Helge Holden

The Camassa-Holm Equation --- A Survey

Abstract: The Camassa-Holm equation u_t-u_{xxt}+ u_x+3u u_x-2u_x u_{xx}-u u_{xxx}=0 has received considerable attention since its discovery in 1993 thanks to its many intriguing mathematical properties, e.g., it is completely integrable and can derived as a model for water waves. In particular, the solution of the Cauchy problem may bifurcate into two distinct solutions when the solution becomes singular, so-called wave breaking. We review the current understanding of this problem, with emphasis on the Lipschitz stability of the solution of the Cauchy problem. Extensions to a two-component generalization of the Camassa-Holm equation will also be discussed. The talk is based on joint work with X. Raynaud (Sintef) and K. Grunert (Norwegian University of Science and Technology).