Nonlinear Partial Differential Equations

Date: August 30, 2022

Time: 09:00AM - 10:00AM

Location: Zoom

Speaker: Tak Kwong Wong, University of Hong Kong

Title: Regularity structure, global-in-time well-posedness and long-time behavior of energy conservative solutions to the Hunter-Saxton equation

Abstract: The Hunter-Saxton equation is an integrable equation in one spatial dimension, and can be used to study the nonlinear instability in the director field of a nematic liquid. In this talk, we will discuss the regularity structure, global-in-time well-posedness and long-time behavior of energy conservative solutions to the Hunter-Saxton equation. In particular, singularities for the energy measure may only appear at at most countably many times, and are completely determined by the absolutely continuous part of initial energy measure. The temporal and spatial locations of singularities are explicitly determined by the initial data as well. The long-time behavior of energy conservative solution is given by a kink-wave that is determined by the total energy of the system only. The analysis is based on using the method of characteristics in a generalized framework that consists of the evolutions of energy conservative solution and its energy measure. This is a joint work with Yu Gao and Hao Liu.