2021/06/17 by Yu Gao, Hao Liu, Gao, Yu +3
Mathematics · Physics and Astronomy · #35B65 #37K10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Photonic Systems #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2106.09228
openalex publication_date 2021/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.