Lipschitz metric for the periodic second-order Camassa-Holm equation

In this study, we consider the Cauchy problem for the second-order Camassa-Holm equation with periodic initial data u0. Using the vanishing viscosity method, the local weak solution of the equation is obtained in the finite energy space. A continuous semigroup of weak conservative solutions in Lagrangian coordinates is constructed. In particular, for any two solutions u(t) and v(t) of the equation, a Lipschitz metric dD is constructed with the property that dD(u(t),v(t))≤eCtdD(u0,v0).