On the Cauchy problem for the Hunter-Saxton equation on the line

2020/12/31 by Weikui Ye, Zhaoyang Yin, Ye, Weikui +1 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Photonic Systems #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2012.15429

openalex publication_date 2020/12/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the Cauchy problem for the Hunter-Saxton (HS) equation on the line. Firstly, we establish the local well-posedness for the integral form of the (HS) equation by constructing some special spaces E^s_p,r, which mix Lebesgue spaces and homogeneous Besov spaces. Then we present a global existence result and provide a sufficient condition for strong solutions to blow up in finite time for the equation. Finally, we give the ill-posedness and the unique continuation of the Hunter-Saxton equation.

Citations

• Global Conservative Solutions of the Camassa–Holm Equation
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• Wave breaking for nonlinear nonlocal shallow water equations
• Ill-posedness of the Camassa–Holm and related equations in the critical space

Cited by

• Well-posedness for the dispersive Hunter-Saxton equation

Related

• Long-time asymptotic behavior of the Hunter-Saxton equation
• Lipschitz metric for the Hunter-Saxton equation
• Asymptotic behavior of conservative solutions to the Hunter-Saxton equation
• Regularity structure of conservative solutions to the Hunter-Saxton equation
• Well-posedness for the dispersive Hunter-Saxton equation