Dissipative solutions for the Camassa-Holm equation

Abstract

We show that the Camassa--Holm equation$u_t-$uxxt+3uux-$2u_xuxx-uuxxx=0possesses a global continuous semigroup of weakdissipative solutions for initial data $u|_{t=0}$in $H^1$. The result is obtained by introducing acoordinate transformation into Lagrangiancoordinates. Stability in terms of $H^1$ and $L^\infty$ norm is discussed.