Global estimates and blow-up criteria for the generalized Hunter-Saxton system

TL;DR

This paper derives a general formula for periodic solutions of the generalized Hunter-Saxton system, enabling detailed analysis of blow-up phenomena and long-term behavior across various parameter settings.

Contribution

It introduces a universal representation formula for solutions, facilitating comprehensive qualitative analysis of blow-up and asymptotic states for the system.

Findings

Derived a general solution formula valid for all parameters

Analyzed conditions leading to finite-time blow-up

Examined convergence to steady states over time

Abstract

The generalized, two-component Hunter-Saxton system comprises several well-known models of fluid dynamics and serves as a tool for the study of one-dimensional fluid convection and stretching. In this article a general representation formula for periodic solutions to the system, which is valid for arbitrary values of parameters $(\lambda,\kappa)\in\mathbb{R}\times\mathbb{R}$, is derived. This allows us to examine in great detail qualitative properties of blow-up as well as the asymptotic behaviour of solutions, including convergence to steady states in finite or infinite time.