Lipschitz metric for conservative solutions of the modified two-component Camassa-Holm system

In this paper, we study the Lipschitz continuous dependence of conservative solutions to a modified two-component Camassa-Holm system. Even for smooth initial data, it is known that the gradient of these solutions can blow up in finite time. When this happens, the flow generated by the equations fails to be Lipschitz continuous w.r.t. the usual distance in the Sobolev space. To cope with this issue, we construct a new distance, generated by a Finsler structure, which renders Lipschitz continuous the semigroup of conservative solutions. Our distance is constructed first on a subset of sufficiently regular solutions, then extended by continuity to the entire space. This is possible thanks to a generic regularity result, which has independent interest. Roughly speaking, we show that, for generic smooth initial data, the solution remains piecewise smooth.