Generic regularity and Lipschitz metric for the Hunter-Saxton type equations

The Hunter-Saxton equation determines a flow of conservative solutions taking values in the space H1(R+). However, the solution typically includes finite time gradient blowups, which make the solution flow not continuous w.r.t. the natural H1 distance. The aim of this paper is to first study the generic properties of conservative solutions of some initial boundary value problems to the Hunter-Saxton type equations. Then using these properties, we give a new way to construct a Finsler type metric which renders the flow uniformly Lipschitz continuous on bounded subsets of H1(R+).