Global Lagrangian solutions of the Camassa-Holm equation

2017/10/16 by Jae Min Lee, Lee, Jae Min
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Algebraic structures and combinatorial models #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.1710.05484

openalex publication_date 2017/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we construct global weak conservative solutions of the Camassa-Holm equation on the periodic domain. We first express the equation in Lagrangian flow variable η and then transform it using a change of variable ρ=√(η_x). The new variable removes the singularity of the CH equation, and we obtain both the global weak conservative solution and global spatial smoothness of the Lagrangian trajectories of the CH equation. This work is motivated by J. Lenells who proved similar results for the Hunter-Saxton equation using the geometric interpretation.

Citations

• Global Conservative Solutions of the Camassa–Holm Equation
• Wave breaking for nonlinear nonlocal shallow water equations

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