Global weak solutions to the Camassa-Holm equation

Zhenhua Guo; Mina Jiang; Zhian Wang; Gao-Feng Zheng

doi:10.3934/dcds.2008.21.883

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2008, Volume 21, Issue 3: 883-906. Doi: 10.3934/dcds.2008.21.883

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Global weak solutions to the Camassa-Holm equation

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

Received: March 31, 2007

Revised: September 30, 2007

Published: August 2008

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Abstract

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 $u_- < u_+$.

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Mathematics Subject Classification: Primary: 35D05, 35L60.

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