Lipschitz metric for the periodic Camassa–Holm equation
Katrin Grunert, Helge Holden and Xavier Raynaud
Abstract: We study stability of conservative solutions of the Cauchy problem for the periodic Camassa–Holm equation ut−uxxt+3uux−2uxuxx−uuxxx=0 with initial data u0. In particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t),v(t))≤eCtdD(u0,v0). The relationship between this metric and usual norms in H1per and L∞per is clarified.