On the orbital stability of the Degasperis-Procesi antipeakon-peakon profile

Abstract

In this paper, we prove an orbital stability result for the Degasperis-Procesi peakon with respect to perturbations having a momentum density that is first negative and then positive. This leads to the orbital stability of the antipeakon-peakon profile with respect to such perturbations.

Introduction

In this paper, we consider the Degasperis-Procesi equation (DP) first derived in [5], usually written as

$$\{\begin{array}{c}{u}_t-u_{txx}+4u{u}_x=3u_x{u}_{xx}+u{u}_{xxx},(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R},\hfill \\ u(0,x)=u_0(x),x\in \mathbb{R}.\hfill \end{array}$$

The DP equation has been proved to be physically relevant for water waves (see [2], [12]) as an asymptotic shallow-water approximation to the Euler equations in some specific regime. It shares a lot of properties with the famous Camassa-Holm equation (CH) that reads (see [1])

$$u_t-u_{txx}=-3u{u}_x+2u_x{u}_{xx}+u{u}_{xxx},(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R}.$$

In particular, it has a bi-hamiltonian structure, it is completely integrable (see [6]) and has got the same explicit peaked solitary waves. These solitary waves are called peakons whenever $c>0$ and antipeakons whenever $c<0$ and are defined by

$$u(t,x)={\phi }_c(x-ct)=c\phi (x-ct)=c{e}^{-|x-ct|},c\in {\mathbb{R}}^{⁎},(t,x)\in {\mathbb{R}}^2.$$

Note that to give a sense to these solutions one has to apply ${(1-{\partial }_x^2)}^{-1}$ to (1), to rewrite it under the form

$$u_t+\frac{1}{2}{\partial }_x(u^2)+\frac{3}{2}{(1-{\partial }_x^2)}^{-1}{\partial }_x(u^2)=0,(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R}.$$

However, in contrast with the CH equation, the DP equation has also shock peaked waves (see for instance [14]) which are given by

$$u(t,x)=-\frac{1}{t+k}\text{sgn}(x)e^{-|x|},k>0(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R}.$$

Another important difference between the CH and the DP equations is due to the fact that the DP conservations laws permit only to control the $L^2$-norm of the solution whereas the $H^1$-norm is a conserved quantity for the CH equation. In particular, without any supplementary hypotheses, the solutions of the DP equation may be unbounded contrary to the CH-solutions. In this paper we will make use of the three following conservation laws of the DP equation:

$$M(u)=\underset{\mathbb{R}}{\int }y,E(u)=\underset{\mathbb{R}}{\int }yv=\underset{\mathbb{R}}{\int }(4v^2+5v_x^2+v_{xx}^2)$$

$$\text{and}F(u)=\underset{\mathbb{R}}{\int }{u}^3=\underset{\mathbb{R}}{\int }(-v_{xx}^3+12v{v}_{xx}^2-48v^2{v}_{xx}+64v^3),$$

where $y=(1-{\partial }_x^2)u\text{ and }v={(4-{\partial }_x^2)}^{-1}u$.

It is worth noticing that these two variables, the momentum density $y=(1-{\partial }_x^2)u$ and the smooth variable $v={(4-{\partial }_x^2)}^{-1}u$ play a crucial role in the DP dynamic. In the sequel we will often make use of the fact that (1) can be rewritten under the form

$$y_t+u{y}_x+3u_{x}y=0,(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R},$$

which is a transport equations for the momentum density as well as under the form

$$v_t=-{\partial }_x{(1-{\partial }_x^2)}^{-1}{u}^2,(t,x)\in {\mathbb{R}}_{+}\times \mathbb{R}.$$

Note that, in the same way as v is associated with u, we will associate with the peakon profile ${\phi }_c$ the so-called smooth-peakon profile ${\rho }_c$ that is given by

$${\rho }_c={(4-{\partial }_x^2)}^{-1}{\phi }_c=\frac{1}{4}{e}^{-2|\cdot |}⁎{\phi }_c=\frac{c}{3}{e}^{-|\cdot |}-\frac{c}{6}{e}^{-2|\cdot |}\ge 0.$$

