Orbital stability of solitary waves for a shallow water equation

Physica D 157 (2001) 75–89

Orbital stability of solitary waves for a shallow water equation Adrian Constantin a,∗ , Luc Molinet b a

b

Department of Mathematics, Lund University, Box 118, S-22100 Lund, Sweden Département de Mathématiques, Institut Galilée, Université Paris 13, 99 Avenue Jean-Baptiste Clément, 93430 Villetaneuse, France Received 10 May 2000; received in revised form 13 March 2001; accepted 18 May 2001 Communicated by C.K.R.T. Jones

Abstract We prove the orbital stability of the solitary waves for a shallow water equation by means of variational methods, considering a minimization problem with an appropriate constraint. © 2001 Published by Elsevier Science B.V. Keywords: Orbital stability; Solitary waves; Shallow water equation; Variational methods; Minimization problem

1. Introduction 1.1. The shallow water equation There are several classical models describing the motion of waves at the free surface of shallow water under the influence of gravity. In shallow water, as depth is assumed to be small compared to the horizontal scale, the vertical velocity component is very small; further, imposing a linear relation between elevation and fluid velocity, the propagation of the modeled waves is restricted in one direction and the obtained evolution equation is one-dimensional. Among these models, we find the celebrated Korteweg–de Vries (KdV) equation [21] ut + 6uux + uxxx = 0,

u(0, x) = u0 (x),

t > 0, x ∈ R.

(1.1)

Here and below u(t, x) represents the wave height above a flat bottom, x is proportional to distance in the direction of propagation, and t is proportional to elapsed time. Eq. (1.1) admits solitary wave solutions, i.e. solutions of the form u(t, x) = φ(x − ct) which travel with fixed speed c (traveling waves), and that vanish at infinity. The solitary waves of the KdV equation retain their individuality under interaction and eventually emerge with their original shapes and speeds (see [15]); for this reason they are called solitons. Moreover, KdV is an integrable infinite-dimensional Hamiltonian system (see [23]). However, the KdV equation does not model the occurrence of breaking for water waves (under wave breaking we understand, cf. [29], the phenomenon that a wave remains bounded, but its slope becomes unbounded in finite time). Indeed, as soon as the initial profile u0 ∈ H 1 (R), 1 the solutions of (1.1) are global in time (cf. [20]), whereas (cf. [29]), some shallow water waves break. ∗ Corresponding author. E-mail address: [email protected] (A. Constantin). 1 H k (R), k ∈ N, stands for the Sobolev space of functions with derivatives up to order k having finite L2 (R) norm.

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An alternative model for KdV is the regularized long-wave equation [3] ut + ux + uux − uxxt = 0,

u(0, x) = u0 (x),

t > 0, x ∈ R.

(1.2)

Eq. (1.2) has better analytical properties than the KdV model but it is not integrable and numerical work suggests that its traveling waves are not solitons (see [14]). As any smooth initial profile with nice decay properties develops into a solution of permanent form for (1.2) cf. [3], the regularized long-wave equation does not model wave breaking. Whitham [29] suggested the equation
u(0, x) = u0 (x), t > 0, x ∈ R (1.3) ut + uux + k0 (x − ξ )ux (t, ξ ) dξ = 0, R

 with the singular kernel k0 (x) = (1/2π) R ((tanh ξ )/ξ )1/2 eiξ x dξ , as another alternative model for shallow water waves and conjectured that it describes the effect of breaking of waves (for a proof, we refer to [9,26]). The numerical calculations carried out for the Whitham equation (1.3) do not support the hypothesis that soliton interaction occurs for its traveling waves (cf. [14]). It is intriguing to know if all these properties like solitons, integrability and breaking waves, may be found in a single equation. Camassa and Holm [4] proposed 2 the following equation: ut − utxx + 3uux = 2ux uxx + uuxxx ,

u(0, x) = u0 (x),

t > 0, x ∈ R.

(1.4)

Camassa and Holm [4] gave a physical derivation for (1.4), obtained the associated isospectral problem and found that the equation has solitary waves that interact like solitons. Numerical simulations [5] support their results. The study of the associated isospectral problem proves the integrability of (1.4) in the periodic case for a large class of initial data [10]. Results on the existence of global solutions and wave breaking 3 were obtained in [8,9], while the existence of weak solutions is addressed in [30,31]. Concerning integrability, let us mention that (cf. [17]) Eq. (1.4) provides a counter-example to the long-lasting Painlevé conjecture on the complete integrability of nonlinear partial differential equations [1]. Let us also point out that, corresponding to the Lagrangian formulation, 4 Eq. (1.4) is a re-expression of the geodesic flow in the group of compressible diffeomorphisms of the line [24]. The connection of the shallow water flow with infinite-dimensional geometry resembles the fact that the Euler equation of hydrodynamics is an expression of the geodesic flow in the group of incompressible diffeomorphisms of R3 [2], and can be used to study for Eq. (1.4) the existence of global solutions as well as the phenomenon of breaking waves (for which the exact blow-up rate can be determined and, in some cases, the blow-up set), cf. [7]. The plentitude of structures tied into the shallow water equation (1.4) explains the many interesting (and physically relevant) phenomena modeled by it. The equation reconciles the properties which were known for different shallow water models. 1.2. Solitary waves for the shallow water equation Camassa and Holm [4] found that the solitary waves for Eq. (1.4) are given by uc (x, t) = c e−|x−ct| , 2

x ∈ R,

(1.5)

