear and the nonlinear parts of *F*′(*u*) in equation (3.8) and integrating by parts, we get(3.9)where the first two terms come from the linear part2 of *F*′(*u*) and the last three ones from the nonlinear part. The first term is already in a suitable form, so let us start by treating the second one. It is actually the most difficult one and we will follow closely (El Dika 2005*a*,*b*). Setting , it holds and integrating by parts, we infer thatTaking advantage of the identity,we observe that setting *c*_{2}=(1−*α*)^{2}*c*_{1}, we haveand thusIt thus follows from (3.3) that for *K*>0, satisfying(3.10)it holds(3.11)From (3.7)–(3.9) and (3.11), we deduce that(3.12)It remains to treat the terms coming from the nonlinear part of *F*′(*u*) in equation (3.9). To estimate *J*_{1}, we divide into two regions relating to the size of |*u*|. For some *B*>0, to be specified later, we write(3.13)Observe that for |*x*−*x*(*t*)|<*B* and *x*_{0}>*B*,and thus(3.14)On the other hand, using the *H*^{1} localization of *u*, one can take *B* large enough, so that(3.15)The term *J*_{2}(*t*) can be estimated in the same way as *J*_{1}(*t*), using that according to equation (3.3) for *K*≥1, . One gets(3.16)It thus remains to estimate *J*_{3}(*t*). For this, we decompose again into two regions relating to the size of |*u*|. First, proceeding as in equation (3.14), we easily check that for any *B*>0,(3.17)since(3.18)Now in the region |*x*−*x*(*t*)|>*B*, noting that *ψ*′ and are non-negative, we get(3.19)On the other hand, from equations (3.3) and (3.18), we infer that for *K*>1,Therefore, for *B* large enough,(3.20)From (3.12), (3.14)–(3.17) and (3.20), we conclude that there exists *C* and *B*>0 only depending on *ϵ*↦*R*_{ϵ}, , *c*_{1}>2*κ*, *α* and *K*, such that for *x*_{0}≥*B*, *K*>1, satisfying (3.10) and *t*≤*t*_{0}, it holds(3.21)Integrating between *t* and *t*_{0}, we deduce that ∀*t*≤*t*_{0},(3.22)

For *u*∈*Y*(]−∞, ∞[), the idea is to use an exterior regularization by a sequence of mollifiers *ρ*_{n}. It is clear that if *u* is *H*^{1}-localized, then *u*_{n}=*ρ*_{n}**u* is also *H*^{1}-localized with the same localizing function *x*↦*x*(*t*) and a radius that converges to *R*_{ϵ} as *n* tends to infinity. Clearly, using (2.1), *u*_{n}∈C^{1}(; *H*^{∞}) and *u*_{n} satisfies (3.7) and (3.8), with an additional commutator term given byBut using lemmas 3 and 4 of Constantin & Molinet (2000), one can easily check that for *u*∈*Y*(]−∞, ∞[), *Λ*_{n}(*t*)→0 uniformly on every bounded set of . Therefore, equation (3.22) holds for *u*_{n} with an additional term *ϵ*_{n}(*t*−*t*_{0}), where . Passing to the limit in *n*, we get (3.22) for *u*∈*Y*(]−∞, ∞[).

Let us show that , which together with (3.22) will clearly lead to(3.23)For *R*_{ϵ}>0, to be specified later, we decompose intoFrom the *H*^{1}-localization hypothesis, for any *ϵ*>0, there exists *R*_{ϵ}>0, such that *I*_{1}(*t*)≤ϵ/2. On the other hand, we observe thatBut *x*_{t}>*c*_{1}>0 obviously implies that *R*_{ϵ}−*x*_{0}−*α*(*x*(*t*_{0})−*x*(*t*))≤*R*_{ϵ}−*x*_{0}−*c*_{1}(*t*_{0}−*t*), which proves our claim since .

Now, it follows from equations (3.5) and (3.23) that for all *t*∈(3.24)The invariance of the CH equation under the transformation (*t*, *x*)↦(−*t*, −*x*) ensures that(3.25)and Sobolev embedding theorem permits to conclude the proof of theorem 3.1.