In [13] (see also [10] for a great simplification) an orbital stability¹ result is proven for the DP peakon by adapting the approach first developed by Constantin and Strauss [4] for the Camassa-Holm peakon. However, in deep contrast to the Camassa-Holm case, the proof in [13] (and also in [10]) crucially use that the momentum density of the perturbation is non negative. This is absolutely required for instance in [[13], Lemma 3.5] to get the crucial estimate on the auxiliary function h (see Section 5 for the definition of h)). Up to our knowledge, there is no available stability result for the Degasperis-Procesi peakons without this requirement on the momentum density and one of the main contribution of this work is to give a first stability result for the DP peakon with respect to perturbations that do not share this sign requirement. At this stage, it is worth noticing that the global existence of smooth solutions to the DP equation is only known for initial data that have either a momentum density with a constant sign or a momentum density that is first non negative and then non positive.

The first part of this paper is devoted to the proof of a stability result for the peakon with respect to perturbations that belong to this second class of initial data. We would like to emphasize that the key supplementary argument with respect to the case of a non negative momentum density is of a dynamic nature. Inspired by similar considerations for the Camassa-Holm equation contained in [17], we study the dynamic of the momentum density $y(t)$ at the left of a smooth curve $x(t)$ such that $u(t,\cdot )-{\phi }_c(\cdot -x(t))$ remains small for all $t\in [0,T]$ with $T>0$ large enough. This is in deep contrast with the arguments in the case $y\ge 0$ and with the common arguments for orbital stability that are of static nature: They only use the conservation laws together with the continuity of the solution.

In a second time, we combine this stability result with some almost monotony results to get the orbital stability of the DP antipeakon-peakon profile and more generally of trains of antipeakon-peakons.

Before stating our results let us introduce some notations and some function spaces that will appear in the statements. For $p\in [1,+\infty ]$ we denote by $L^p(\mathbb{R})$ the usual Lebesgue spaces endowed with their usual norm ${\Vert \cdot \Vert }_{L^p}$. We notice that by integration by parts, it holds

$${\Vert u(t,\cdot )\Vert }_{L^2}^2=\underset{\mathbb{R}}{\int }{(4v-v_{xx})}^{2}dx=\underset{\mathbb{R}}{\int }(16v^2+8v_x^2+v_{xx}^2)dx$$

and thus

$$E(u)\le {\Vert u\Vert }_{L^2}^2\le 4E(u).$$

Therefore, $E(\cdot )$ is equivalent to ${\Vert \cdot \Vert }_{L^2(\mathbb{R})}^2$ and in the sequel of this paper we set

$${\Vert u\Vert }_{\mathcal{H}}=\sqrt{E(u)}\text{ so that }{\Vert u\Vert }_{\mathcal{H}}\le {\Vert u\Vert }_{L^2}\le 2{\Vert u\Vert }_{\mathcal{H}}$$

As in [3], we will work in the space Y defined by

$$Y:=\{u\in L^2(\mathbb{R})\text{with}u-u_{xx}\in \mathcal{M}(\mathbb{R})\}$$

where $\mathcal{M}(\mathbb{R})$ is the space of finite Radon measure on $\mathbb{R}$ that is endowed with the norm ${\Vert \cdot \Vert }_{\mathcal{M}}$ where

$${\Vert y\Vert }_{\mathcal{M}}:=\underset{\phi \in C(\mathbb{R}),{\Vert \phi \Vert }_{L^{\infty }}\le 1}{\sup }|〈y,\phi 〉|.$$

Hypothesis 1

We will say that $u_0\in Y$ satisfies Hypothesis 1 if there exists $x_0\in \mathbb{R}$ such that its momentum density $y_0=u_0-u_{0,xx}$ satisfies

$$\text{supp }{y}_0^{-}\subset ]-\infty ,x_0]\text{ and }\text{supp }{y}_0^{+}\subset [x_0,+\infty [,$$

where $y_0^{+}$ and $y_0^{-}$ are respectively the positive and the negative part of the Radon measure $y_0$.

