Global weak solutions to the Camassa-Holm equation

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August 2008, 21(3): 883-906. doi: 10.3934/dcds.2008.21.883

Global weak solutions to the Camassa-Holm equation

Zhenhua Guo ^1, , Mina Jiang ^2, , Zhian Wang ^3, and Gao-Feng Zheng ^2,

1.	Center for Nonlinear Studies and Department of Mathematics, Northwest University, Xi’an 710069, China
2.	Department of Mathematics, Huazhong Normal University, Wuhan, 430079, China, China
3.	Department of Mathematical and Statistical Sciences, 632CAB, University of Alberta, Edmonton, Alberta T6G 2G1, Canada

Received April 2007 Revised October 2007 Published April 2008

Abstract
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The existence of a global weak solution to the Cauchy problem for a one-dimensional Camassa-Holm equation is established. In this paper, we assume that the initial condition $u_0(x)$ has end states $u_{\pm}$, which has much weaker constraints than that $u_0(x) \in H^1(\mathbb R)$ discussed in [30]. By perturbing the Cauchy problem around a rarefaction wave, we obtain a global weak solution as a limit of viscous approximation under the assumption $u_- < u_+$.

Keywords: vanishing viscosity method, Young measure., Camassa-Holm equation, global weak solution.

Mathematics Subject Classification: Primary: 35D05, 35L6.

Citation: Zhenhua Guo, Mina Jiang, Zhian Wang, Gao-Feng Zheng. Global weak solutions to the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883

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