On the $L^p$ regularity of solutions to the generalized Hunter-Saxton system

TL;DR

This paper investigates the global regularity and finite-time blowup of solutions to the generalized Hunter-Saxton system in various boundary conditions, extending previous results in fluid dynamics models.

Contribution

It extends the analysis of solution regularity for the generalized Hunter-Saxton system to broader parameter ranges and boundary conditions, including finite-time blowup scenarios.

Findings

Global regularity results in $L^p$ spaces for certain parameters.

Finite-time blowup under specific boundary conditions.

Improved understanding of boundary effects on solution behavior.

Abstract

The generalized Hunter-Saxton system comprises several well-known models from fluid dynamics and serves as a tool for the study of fluid convection and stretching in one-dimensional evolution equations. In this work, we examine the global regularity of periodic smooth solutions of this system in $L^p$, $p \in [1,\infty)$, spaces for nonzero real parameters $(\lambda,\kappa)$. Our results significantly improve/extend those by Wunsch et al. [27-29] and Sarria [21]. Furthermore, we study the effects that different boundary conditions have on the global regularity of solutions by replacing periodicity with a homogeneous three-point boundary condition and establish finite-time blowup of a local-in-time solution of the resulting system for particular values of the parameters.