# American Institute of Mathematical Sciences

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

## Global weak solutions to the Camassa-Holm equation

 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 $u_- < u_+$.
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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