Lipschitz metric for the periodic Camassa–Holm equation

We study the stability of conservative solutions of the Cauchy problem for the Camassa–Holm equation ut−uxxt+κux+3uux−2uxuxx−uuxxx=0 with periodic 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 and is clarified.