Eq. (1.4) and its bi-Hamiltonian structure were written down earlier by Fokas and Fuchssteiner [16]. A formal proof that (1.4) has solutions which blow-up in finite time can be found in [5]; in [7], it is proved that the only way singularities can develop in a classical solution of (1.4) is in the form of breaking waves. 4 The spatial (Eulerian) equation (1.4) describes the water motion from the viewpoint of a fixed observer. In material (Lagrangian) coordinates, one describes the motion as seen from one of the particles of the fluid (the observer follows the fluid). 3

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where c > 0 is the speed. They discovered that these peaked solitary waves 5 are solitons. The solitons are not classical solutions of the shallow water equation, they have to be understood (see [31]) as weak solutions of this equation due to the fact that one can write (1.4) as a conservation law ut + uux + ∂x p ∗ [u2 + 21 u2x ] = 0,

(1.6)

where ∗ stands for convolution with respect to the spatial variable x ∈ R and p(x) :=

1 −|x| , 2e

x ∈ R.

(1.7)

In the form (1.6), one can observe a similarity with (1.1) and (1.3). The existence and stability of solitary waves is one of the fundamental qualitative properties of the solutions of nonlinear wave equations (cf. [27]). Due to the fact that the solitons hardly interact with each other at all, it is reasonable to expect that they are stable. It is the aim of this paper to prove the stability of the solitary waves for the shallow water equation (1.4) by using a variational approach: we show that the solitary waves are minima of the energy with an appropriate constraint, deducing from this their stability. We would like to point out that the stability of the solitons was proved by an entirely different approach in [11]. The present method has the advantage of ensuring that (1.4) satisfies the physical principle asserting that ground states are stable. 6 This paper is organized as follows. In Section 2, we sketch the steps in our proof of the stability of the solitary waves: for the full details in the justification of the several delicate technical points, see Section 3. For convenience, we collect in Appendix A some results on the shallow water equation that are used throughout the paper.

2. Outline of the variational approach For the shallow water equation (1.4), we have conservation of energy [4], i.e. if u is a solution of the equation, then the quantity
1 [u2 (t, x) + u2x (t, x)] dx 2 R is preserved as long as the solution exists. If it is present, a state of lowest energy (with an appropriate constraint) is called a ground state. A standard principle in physics asserts that ground states are stable. This consideration suggests to characterize the solitary waves as ground states and to use a variational approach to prove their stability. Before giving the exact definition, let us briefly explain what kind of stability is to be expected. Eq. (1.4) is a wave equation so that its solutions tend to be oscillations which spread out spatially in a quite complicated way and therefore in general for two nearby initial profiles the resulting solutions do not remain nearby for all future times. 7 The spreading can be counteracted if we deal with special solutions where the wave is concentrated so that it locally forms peaks. Also, for a typical nonlinear wave, the phase, speed, and amplitude are coupled. If we define the orbit of a function ϕ to be the set O(ϕ) = {ϕ(· + x0 ), x0 ∈ R}, a solitary wave of (1.4) is called orbitally stable if profiles near its orbit remain at all later times near the orbit. To make this precise, consider the following definition. 5

Interestingly (cf. [29]), the solitary waves which are found in shallow water have a symmetrical peaking of the crests with a finite angle there. We should also point out that the approach used in [11] is peculiar in that it is apparently limited to this special equation. 7 This can be easily seen in the case of Eq. (1.4) by looking at the time evolution of two solitary waves which initially have their peaks located at the same point. At time zero their distance, measured in L∞ (R), equals the difference of the speeds but as time elapses the distance gets close to the largest of the two speeds as the amplitude of a solitary wave (1.5) equals to its speed (a picture helps). 6

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Definition. Let ϕc be the profile of the solitary wave of (1.4) traveling with speed c > 0. Then ϕc is orbitally stable if for all ε > 0, there is a δ > 0 such that for any u0 ∈ H 3 (R) with u0 − ϕc H 1 (R) ≤ δ and such that u0 − u0,xx is a positive integrable function with integral less than 4c, the corresponding solution u(t) of (1.4) with u(0) = u0 satisfies sup inf u(t) − ϕc (· − r)H 1 (R) ≤ ε. t≥0 r∈R