## 4. Orbital stability of the train of *N* solitons

The aim of this section is to show the orbital stability of the train of *N* solitary waves of different speeds that are arranged in increasing order and sufficiently decoupled. The idea, introduced by Martel *et al*. (2002), is to combine the classical arguments of the stability of one solitary wave with the almost monotonicity of the *N*−1 functionals that measure the energy at the right of the *N*−1 first bumps (counted from left to right) of *u*.

### (a) Statement of the result

*Let there be given N velocities c*_{1}, …, *c*_{N}, *such that* 0<2*κ*<*c*_{1}<*c*_{2}<⋯<*c*_{N}. *There exist γ*_{0}, *A*>0, *L*_{0}>0 *and ϵ*_{0}>0, *such that if u*∈*Y*([0, *T*]) *is a solution of the CH equation satisfying*

(4.1)*for some* 0<*ϵ*<*ϵ*_{0} *and x*_{j}−*x*_{j−1}≥*L*, *with L*>*L*_{0}, *then there exist x*_{1}(*t*), …, *x*_{N}(*t*), *such that*(4.2)

Theorem 4.1 actually holds with *ϵ* instead of in equation (4.2). The proof requires to change a little the height or the speed of the solitary waves at time *t*=0 to impose additional orthogonality conditions (Martel *et al*. 2002; El Dika & Martel 2004). In this paper, in order to be more concise (for the reader's convenience), we will only prove (4.2).

We recall that the CH equation admits *N*-solitons solution (Matsuno 2005; Constantin *et al*. 2006) and that as time passes by, the *N*-solitons solution evolves in such a way that the individual components become ordered according to their height or speed and they move apart. Theorem 4.1 is thus a very interesting intermediate result towards the proof of the stability of the *N*-solitons solution.

Note that the constants *γ*_{0}, *A*, *L*_{0} and *ϵ*_{0} do not depend on *T* in theorem 4.1. In particular, for global solutions, the stability result holds on _{+}, i.e. estimate (4.2) holds for *t*∈[0, +∞[.

For *α*>0 and *L*>0, we define the following neighbourhood of all the sums of *N* solitary waves of speed *c*_{1}, …, *c*_{N} with spatial shifts *x*_{j} that satisfied *x*_{j}−*x*_{j−1}≥*L*,

(4.3)

By the continuity of the map *t*↦*u*(*t*) from [0, *T*] into *H*^{1}(), to prove theorem 4.1, it suffices to prove that there exist *A*>0, *γ*_{0}>0, *ϵ*_{0}>0 and *L*_{0}>0, such that ∀*L*>*L*_{0} and 0<*ϵ*< *ϵ*_{0}, if *u*_{0} satisfies (4.1) and for some 0<*t*_{0}<*T*, on [0, *t*_{0}], then . In the sequel, *σ*_{0}>0 denotes a positive real number that depends on *κ*, , *c*_{1} and *c*_{N}. In particular, we require that(4.4)

### (b) Modulation

We first prove that as long as *u* stays in some neighbourhood *U*(α, *L*/2) of the sum of *N* solitary waves, we can decompose *u* as the sum of *N* modulated solitary waves plus a function *v*(*t*) that remains small in *H*^{1}(), in the following way:

with the *N* following orthogonality relations:

*There exist ϵ*_{0}>0, *α*_{0}>0 *and L*_{0}>0, *such that* ∀*ϵ*∈]0, *ϵ*_{0}[, ∀*L*>*L*_{0} *and* ∀*α*∈]0, *α*_{0}[, *if the solution u*(*t*)∈*U*(α, *L*/2) *on* [0, *t*_{0}] *with u*∈*Y*([0, *T*]) *satisfying* *(4.1)**, then there exist N C*^{1} *functions x*_{i} : [0, *t*_{0}]→, *i*=1, …, *N*, *x*_{j}(*t*)−*x*_{j−1}(*t*)>2*L*/3 *and a constant C*_{0}>0, *such that*(4.5)(4.6)(4.7)

