Global existence for the generalized two-component 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].

22 pages; submitted in modified form on August 7, 2010

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Cited by in corpus (10)

• Geometric Hydrodynamics via Madelung Transform
• Geometric hydrodynamics and infinite-dimensional Newton's equations
• Geometry of the Madelung transform
• Blow-up phenomena and global existence for a periodic two-component Hunter-Saxton system
• On a two-component $\pi$-Camassa--Holm system
• On initial boundary value problems for variants of the Hunter-Saxton equation
• On the weakly dissipative Camassa-Holm, Degasperis-Procesi, and Novikov equations
• Geometric numerical integrators for Hunter-Saxton-like equations
• Multi-soliton solution to the two-component Hunter-Saxton equation
• Global estimates and blow-up criteria for the generalized Hunter-Saxton system