In order to explain the definition, we have to clarify some points. First of all, the form of the partial differential equation (1.4) suggests to define a strong solution as a function in C([0, T ]; H 3 (R))∩C 1 ([0, T ]; H 2 (R)) satisfying (1.4) in C([0, T ]; L2 (R)) for some T > 0 (we deal with a quasilinear hyperbolic equation, cf. [8]). For an initial profile u0 ∈ H 3 (R), there exists a maximal interval on which the strong solution of (1.4) with initial data u(0) = u0 is defined (see Appendix A). Moreover, if in addition u0 − u0,xx ≥ 0 on R, the corresponding solution is global in time and if u0 ∈ H 4 (R) we deal with a classical solution, i.e. a C 1 -function of time and C 3 -function of the spatial variable satisfying (1.4) pointwise. Orbital stability of the solitary wave ϕc means that any solution starting close to ϕc remains close to some translate of the latter at any later time. But the solitary waves of (1.4) are peaked waves and therefore not strong solutions. At time zero ϕc − ∂x2 ϕc = 2cδ (here δ is the Dirac measure) so that the hypothesis on the function u0 − u0,xx arises quite naturally if we are interested in classical initial profiles approximating ϕc , as L1 (R) is isometrically embedded in M(R), the space of Radon measures of finite total mass. The standard approach to stability problems is to linearize, as it is commonly assumed that stability is governed by the linearized equation. If the evolution equation is weakly nonlinear, this idea is conclusive for orbital stability of the solitary waves for a large variety of equations (see [27]). A different approach 8 consists in proving directly that the solitary waves are minima of constrained energy, provided that this minimum is unique up to translations. Without uniqueness, one only gets the stability of the set of minima. To clarify whether the ground states are unique up to translations or not is a delicate matter and for this reason some results on orbital stability concern the stability of the set of ground states 9 and not the stability of the ground states themselves (see [12,13]). In the present model (1.4), the nonlinearity plays a dominant role rather than being a higher-order correction. For this reason, we expect that in our case the passage from the linear to the nonlinear theory is not an easy task, and may even be false. Our approach is to prove directly that the solitary waves are minima of the energy with an appropriate constraint, deducing from this their stability. We consider the variational problem   (u2 + u2x ) 4 R  I := inf . (2.1) = 3 2 3 u∈H 1 (R) R (u + uux ) In Section 3, we will prove that the above minimization problem has, up to translation, a unique solution ϕ(x) = e−|x| , x ∈ R. We call ground states the family of translates of the function ϕ, x → e−|x−x0 | for some x0 ∈ R, as they are the states of lowest energy. If the ground state ϕ is not orbitally stable, there are initial profiles un0 ∈ H 3 (R) satisfying
un0 − un0,xx ≥ 0 on R with (un0 − un0,xx ) ≤ 4, R

that approximate ϕ in H 1 (R) such that for the corresponding global solutions un (t) of (1.4), we can find positive

8 9

Used for the ground states of the nonlinear Schrödinger equation [6]. Set which might be larger than the family of translates of a solitary wave.

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times tn ≥ 0 with inf un (tn , x) − ϕ(x − r)H 1 (R) ≥ ε,

r∈R

n≥1

(2.2)

for some ε > 0. Note that (see Theorem A.1)

F (u) := u3 + uu2x , E(u) := u2 + u2x , R

u ∈ H 1 (R)

R

(2.3)

are conservation laws for the shallow water equation (1.4). From un0 → ϕ in H 1 (R), we deduce that F (un (tn , ·)) = F (un0 ) → F (ϕ) = 43 , while E(un (tn , ·)) = E(un0 ) → E(ϕ) = 2. The above relations imply that the sequence vn (x) := αn un (tn , x),

x∈R

with


3αn3

R

([un (tn , x)]3 + un (tn , x)[unx (tn , x)]2 ) dx = 4,

(2.4)

defined for n ≥ 1, is a minimizing sequence for our variational problem (2.1). Using the principle of concentrated 2 } compactness, we prove that there is a subsequence {vn3k + vnk vn2k ,x } of {vn3 + vn vn,x n≥1 and suitable points {xnk } in order that the translated measures ρnk := {vn3k (· + xnk ) + vnk (· + xnk )vn2k ,x (· + xnk )}

(2.5)

are such that for every > 0, each measure ρnk has mass less than outside some fixed ball centered at zero. As {wnk } := {vnk (· + xnk )} is also a minimizing sequence for the variational problem (2.1), we can extract a subsequence of it converging weakly in H 1 (R) to some function w ∈ H 1 (R). The information on the concentration of the measures ρnk combined with the method of compensation-compactness enables us to show that w satisfies the constraint. This forces the subsequence to converge actually strongly to w in H 1 (R) and w is a solution of our variational problem. Therefore w is a ground state. This is in contradiction with relation (2.2) and hence the ground state ϕ must be orbitally stable. The full details and the justification of the technical points in the above approach are given in Section 3. We considered the solitary wave (x, t) → e−|x−t| for convenience. The same method (modulo an appropriate rescaling of the variational problem) applies to the solitary wave (x, t) → c e−|x−ct| , c > 0.