Since by assumption *u*(.)∈*U*(*α*, *L*/2) on [0, *T*], there exist *N C*^{1} functions *z*_{i} : [0, *t*_{0}]→, such that *z*_{i}−*z*_{i−1}>*L*/2 and(4.8)For *Z*=(*z*_{1}, …, *z*_{N})∈^{N} fixed, such that *z*_{i}−*z*_{i−1}>*L*/2, we setFor 0<*α*<*α*_{0}, we define the functionwith*Y* is clearly of class *C*^{1}. For *i*=1, …, *N*,For ∀*j*≠*i*Hence, using the exponential decay of *φ*_{c} and that *z*_{i}−*z*_{i−1}>*L*/2, we infer that for *L*_{0} large enough (recall that *L*>*L*_{0}),(4.9)and for *j*≠*i*,(4.10)We deduce that , where *D* is an invertible diagonal matrix with ‖*D*^{−1}‖≤(*C*_{2})^{−1} and . Note that *C*_{2} depends only on and *L*_{0}. Hence, there exists *L*_{0}>0, such that for *L*>*L*_{0}, is invertible with an inverse matrix of norm smaller than 2 (*C*_{2})^{−1}. From the implicit function theorem, we deduce that there exists *α*_{0}>0 and *C*^{1} functions (*y*_{1},…,*y*_{N}) from *B*(*R*_{Z}, *α*_{0}) to a neighbourhood of (0,…,0), which are uniquely determined, such thatIn particular, there exits *C*_{0}>0, such that if *u*∈*B*(*R*_{Z}, *α*), with 0<*α*≤*α*_{0}, then(4.11)Note that *α*_{0} and *C*_{0} depend only on *c*_{1} and *L*_{0} and not on the point (*z*_{1}, …, *z*_{N}). For *u*∈*B*(*R*_{Z}, *α*_{0}), we set *x*_{i}(*u*)=*z*_{i}+*y*_{i}(*u*). Assuming that *α*_{0}≤*L*_{0}/(8*C*_{0}), (*x*_{1}, …, *x*_{N}) are *C*^{1} functions on *B*(*R*_{Z}, *α*), satisfying(4.12)For *L*≥*L*_{0} and 0<*α*<*α*_{0} to be chosen later, we define the modulation of *u*∈*U*(*α*, *L*/2) in the following way: we cover *U*(*α*, *L*/2) by a finite number of ballwhere *α*_{0}/2≤*ρ*_{0}<*α*_{0}, so that the functions *x*_{j}(*u*) are uniquely determined for . We can thus define the functions *t*↦*x*_{j}(*t*) on [0, *t*_{0}] by setting *x*_{j}(*t*)=*x*_{j}(*u*(*t*)). The orthogonality relations (4.6) are clearly satisfied by construction. On the other hand, by equations (4.1), (4.11), the triangular inequality and the smoothness of *φ*_{c}, it is not too hard to prove equation (4.5). It remains to show (4.7) and the estimate on *x*_{j}(*t*)−*x*_{j−1}(*t*). We setNote that on account of (4.11) and the smoothness of *φ*_{c}, it holds(4.13)Moreover, differentiating equation (4.6) with respect to time, we getand thus(4.14)Substituting *u* by in the equation and using that *R*_{j} satisfieswe infer that *v*(*t*) satisfies3 on [0, *t*_{0}],(4.15)Taking the *L*^{2}-scalar product with ∂_{x}*R*_{i}, integrating by parts, using the decay of *R*_{j} and its derivatives, (4.13), (4.14) and (4.12), we find(4.16)Taking *α*_{0} small enough and *L*_{0} large enough, we get and thus for all 0<*α*<*α*_{0} and *L*≥*L*_{0}>3*C*_{0}*ϵ*, it follows from (4.1), (4.11), (4.16) and (4.4) that(4.17)

