Global Conservative Solutions of the Camassa–Holm Equation—A Lagrangian Point of View

Abstract

We show that the Camassa–Holm equation ut − uxxt + 3uux − 2uxuxx − uuxxx = 0 possesses a global continuous semigroup of weak conservative solutions for initial data u|t=0 in H1. The result is obtained by introducing a coordinate transformation into Lagrangian coordinates. To characterize conservative solutions it is necessary to include the energy density given by the positive Radon measure μ with . The total energy is preserved by the solution.