On initial boundary value problems for variants of the Hunter-Saxton equation

TL;DR

This paper investigates initial boundary value problems for modified and two-component variants of the Hunter-Saxton equation, establishing well-posedness and blow-up results to understand wave behavior in nematic liquid crystals.

Contribution

It introduces and analyzes a modified Hunter-Saxton equation and a two-component system, providing new well-posedness and blow-up results for these variants.

Findings

Established well-posedness for the modified Hunter-Saxton equation.

Proved blow-up phenomena for certain initial boundary conditions.

Extended analysis to a two-component Hunter-Saxton system.

Abstract

The Hunter-Saxton equation serves as a mathematical model for orientation waves in a nematic liquid crystal. The present paper discusses a modified variant of this equation, coming up in the study of critical points for the speed of orientation waves, as well as a two-component extension. We establish well-posedness and blow-up results for some initial boundary value problems for the modified Hunter-Saxton equation and the two-component Hunter-Saxton system.