### (c) Monotonicity property

In this section we prove the almost monotonicity of functionals that are very close to the energy at the right of the *i*th bump, *i*=1, …, *N*−1 of *u*. The proof is very similar to that of lemma 3.1. However, since we do not try this time to get an optimal decay rate as in theorem 3.1, we change the function *Ψ* a little for convenience. Let *Ψ* be a *C*^{∞} function, such that 0<*Ψ*≤1, *Ψ*′>0 on , |*Ψ*‴|≤10|*Ψ*′| on [−1/2, 1/2],Setting *Ψ*_{K}=Ψ(./*K*), we introduce, for *j*∈{2, …, *N*},where *Ψ*_{j,K}(*x*)=Ψ_{K}(*x*−*y*_{j}(*t*)), with *y*_{j}(*t*)=(*x*_{j−1}(*t*)+*x*_{j}(*t*))/2 for *j*=2, …, *N*. Note that *I*_{j}(*t*) is close to and thus measures the energy at the right of the (*j*−1)th bump of *u*.

*Let u*∈*Y*([0, *T*]) *be a solution of the CH equation, such that u*(*t*)∈*U*(α, *L*/2) *on* [0, *t*_{0}], *where the x*_{j}(.) *are defined in* *§4b*. *There exist α*_{0}>0 *and L*_{0}>0 *depending only on σ*_{0}, *such that if* 0<*α*<*α*_{0} *and L*≥*L*_{0}, *then*(4.18)

We note that thanks to equations (4.7) and (4.4) for all *t*∈[0, *t*_{0}],As in equations (3.7)–(3.9), we deduce that(4.19)where *J*_{1}, *J*_{2} and *J*_{3} are as in equation (3.9) with *Ψ*_{K} replaced by *Ψ*_{j,K}. The second term of equation (4.19) can be treated exactly as in §3 to obtain that for ,We give only the modifications to treat *J*_{1}, since the ones for *J*_{2} and *J*_{3} can be obtained in the same way. Recall thatWe divide into two regions *D*_{j} and withFirst, since from equation (4.17) for ,we get as in equation (3.14),On the other hand, in the region *D*_{j}(*t*), we note thatTherefore, for *α* small enough and *L* large enough, we get as in (3.15),

### (d) Local coercivity

Let us recall that the second differential operator of (*cE*−*F*) around a solitary wave *φ*_{c} is given by(4.20)

*There exist δ*>0, *C*_{δ}>0 *and C*>0 *depending only on c*_{1}>2*κ*, *such that for all c*≥*c*_{1}, *Θ*∈*C*^{2}()>0 *and v*∈*H*^{1}(), *satisfying*(4.21)*and*(4.22)*it holds*(4.23)

We note that(4.24)On the other hand, on account of the results on the spectrum of *H*_{c} derived by Constantin & Strauss (2002), it can be easily seen that there exist *δ*>0 and *C*_{δ}>0, such that if for *c*≥*c*_{1},(4.25)then(4.26)Note also that by easy calculations,(4.27)Recall that according to Constantin & Strauss (2002), *φ*_{c} takes values in [0, *c*−2*κ*] and *φ*_{c}′ takes values in [−*c*,*c*]. Thus, equation (4.23) follows from (4.24)–(4.26) under the hypotheses (4.21) and (4.22).

### (e) Proof of the orbital stability

For some *A*>0 and *γ*_{0}>0, to be specified later, let us assume that on [0, *t*_{0}], with and *L*≥*L*_{0}, so that lemmas 4.1 and 4.2 hold for *u*. Let us recall that we want to prove that we can choose *A* and *γ*_{0}, such that .