3. Stability of the solitary waves In this section, we prove (the precise definition of orbital stability of a solitary wave for (1.4) is given in Section 2) the following theorem. Theorem 3.1. The solitary waves of the shallow water equation (1.4) are orbitally stable. For the reader’s convenience, we first recall the principle of concentration-compactness and the results from the theory of compensated compactness that will be needed as well as a lower semicontinuity result for variational integrals suited for our approach.

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Lemma 3.2  (Concentration-compactness [22]). Suppose {µn }n≥0 is a sequence of probability measures on R : µn ≥ 0, R dµn = 1. There is a subsequence {µnk } such that one of the following three conditions holds: 1. (Concentration) There exists a sequence {xnk } ⊂ R such that for any ε > 0, there is a radius R > 0 with the property that
dµnk (x) ≥ 1 − ε |x−xnk |
for all nk . 2. (Vanishing) For all R > 0, there holds 
 lim sup dµnk (y) = 0. nk →∞

x∈R |y−x|
3. (Dichotomy) There exists a number λ ∈ (0, 1) such that for any ε > 0, there is a number R > 0 and a sequence {xnk } with the following property: given R  > R there are non-negative measures µ1nk , µ2nk , such that 0 ≤ µ1nk + µ2nk ≤ µnk , supp(µ1nk ) ⊂ {|x − xnk | < R},    

    lim sup λ − dµ1nk  + (1 − λ) − dµ2nk  ≤ ε. n →∞ k

R

supp(µ2nk ) ⊂ {|x − xnk | > R  },

R

The theory of compensated compactness enables one to prove continuity of nonlinear expressions under weak convergence if appropriate conditions hold (see [18,28]). We will need a specific result in this direction. Before stating the result, we remind some definitions. Given q > 1, let −1 Wq,loc (R) = {v ∈ D (R) : ηv ∈ Wq−1 (R) for all η ∈ D(R)},

where D(R) are the test functions, D (R) the space of all distributions, for r ≥ 1, Wr1 (R) the space of all Lr (R)-functions with distributional derivative in Lr (R), and for q > 1, Wq−1 (R) the set of distributions w which can be written in the form w = w0 + ∂x w1 where w0 , w1 ∈ Lq (R) (the dual of Wr1 (R) where (1/r) + 1/q = 1). The weak topology on the space of measures M(R) is defined as follows: µn * µ iff R η dµn → R η dµ for all continuous η : R → R with compact support. Lemma 3.3 (Hörmander [18]). Let {un }n≥1 and {vn }n≥1 be sequences of functions in L2loc (R) such that un * −1 u, vn * v in L2loc (R) (weakly) and dun → du, dvn → dv in W2,loc (R). Then un vn * uv in the weak topology of measures. To verify the hypotheses of Lemma 3.3, the following result of Murat will be useful. −1 Lemma 3.4 (Murat [25]). If E is a bounded subset of M(R) which is also bounded in the space Wq,loc (R) for some −1 (R). q > 2, then E is relatively compact in the space W2,loc

A particular case of a classical weak lower semicontinuity theorem of Tonelli is also needed. Lemma 3.5 (Struwe [28]). Let G : R × R → R be a continuous non-negative function with G(u, ·) convex for 1,1 every u ∈ R. Then, if un , u ∈ Wloc (R) and un → u in L1 (Ω), un,x * ux weakly in L1 (Ω) for all bounded Ω ⊂ R, it follows that





R

G(u, ux ) dx ≤ lim inf n→∞

R

A. Constantin, L. Molinet / Physica D 157 (2001) 75–89

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G(un , un,x ) dx.

Our first result is as follows. Proposition 3.6. The solutions of the variational problem (2.1) are exactly all translates of ϕ, where ϕ(x) := e−|x| , x ∈ R. Proof. Recall the notation introduced in (2.3). To f ∈ H 1 (R), we associate the L2 (R)-function  f (x) − fx (x), x < x0 , g(x) := f (x) + fx (x), x ≥ x0 ,

(3.1)

where x0 ∈ R is some point where the function f takes its maximal value. A straightforward calculation shows that

(f 2 (x) + fx2 (x)) dx = 2f 2 (x0 ) + g 2 (x) dx, R

R




R

(f

3

(x) + f (x)fx2 (x)) dx


4 3 = f (x0 ) + f (x)g 2 (x) dx. 3 R

(3.2)

Let us first show that if an H 1 (R)-function f satisfying the constraint is such that f (x0 ) < 1, then E(f ) > 2. Choose ε ∈ (0, 1) such that f (x0 ) = 1 − ε (note that ε ≥ 1 forces f to be non-positive and as such it cannot satisfy the constraint). As F (f ) = 43 ,
4 f (x)g 2 (x) dx = 4ε(1 − ε) + ε 3 , 3 R by the second relation in (3.2). But f (x0 ) is a maximum, so that

f (x)g 2 (x) dx ≤ (1 − ε) g 2 (x) dx R

R

and from the above calculation, we obtain
4ε 3 4ε + ≤ g 2 (x) dx. 3(1 − ε) R From this, f 2 (x0 ) = (1 − ε)2 and the first identity in (3.2), we infer E(f ) ≥ 2 + 2ε2 +

