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

Li Yang; Zeng Rong; Shouming Zhou; Chunlai Mu

doi:10.3934/dcds.2018230

Article Contents

2018, Volume 38, Issue 10: 5205-5220. Doi: 10.3934/dcds.2018230

This issue Previous Article A quantitative shrinking target result on Sturmian sequences for rotations Next Article Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation

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

1.
College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
2.
Department of Mathematics, Southwestern University of Finance and economics, Sichuan 611130, China
3.
College of Mathematics Science, Chongqing Normal University, Chongqing 401331, China
4.
College of Mathematics and statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: Zeng Rong and Shouming Zhou
* Corresponding author: Zeng Rong and Shouming Zhou

Received: February 2018

Published: October 2018

The first author (Yang) is supported by Chongqing Normal University (Grant No. YKC18038). The third author (Zhou) is partly supported by the Science and Technology Research Program of Chongqing Municipal Education Commission, Natural Science Foundation of Chongqing. The fourth author (Mu) is supported by NSFC (Grant No. 11571062 and 11771062), the Basic and Advanced Research Project of CQC-STC (Grant No. cstc2015jcyjBX0007) and the Fundamental Research Funds for the Central Universities(Grant No. 106112016CDJXZ238826).

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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 $u = u(t,x)$ , an equation is introduced to single out a unique characteristic curve through each initial point. By analyzing the evolution of the quantities $u$ and $v = 2 \arctan u_x$ along each characteristic, it is obtained that the Cauchy problem with general initial data $u_0∈ H^1(\mathbb{R})$ has a unique global conservative solution.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. The Lipschitz continuous text function $\varphi^\epsilon$ introduced at (32)

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