We define the function *Φ*_{i}=*Φ*_{i}(*t*) by *Φ*_{1}=1−*Ψ*_{2,K}=1−*Ψ*_{K}(.−*y*_{2}(*t*)), *Φ*_{N}=*Ψ*_{N,K}=*Ψ*_{K}(.−*y*_{N}(*t*)) and for *i*=2, …, *N*−1where *Ψ*_{K} and the *y*_{i}s are defined in §4*c*. We fix *K*>0 large enough, so that *Φ*_{i} satisfies the assumption (4.22) of lemma 4.3 (note that *K* depends only on *κ*, *c*_{1} and *c*_{N}). It is easy to check that ,(4.28)and(4.29)We will use the following localized version of *E* and *F* defined for *i*∈{1, …, *N*}:(4.30)Let us note that the second differential operator of around *v*∈*H*^{1} is given by(4.31)In the sequel, we set *u*=*u*(*t*_{0}), X=(*x*_{1}, …, *x*_{N})=(*x*_{1}(*t*_{0}), …, *x*_{N}(*t*_{0})), and .

For *i*∈{1, …, *N*}, we define the function *w*_{i}∈*H*^{1}() by(4.32)whereNote that *w*_{i} are well defined, since clearly from equations (4.28), (4.29) and (1.4),(4.33)We set *v*=*u*−*R*_{X} and in the sequel we assume that(4.34)since otherwise the desired result is proved with *A*=2 and *γ*_{0}=*σ*_{0}/2.

*Step 1*. Observing that, according to equation (4.13),(4.35)we start by establishing the following estimates on the coefficients *a*_{i}:(4.36)To prove equation (4.36), we first note that by the definitions of *I*_{i} and *E*_{i}, for all *v*∈*H*^{1}(), we have(4.37)On the other hand, by Taylor formula, equations (4.32) and (4.34),(4.38)since from equations (1.4), (4.28) and (4.29), it is easy to check that(4.39)Noting that according to lemma 4.2 and (4.1),and thatwe thus infer from equations (4.37), (4.38) and (4.34) that(4.40)Now, from the definition of *F*_{i}(*u*), Taylor formula and equations (4.28) and (4.29), we have(4.41)(4.42)On the other hand, on account of the conservation of *F*(.),(4.43)Since, by the identity *F*′(*φ*_{c})=c*E*′(*φ*_{c}), equations (4.28) and (4.29),(4.44)we deduce from equations (4.32), (4.34) and (4.42)–(4.44) thatand using Abel's transformation, it follows that(4.45)Combining this last estimate with (4.40), we infer thatand according to equation (4.33), this leads to (4.36). Note that equation (4.32) and (4.36) ensure that(4.46)

*Step 2*. With equations (4.36) and (4.46) in hand, we claim that for *ϵ*_{0} small enough and *L* large enough, *w*_{i}(*t*) satisfies (4.21) with *Θ*=Φ_{i} and . Therefore, according to lemma 4.3 and equation (4.31),(4.47)Indeed, from equation (4.6),Hence, using equations (4.28), (4.36), (4.35) and (4.46),In the same way, by equation (4.32),

*Step 3*. Let us show that there exists *C*>0 independent of *A* and *γ*_{0}, such that for *L*≥*L*_{0}, with *L*_{0} large enough,(4.48)which will conclude the proof of theorem 4.1 by taking *γ*_{0}=*σ*_{0}/2 and . Note that, using Abel's formula, it follows from equation (4.37), lemma 4.2, the conservation of energy and equation (4.39) that(4.49)On the other hand, Taylor's formula, equations (4.32) and (4.44) giveCombining this last relation with equation (4.49), we infer that(4.50)But from equations (4.46), (4.31) and (4.36),Moreover, it is not too hard to check that for *z*∈*H*^{1}(),and thus (4.50) and (4.47) lead to(4.51)But again by the definition of *w*_{i} and equation (4.36), it is easy to check thatTherefore, equations (4.51), (4.36) and (4.43) lead to(4.52)which concludes the proof of theorem 4.1.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

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*W*^{1,1}() is the space of*L*^{1}() functions with derivatives in*L*^{1}() and*BV*() is the space of function with bounded variation.↵Note that these terms vanish in the limit case

*κ*=0.↵Here, again, we assume that

*u*∈*C*([0,*T*];*H*^{∞}()). The result for*u*∈*Y*([0,*T*]) can be obtained in the same way by using an exterior regularization and showing that the commutator term tends to 0 as in remark 3.2.- © 2007 The Royal Society