Regularity structure of conservative solutions to the Hunter-Saxton equation

TL;DR

This paper characterizes the regularity and singularity structure of conservative solutions to the Hunter-Saxton equation, establishing global existence, uniqueness, and detailed behavior of energy measure singularities over time.

Contribution

It introduces a precise characterization of energy measure singularities and their temporal-spatial locations, advancing understanding of solution regularity for the Hunter-Saxton equation.

Findings

Singularities occur at at most countably many times.

Singularities are determined by the initial energy measure.

The method of characteristics provides a clear semi-group property.

Abstract

In this paper we characterize the regularity structure, as well as show the global-in-time 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 solution to the Hunter-Saxton equation and the energy measure. This method also provides a clear description of the semi-group property for the solution and energy measure for all times.