Theorem 1 Stability of a single peakon

There exists $0<{\epsilon }_0<1$ such that for any $c>0$, $A>0$ and $0<\epsilon <{\epsilon }_0\frac{1\wedge c^2}{{(2+c)}^3}$, there exists $0<\delta =\delta (A,\epsilon ,c)\le {\epsilon }^4$ such that for any $u_0\in Y$ satisfying Hypothesis 1 with

$${\Vert u_0-{\phi }_c\Vert }_{\mathcal{H}}\le \delta \le {\epsilon }^4$$

and

$${\Vert u_0-u_{0,xx}\Vert }_{\mathcal{M}}\le A,$$

the emanating solution of the DP equation satisfies

$${\Vert u(t,\cdot )-{\phi }_c(\cdot -\xi (t))\Vert }_{\mathcal{H}}\le 2(2+c)\epsilon ,\forall t\in {\mathbb{R}}_{+}$$

and

$${\Vert u(t,\cdot )-{\phi }_c(\cdot -\xi (t))\Vert }_{L^{\infty }}\le 8{(2+c)}^2{\epsilon }^{2/3},\forall t\in {\mathbb{R}}_{+},$$

where $\xi (t)\in \mathbb{R}$ is the only point where the function $v(t,\cdot )={(4-{\partial }_x^2)}^{-1}u(t,\cdot )$ reaches its maximum on $\mathbb{R}$.

Combining the above stability of a single peakon with the general framework first introduced in [16] and more precisely following [7]-[8] we obtain the stability of a train of well-ordered antipeakons and peakons. This contains in particular the stability of the antipeakon-peakon profile.

Theorem 2

Let be given $N_{-}\in {\mathbb{N}}^{⁎}$ negative velocities $c_{-N_{-}}<..<c_{-1}<0$, $N_{+}\in {\mathbb{N}}^{⁎}$ positive velocities $0<c_1<..<c_{N_{+}}$ and $A>0$. There exist $B=B(\overrightarrow{c})>0$, $L_0=L_0(A,\overrightarrow{c})>0$ and $0<{\epsilon }_0={\epsilon }_0(\overrightarrow{c})<1$ such that for any $0<\epsilon <{\epsilon }_0(\overrightarrow{c})$ there exists $0<\delta (\epsilon ,A,\overrightarrow{c})<{\epsilon }^4$ such that if $u\in C({\mathbb{R}}_{+};H^1)$ is the solution of the DP equation emanating from $u_0\in Y$, satisfying Hypothesis 1 with

$${\Vert u_0-u_{0,xx}\Vert }_{\mathcal{M}}\le A,$$

and

$${\Vert u_0-\sum _{\genfrac{}{}{0ex}{}{j=-N_{-}}{j\ne 0}}^{N_{+}}{\phi }_{c_j}(\cdot -z_j^0)\Vert }_{\mathcal{H}}\le \delta \le {\epsilon }^4$$

for some $z_{-N_{-}}^0<..<z_{-1}^0<z_1^0<\cdot \cdot \cdot <z_{N_{+}}^0$ such that

$$z_j^0-z_q^0\ge L\ge L_0,\forall (j,q)\in {([[-N_{-},N_{+}]]\setminus \{0\})}^2,j>q,$$

then there exist $N_{-}+N_{+}$ functions ${\xi }_{-N_{-}}(\cdot ),..,{\xi }_{-1}(\cdot ),{\xi }_1(\cdot ),..,{\xi }_{N_{+}}(\cdot )$ such that

$$\underset{t\in \mathbb{R}+}{\sup }{\Vert u(t,\cdot )-\sum _{\genfrac{}{}{0ex}{}{j=-N_{-}}{j\ne 0}}^{N_{+}}{\phi }_{c_j}(\cdot -{\xi }_j(t))\Vert }_{\mathcal{H}}<B(\epsilon +L^{-1/8})$$

and

$$\underset{t\in \mathbb{R}+}{\sup }{\Vert u(t,\cdot )-\sum _{\genfrac{}{}{0ex}{}{j=-N_{-}}{j\ne 0}}^{N_{+}}{\phi }_{c_j}(\cdot -{\xi }_j(t))\Vert }_{L^{\infty }}\lesssim {\epsilon }^{2/3}+L^{-\frac{1}{12}}.$$

Moreover, for any $t\ge 0$ and $i\in [[1,N_{+}]]$ (resp. $i\in [[-N_{-},-1]])$, ${\xi }_i(t)$ is the only point of maximum (resp. minimum) of $v(t)$ on $[{\xi }_i(t)-L/4,{\xi }_i(t)+L/4]$.