The regularity of solution for a generalized Hunter–Saxton type equation

Abstract

The cusp singularity, with only Hölder continuity, is a typical singularity formed in the quasilinear hyperbolic partial differential equations, such as the Hunter–Saxton and Camassa–Holm equations. We establish the global existence of Hölder continuous energy conservative weak solution for a family of Hunter–Saxton type equations, where the regularity of solution varies with respect to a parameter. This result can help us predict regularity of cusp singularity for many other models.