Well-posedness for the dispersive Hunter-Saxton equation

TL;DR

This paper establishes local and global well-posedness for the dispersive Hunter-Saxton equation, a model relevant to nematic liquid crystals, using normal form transformations and frequency envelope techniques.

Contribution

It introduces a novel approach combining normal forms and frequency envelopes to prove well-posedness for a quasilinear dispersive PDE.

Findings

Proved local well-posedness of the dispersive Hunter-Saxton equation.

Established global well-posedness under certain conditions.

Demonstrated continuous dependence on initial data.

Abstract

This article represents a first step towards understanding the well-posedness for the dispersive Hunter-Saxton equation. This problem arises in the study of nematic liquid crystals, and although the equation has formal similarities with the KdV equation, the lack of $L^2$ control gives it a quasilinear character, with only continuous dependence on initial data. Here, we prove the local and global well-posedness of the Cauchy problem using a normal form approach to construct modified energies, and frequency envelopes in order to prove the continuous dependence with respect to the initial data.