Global existence for the generalized two-component Hunter-Saxton system

TL;DR

This paper investigates the conditions for global existence and blow-up of solutions to a two-component generalized Hunter-Saxton system, providing new results that extend previous research in the field.

Contribution

It establishes new criteria for global solutions and blow-up scenarios for the generalized Hunter-Saxton system with specific parameter choices, improving upon prior results.

Findings

Identified conditions for global existence of solutions.

Derived blow-up scenarios for certain parameter values.

Extended previous results on the Hunter-Saxton system.

Abstract

We study the global existence of solutions to a two-component generalized Hunter-Saxton system in the periodic setting. We first prove a persistence result of the solutions. Then for some particular choices of parameters $(\alpha, \kappa)$, we show the precise blow-up scenarios and the existence of global solutions to the generalized Hunter-Saxton system under proper assumptions on the initial data. This significantly improves recent results obtained in [M. Wunsch, DCDS Ser. B 12 (2009), 647-656] and [M. Wunsch, SIAM J. Math. Anal. 42 (2010), 1286-1304].