4ε 3 > 2. 3(1 − ε)

As E(ϕ) = 2, this proves that in order to solve the variational problem (2.1), it is enough to restrict ourselves to the case f (x0 ) ≥ 1. Assuming that there exists such a function f with E(f ) < 2, we obtain a contradiction with the first relation in (3.2). Therefore I = 2 and ϕ is a solution of (2.1). If f is another minimizing function we must have f (x0 ) ≥ 1 by the above discussion. Then the first relation in (3.2) forces g ≡ 0 and f (x0 ) = 1, as I = 2. But this implies f (x) = e−|x−x0 | , x ∈ R, and the proof is complete. 䊐 We have now all necessary means to proceed with the following proof.

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Proof of Theorem 3.1. Assuming instability, we will show that this leads to a contradiction. If ϕ is not orbitally stable, there is an ε > 0 and a sequence {un0 }n≥1 in H 3 (R) with
un0 − un0,xx ≥ 0 on R, (un0 − un0,xx ) ≤ 4, R

and un0 → ϕ

in

H 1 (R),

such that for the corresponding global solutions un (t) of (1.4), we can find positive times tn ≥ 0 with inf un (tn , x) − ϕ(x − r)H 1 (R) ≥ ε,

r∈R

n ≥ 1.

Since un0 → ϕ in H 1 (R), we have E(un0 ) → E(ϕ) = 2 as n → ∞ and lim E(un (tn , ·)) = 2

n→∞

as E is a conservation law for Eq. (1.4), cf. Theorem A.1. From un0 → ϕ in H 1 (R), we infer un0 → ϕ in L∞ (R) and ((un0 )2 + (∂x un0 )2 ) → ϕ 2 + ϕx2 in L1 (R) so that F (un0 ) → F (ϕ) = 43 . The fact that F is also a conservation law for (1.4), cf. Theorem A.1, implies now that lim F (un (tn , ·)) = 43 .

n→∞

Defining the sequence {vn }n≥1 of H 1 (R)-functions as in (2.4), the previous relations read F (vn ) = 43 ,

n ≥ 1,

(3.3)

E(vn ) → 2,

n ≥ 1,

(3.4)

and

so that {vn }n≥1 is a minimizing sequence for the variational problem (2.1). Using relation (A.2) and Lemma A.3(a) from Appendix A, we deduce that all un (tn , ·) and therefore all vn are non-negative functions. We proceed now in several steps toward the conclusion. Step 1. There are suitable points xnk ∈ R such that for any > 0 there is a radius R > 0 with the property that
dρnk (x) ≥ 1 − , |x|
where ρnk are measures defined by (2.5). We will prove this claim by using the concentration-compactness principle for the probability measures µn = 3 3 2 4 (vn + vn vn,x ), n ≥ 1. 1. Let us first show that vanishing is not possible. Otherwise, we would have 
 lim sup dµnk (y) = 0 n→∞

x∈R |x−y|<1/2

for some subsequence {µnk }.

(3.5)

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Note that for f ∈ H 1 (R) non-negative, 1 2




x+1/2 x−1/2


(f 3 + f f 2x ) ≥

x+1/2

x−1/2

1 f 2 |fx | ≥ 3



 sup

|x−y|<1/2

{f 3 (y)} −

inf

|x−y|<1/2

{f 3 (y)} ,

and 1 2 so that





x+1/2 x−1/2

x+1/2

x−1/2

(f 3 + f f 2x ) ≥

(f 3 + f f 2x ) ≥

1 inf {f 3 (y)}, 2 |x−y|<1/2

1 sup {f 3 (y)}. 3 |x−y|<1/2

The above inequality yields that for all x ∈ R,

3 dµnk (y) = (v 3 + vnk vn2k ,x ) dy 4 |x−y|<1/2 nk |x−y|<1/2 
1/3
3 2 ≤3 (vnk + vnk vnk ,x ) dy |x−y|<1/2

|x−y|<1/2

(vn2k + vn2k ,x ) dy.

Sum over x ∈ Z to get
1=

R


dµnk ≤ 3 sup

1/3


x∈Z |x−y|<1/2

dµnk (y)

R

(vn2k + vn2k ,x ) dy.

