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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

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 and Continuous Dynamical Systems, 2008, 21 (3) : 883-906. doi: 10.3934/dcds.2008.21.883
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