Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics

Shouming Zhou,Li Yang,Chunlai Mu,Zeng Rong

doi:10.3934/dcds.2018230

Abstract

It was showed that the generalized Camassa-Holm equation possible development of singularities in finite time, and beyond the occurrence of wave breaking which exists either global conservative or dissipative solutions. In present paper, we will further investigate the uniqueness of global conservative solutions to it based on the characteristics. From a given conservative solution \begin{document}$u = u(t,x)$ \end{document} , an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities \begin{document}$u$ \end{document} and \begin{document}$v = 2 \arctan u_x$ \end{document} along each characteristic, it is obtained that the Cauchy problem with general initial data \begin{document}$u_0∈ H^1(\mathbb{R})$ \end{document} has a unique global conservative solution.

Highlights

In this paper, we consider the generalized Camassa-Holm equation, ut − uxxt + [g(u)]x = γ(2uxuxx + uuxxx), x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R, (1)2010 Mathematics Subject Classification
Bressan and Constantin showed that after wave breaking the solution of the solution of the Camassa-Holm equation can be continued uniquely as either global conservative or global dissipative solutions(cf.[1, 2])
The aim of this paper is to further prove that, the uniqueness of the global conservative weak solutions of equation (1) by suitable modifying recent result in [3] for the Camassa-Holm equation

Summary

Introduction

We consider the generalized Camassa-Holm equation, ut − uxxt + [g(u)]x = γ(2uxuxx + uuxxx), x ∈ R, t > 0, u(x, 0) = u0(x), x ∈ R,. The generalized Camassa-Holm equation, global conservative solutions, uniqueness. Bressan and Constantin showed that after wave breaking the solution of the solution of the Camassa-Holm equation can be continued uniquely as either global conservative or global dissipative solutions(cf.[1, 2]). The uniqueness of conservation solution to the Camassa-Holm equation was direct proved in [3]. The aim of this paper is to further prove that, the uniqueness of the global conservative weak solutions of equation (1) by suitable modifying recent result in [3] for the Camassa-Holm equation. Let u0 ∈ H1(R), the generalized Camassa-Holm equation (1) has a global conservative solution u = u(t, x). Combing the following Lemma 3.1 with Lemma 3.3, we establish the Lipschitz continuity of x and u as functions of the variables t, β

Lemma the maps

Defining the function

Lemma the map

Gβ uβ

This implies