As {vnk } is a bounded sequence in H 1 (R), being a minimizing sequence for the variational problem (2.1), the right-hand side of the above inequality tends to zero as n → ∞ by (3.5) and we obtained a contradiction. 2. We now rule out dichotomy. Assume that situation of Lemma 3.2(3) holds. Choosing a sequence m → 0, corresponding Rm > 0, and passing to a subsequence of {µn }n≥1 , we can achieve that supp(µ1m ) ⊂ {|x − xm | < Rm }, and

supp(µ2m ) ⊂ {|x − xm | > 2Rm },

   

    1 2   lim sup λ − dµm  + (1 − λ) − dµm  = 0. m→∞ R

R

(3.6)

(3.7)

Moreover, we may suppose that Rm → ∞ as m → ∞. Choose Φ ∈ C 1 (R, [0, 1]) such that Φ(x) = 1 for |x| ≤ 1, Φ(x) = 0 for |x| ≥ 2 and 0 < Φ(x) < 1 for 1 < |x| < 2, with |Φx (x)| +

|Φx (x)| 1 − Φ 2 (x)

≤ K,

|x| > 1

(3.8)

for some K > 0. It suffices to take Φ(x) = 1 − (|x| − 1)6 for x close to ±1 with |x| > 1 (we require (3.8) as below √ 1 − Φ 2 ∈ C 1 (R) is needed).

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A. Constantin, L. Molinet / Physica D 157 (2001) 75–89

Let Φm (x) := Φ((x − xm )/Rm ) and define  1 2 2. = vm Φm , vm = vm 1 − Φm vm A straightforward calculation shows   2

1 2 2 2 (x) ) + E(vm ) − (vm )2 Φm,x − (vm )2 ∂x 1 − Φm . E(vm ) = E(vm R

R

(3.9)

Note that by (3.8) and the way we defined Φm , we have      K 2 (x) ≤ |Φm,x (x)| + ∂x 1 − Φm  R , x ∈ R. m As {vm } is a minimizing sequence for the variational problem (2.1), it must be bounded in L2 (R). But Rm → ∞ for m → ∞ so that the above estimate yields 
 
  2     2 2 2 2 (x)  = 0. lim  (vm ) Φm,x  +  (vm ) ∂x 1 − Φm m→∞  R   R We proved that 1 2 lim E(vm ) = lim (E(vm ) + E(vm )).

m→∞

(3.10)

m→∞

1 , v 2 were chosen such that Note that the non-negative functions vm m 1 vm ≡ vm

on {|x − xm | < Rm },

2 vm ≡ vm

on {|x − xm | > 2Rm },

and recall that 0 ≤ µ1m + µ2m ≤ µm . Using (3.6) and (3.7), we obtain

3 3 xm +Rm 1 3 1 3 1 1 2 1 1 (vm ) + vm (vm,x ) ≥ lim sup (vm ) + vm (vm,x )2 lim sup m→∞ 4 R m→∞ 4 xm −Rm


3 xm +Rm = lim sup (vm )3 + vm (vm,x )2 = lim sup dµm ≥lim sup dµ1m = λ. m→∞ 4 xm −Rm m→∞ |x−xm |2Rm

= lim sup dµm ≥ lim sup dµ2m = 1 − λ. m→∞

|x−xm |>2Rm

m→∞

|x−xm |>2Rm

1 , v 2 ∈ H 1 (R) so that the renormalized functions Clearly vm m

x →

3

1 (x) vm 1) (3/4)F (vm

,

x →

3

2 (x) vm 2) (3/4)F (vm

,

satisfy the constraint of the variational problem (2.1) — recall definition (2.3). Therefore   1 (x) 2 (x) vm vm E

≥ 2, E

≥ 2, 3 3 1) 2) (3/4)F (vm (3/4)F (vm

(3.11)

(3.12)

A. Constantin, L. Molinet / Physica D 157 (2001) 75–89

85

and as the functional E is homogeneous of degree 2, we infer from the above and from (3.11) and (3.12) that for every ε ∈ (0, min{λ, 1 − λ}), we have 1 1 2/3 ) ≥ 2( 43 F (vm )) ≥ 2(λ − ε)2/3 , E(vm

2 2 2/3 E(vm ) ≥ 2( 43 F (vm )) ≥ 2(1 − λ − ε)2/3

for all m large enough. Combining (3.10) with the above two estimates and recalling that {vm } is a minimizing sequence for the variational problem (2.1), we deduce that for all ε > 0 small enough 1 2 ) + E(vm )) ≥ 2((λ − ε)2/3 + (1 − λ − ε)2/3 ). 2 = lim E(vm ) = lim (E(vm m→∞

m→∞

But for 0 < λ < 1, we have λ2/3 + (1 − λ)2/3 > 1. Letting ε → 0 in the previous inequality, we obtain the desired contradiction that rules out dichotomy. As vanishing and dichotomy do not hold, Step 1 is finished. Step 2. There is a subsequence of wnk := vnk (· + xnk ) converging weakly in H 1 (R) to some w satisfying the constraint in (2.1), if the translates are chosen such that the assertion from Step 1 holds. As the above-defined {wnk } is a minimizing sequence for the variational problem (2.1), we can extract a subsequence, denoted for the sake of simplicity again by {wnk }, such that wnk converges weakly in H 1 (R) to some  function w ∈ H 1 (R). Using the method of compensated compactness we will prove that R (w 3 + wwx2 ) = 43 . First of all, observe that, just like the functions vnk , all wnk are non-negative. Applying Lemma 3.5 with  3 x + xy2 , x ≥ 0, y ∈ R, G : R × R → R, G(x, y) = 0, x ≤ 0, y ∈ R, we deduce that

4 (w 3 + wwx2 ) ≤ lim inf (wn3k + wnk wn2k ,x ) = n →∞ 3 k R R

(3.13)

(the hypotheses of Lemma 3.5 can easily be checked by standard Sobolev imbeddings). Let us first prove that wnk (∂x wnk ) * wwx

weakly in L2 (R).

