Asymptotic behavior of conservative solutions to the Hunter-Saxton equation

TL;DR

This paper investigates the long-term behavior of conservative solutions to the Hunter-Saxton equation, showing they asymptotically resemble kink-waves determined by total energy, with detailed error estimates and convergence properties.

Contribution

It provides a rigorous analysis of the asymptotic expansion, convergence, and error estimates for conservative solutions of the Hunter-Saxton equation, including the behavior of energy measures.

Findings

Conservative solutions asymptotically approach kink-waves determined by total energy.

Singular part of energy measure converges to zero at infinity.

Explicit error estimates and pointwise growth rates are established.

Abstract

In this paper we study the large time asymptotic behavior of (energy) conservative solutions to the Hunter-Saxton equation in a generalized framework that consists of the evolutions of solution and its energy measure. We describe the large time asymptotic expansions of the conservative solutions, and rigorously verify the validity of the leading order term in $L^{\infty}(\mathbb{R})$ and ${\dot{H}}^1(\mathbb{R})$ spaces respectively. The leading order term is given by a kink-wave that is determined by the total energy of the system only. As a corollary, we also show that the singular part of the energy measure converges to zero, as the time goes to either positive or negative infinity. Under some natural decay rate assumptions on the tails of the initial energy measure, we rigorously provide the optimal error estimates in $L^{\infty}(\mathbb{R})$ and ${\dot{H}}^1(\mathbb{R})$. As the time goes to infinity, the pointwise convergence and pointwise growth rate for the solution are also obtained under the same assumptions on the initial data. The proofs of our results rely heavily on the elaborate analysis of the generalized characteristics designed for the measure-valued initial data, and explicit formulae for conservative solutions.