Periodic conservative solutions of the Camassa–Holm equation
Helge Holden and Xavier Raynaud
Abstract: We show that the periodic 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 H1per. 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. The total energy is preserved by the solution.