Regularity Structure of Conservative Solutions to the Hunter--Saxton Equation

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