(3.14)

As the sequence {wnk } is bounded in H 1 (R), we deduce that it is also bounded in L∞ (R) and that it converges pointwise a.e. (almost everywhere) to w. Therefore, the sequence {wn2k } is bounded in H 1 (R) and it converges a.e. to w2 , which forces wn2k * w 2 in H 1 (R). This on its turn implies (3.14). We will show now that there are subsequences of {∂x wnk }, {wnk (∂x wnk )}, denoted for simplicity by the same subscripts, such that wnk (∂x wnk )2 → wwx2

in D (R).

(3.15)

Note that wnk = unk (tnk , · + xnk ) and let us denote by µM the total variation of a Radon measure µ on R. Using Lemma A.3, we deduce that ∂x2 wnk M ≤ (1 − ∂x2 )unk (tnk , · + xnk )L1 (R) + unk (tnk , · + xnk )L1 (R)

= (1 − ∂x2 )unk (0, · + xnk )L1 (R) + unk (0, · + xnk )L1 (R) = 2un0 k (· + xnk ) − ∂x2 un0 k (· + xnk )L1 (R) ≤ 8,

recalling the conditions on the initial profiles {un0 }n≥1 (see the beginning of the proof). This means that {∂x2 wnk }

is bounded in M(R).

(3.16)

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A. Constantin, L. Molinet / Physica D 157 (2001) 75–89

Clearly ∂x [wnk (∂x wnk )] = (∂x wnk )2 + wnk (∂x2 wnk ). The L1 (R)-norm of {(∂x wnk )2 } and the L∞ (R)-norm of {wnk } are bounded as {wnk } is a minimizing sequence for (2.1). Together with (3.16), this ensures that {∂x [wnk (∂x wnk )]}

is bounded in M(R).

(3.17)

−1 (R). For this, pick f ∈ D(R) Let us show now that {∂x2 wnk } and {∂x [wnk (∂x wnk )]} are bounded in the space W∞,loc −1 (R) × W 1 (R), we have and g ∈ W11 (R). Then, using the definition of the duality pairing in W∞ 1
f (∂x2 wnk ), g = fg(∂x2 wnk ) R
= − (fx g + fgx )(∂x wnk ) ≤ C1 (f )∂x wnk L∞ (R) gW 1 (R) ≤ C2 (f )gW 1 (R)

R

1

1

using Lemma A.4 and the boundedness of {wnk } in L∞ (R). This shows that {∂x2 wnk }

−1 is bounded in W∞,loc (R).

(3.18)

The boundedness of {∂x wnk } and {wnk } in L∞ (R) (already explained) and a calculation along the same lines as the one we just did ensure that {∂x [wnk (∂x wnk )]}

−1 is bounded in W∞,loc (R).

(3.19)

In view of (3.16)–(3.19) and Lemma 3.4, we have that both sequences {∂x2 wnk } and {∂x [wnk (∂x wnk )]} are relatively −1 compact in the space W2,loc (R). As wnk * w weakly in L2 (R) and (3.14) holds, we deduce relation (3.15) by Lemma 3.3. We already proved that the sequences {wnk } and {∂x wnk } are bounded in L∞ (R) so that {wnk (∂x wnk )2 } is bounded in L2 (R), using the information that {wnk } is a minimizing sequence for our variational problem. We deduce from here that {wnk (∂x wnk )2 } must have a subsequence converging weakly in L2 (R) and relation (3.15) enables us to identify the limit as wwx2 . As wnk * w in H 1 (R), a Sobolev imbedding theorem ensures that wn3k → w 3

in L2loc (R).

In conclusion, we know that wn3k + wnk (∂x wnk )2 * w 3 + wwx2

in L2loc (R).

(3.20)

In view of Step 1 for every ε > 0, there is some Rε > 0 such that
3 (w 3 + wnk wn2k ,x ) > 1 − ε 4 |x| 0 and the fact that all functions wnk are non-negative, we conclude that
4 (w 3 + wwx2 ) ≥ . 3 R

A. Constantin, L. Molinet / Physica D 157 (2001) 75–89

87

Together with (3.13), this shows that in the above relation we actually have equality and we are done with the proof of Step 2. Step 3. The sequence {wnk } found in Step 2 converges strongly to w ∈ H 1 (R) which is a solution of our variational problem. By Lemma 3.5, we have E(w) ≤ lim inf E(wnk ) = 2 nk →∞

with the notation introduced in (2.3). Step 2 guarantees that w ∈ H 1 (R) satisfies the constraint so that w is a solution of (2.1). A simple calculation shows that

