Generic regularity and Lipschitz metric of the global conservative solutions for a Camassa–Holm-type equation with cubic nonlinearity

This paper focuses on a Camassa–Holm-type equation with cubic nonlinearity which can be derived as asymptotic models from shallow-water approximation to the 2D incompressible Euler equations. We would like to address the issue of Lipschitz metric by establishing a Finsler-type distance on the tangent space. By an application of Thom’s transversality theorem, we prove that when the initial data are in an open dense set of [Formula: see text], the solution is piecewise smooth. It is extended that the Lipschitz continuous metric to the general weak solutions with the help of this generic regularity result. Moreover, we compare this Lipschitz metric with other metrics.