Regularity structure of conservative solutions to the Hunter-Saxton equation

Abstract

In this paper we characterize the regularity structure, as well as show the global-intimeexistence and uniqueness, of (energy) conservative solutions to the Hunter--Saxton equation byusing the method of characteristics. The major difference between the current work and previouresults is that we are able to characterize the singularities of energy measure and their nature in avery precise manner. In particular, we show that singularities, whose temporal and spatial locationsare also explicitly given in this work, may only appear at at most countably many times, and arecompletely determined by the absolutely continuous part of initial energy measure. Our mathematicalanalysis is based on using the method of characteristics in a generalized framework that consists ofthe evolutions of solutions to the Hunter--Saxton equation and the energy measure. This methodalso provides a clear description of the semigroup property for the solution and energy measure forall times.