2 w − wnk H 1 (R) = E(wnk ) − 2 wwnk − 2 wx wnk ,x + E(w). R

R

w in H 1 (R) together with the fact that {w

But E(w) = 2 and wnk * nk } is a minimizing sequence for our variational problem imply that the right-hand of side of the above identity tends to zero as nk → ∞. Step 4. The contradiction with the assumption of instability. From Step 3 and Proposition 3.6, we know that w is some translate of ϕ. Recalling what wnk stands for, we have lim unk (tnk , · + xnk ) − ϕ(· − r)H 1 (R) = 0

nk →∞

for some r ∈ R. The translation invariance of the H 1 (R)-norm enables us to move xnk to the ground state ϕ in the above relation and in the new form the incompatibility with (2.2) is obvious. 䊐

Acknowledgements We thank Prof. W.A. Strauss for many useful discussions. We are also grateful to Prof. C.K.R.T. Jones and the referees for suggestions and comments.

Appendix A For the reader’s convenience, we summarize briefly some results about Eq. (1.4) that were used in the previous two sections. Associating to a solution of (1.4) the potential y := u − uxx , one can write Eq. (1.4) in the following equivalent form: yt + yx u + 2yux = 0,

y(0, x) = y0 (x),

t > 0, x ∈ R,

a quasilinear evolution equation of hyperbolic type dy + A(y)y = 0 dt

in

L2 (R),

y(0) = y0 ,

where, for y ∈ H 1 (R), A(y) is the operator A(y) := ((1 − ∂x2 )−1 y)∂x + 2((1 − ∂x2 )−1 y)x Id with domain {v ∈ L2 (R) : ((1 − ∂x2 )−1 y)v ∈ H 1 (R)}.

t > 0,

(A.1)

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Eq. (A.1) can be analyzed with Kato’s semigroup approach to the Cauchy problem for quasilinear hyperbolic evolution equations [19] in order to obtain the following well-posedness result. Theorem A.1 (Constantin and Escher [8]). Given an initial data u0 ∈ H 3 (R), there exists a maximal time T = T (u0 ) > 0 so that, on [0, T ), Eq. (1.4) admits a unique solution u = u(·, u0 ) ∈ C([0, T ); H 3 (R)) ∩ C 1 ([0, T ); H 2 (R)). Moreover,

2 2 [u (t, x) + ux (t, x)] dx and [u3 (t, x) + u(t, x)u2x (t, x)] dx R

R

are conserved on [0, T ) and the solution depends continuously on the initial data. Further, if T < ∞, then lim supt↑T u(t)H 3 (R) = ∞. If u0 ∈ H 4 (R), then the solution possesses additional regularity, u ∈ C([0, T ); H 4 (R)) ∩ C 1 ([0, T ); H 3 (R)) and is a classical solution. Regarding the existence of global solutions, we have the following theorem. Theorem A.2 (Constantin and Escher [8]). Assume that u0 ∈ H 3 (R) is such that y0 := u0 − u0,xx ∈ L1 (R) and is non-negative. Then the solution u(t) to (1.4) with initial data u0 exists globally in time. We will also need the following lemma. Lemma A.3 (Constantin and Escher [8]). Let u0 ∈ H 3 (R) and assume that y0 = u0 − u0,xx is non-negative and belongs to L1 (R). If u(t) denotes the corresponding global solution in H 3 (R) to (1.4), we have for all t ≥ 0 that y(t, ·) := u(t, ·) − uxx (t, ·) ≥ 0

R,

on

u(t, ·)L1 (R) = y(t, ·)L1 (R) = y0 L1 (R) .

Lemma A.4. If u0 ∈ H 3 (R) is such that u0 − u0,xx is non-negative and belongs to L1 (R), then u2x (t, x) ≤ u2 (t, x),

t ≥ 0, x ∈ R,

where u(t) denotes the global solution of (1.4) with initial data u0 . Proof. By Lemma A.3, we have that y(t, x) := u(t, x) − uxx (t, x) ≥ 0,

t ≥ 0, x ∈ R.

Note that u(t, x) is given by the convolution u(t, x) = p ∗ y with p(x) := 21 e−|x| , x ∈ R, and therefore 1 u(t, x) = e−x 2




x

1 e y(t, ξ ) dξ + ex 2 −∞ ξ




∞ x

e−ξ y(t, ξ ) dξ,

t ∈ [0, T ), x ∈ R,

(A.2)

from where we infer 1 ux (t, x) = − e−x 2




x

1 eξ y(t, ξ ) dξ + ex 2 −∞




∞ x

e−ξ y(t, ξ ) dξ,

t ∈ [0, T ), x ∈ R.

(A.3)

As the function y(t, ·) is non-negative for all t ≥ 0, the assertion follows by simply comparing the expressions 䊐 obtained by squaring (A.2) and (A.3).

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89

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