Global Conservative Solutions of the Camassa–Holm Equation – A Lagrangian Point of View
Helge Holden and Xavier Raynaud
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 μac=(u2+ux2)dx.