REGULARITY STRUCTURE OF CONSERVATIVE SOLUTIONS TOTHE HUNTER--SAXTON EQUATION

Yu Gao, Hao Liu, Tak Kwong Wong

Research output: Journal article publicationJournal articleAcademic researchpeer-review

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 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.
Original languageEnglish
Pages (from-to)423-452
Number of pages30
JournalSIAM Journal on Mathematical Analysis
Volume54
Issue number1
DOIs
Publication statusPublished - 13 Jan 2022

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