On the orbital stability of the Degasperis-Procesi antipeakon-peakon profile
Introduction
In this paper, we consider the Degasperis-Procesi equation (DP) first derived in [5], usually written as 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]) 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 and antipeakons whenever and are defined by Note that to give a sense to these solutions one has to apply to (1), to rewrite it under the form However, in contrast with the CH equation, the DP equation has also shock peaked waves (see for instance [14]) which are given by 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 -norm of the solution whereas the -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: where .
It is worth noticing that these two variables, the momentum density and the smooth variable 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 which is a transport equations for the momentum density as well as under the form Note that, in the same way as v is associated with u, we will associate with the peakon profile the so-called smooth-peakon profile that is given by 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 at the left of a smooth curve such that remains small for all with large enough. This is in deep contrast with the arguments in the case 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 we denote by the usual Lebesgue spaces endowed with their usual norm . We notice that by integration by parts, it holds and thus Therefore, is equivalent to and in the sequel of this paper we set As in [3], we will work in the space Y defined by where is the space of finite Radon measure on that is endowed with the norm where Hypothesis 1 We will say that satisfies Hypothesis 1 if there exists such that its momentum density satisfies where and are respectively the positive and the negative part of the Radon measure . Theorem 1 Stability of a single peakon There exists such that for any , and , there exists such that for any satisfying Hypothesis 1 with and the emanating solution of the DP equation satisfies and where is the only point where the function reaches its maximum on . Theorem 2 Let be given negative velocities , positive velocities and . There exist , and such that for any there exists such that if is the solution of the DP equation emanating from , satisfying Hypothesis 1 with and for some such that then there exist functions such that and Moreover, for any and (resp. , is the only point of maximum (resp. minimum) of on .
Section snippets
Global well-posedness results
We first recall some obvious estimates that will be useful in the sequel of this paper. Noticing that satisfies for any we easily get and which ensures that It is also worth noticing that for , satisfying Hypothesis 1, and so that for we get
Some uniform -estimates
In [15] it is proven that as far as the solution to the DP equation stays smooth, its -norm can be bounded by a polynomial function of time with coefficients that depend only on the and -norm of the initial data. In this section we first improve this result under Hypothesis 1 by showing that the solution is then bounded in positive times by a constant that only depends on the -norm of the initial data. This result is not directly needed in our work but we think that it has its own
A dynamic estimate on
In this section we assume that for some and some small enough. Then we can construct a -function such that and we study the behavior of in an growing with time interval at the left of . Lemma 4 There exist , and such that if a solution to (4) satisfies for some and some function , then there exist a unique function
Proof of Theorem 1
Before starting the proof, we need the two following lemmas that will help us to rewrite the problem in a slightly different way. The next lemma ensures that the distance in to the translations of is minimized for any point of maximum of . Lemma 5 For any and , it holds where and ξ is any point where v reaches its maximum.Quadratic identity [13]
We will also need the following lemma that is implicitly contained in [10]. Lemma 6 Let such that
Stability of a train of well-ordered antipeakons-peakons
In this section, we generalize the stability result to the sum of well ordered trains of antipeakons-peakons (see Fig. 1). Let be given ordered speeds with We set where to simplify the notations we set For and and satisfying (117)-(118), we define the following neighborhood of all the sums of well-ordered antipeakons and peakons of speed
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