November  2009, 24(4): 1047-1112. doi: 10.3934/dcds.2009.24.1047

Dissipative solutions for the Camassa-Holm equation

1. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491 Trondheim

2. 

Department of Mathematical Sciences, Norwegian University of Science and Technology, NO--7491 Trondheim

Received  June 2008 Revised  November 2008 Published  May 2009

We show that the Camassa--Holm equation $u_t-$uxxt+3uux-$2u_xuxx-uuxxx=0 possesses a global continuous semigroup of weak dissipative solutions for initial data $u|_{t=0}$ in $H^1$. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. Stability in terms of $H^1$ and $L^\infty$ norm is discussed.
Citation: Helge Holden, Xavier Raynaud. Dissipative solutions for the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1047-1112. doi: 10.3934/dcds.2009.24.1047
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