On uniqueness of dissipative solutions of the Camassa-Holm equation

2016/11/01 by Grzegorz Jamróz, Jamróz, Grzegorz
Earth and Planetary Sciences · Mathematics · Physics and Astronomy · #35L65 #37K10 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons #Ocean Waves and Remote Sensing

paper · pdf · doi:10.48550/arxiv.1611.00333

openalex publication_date 2016/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that dissipative weak solutions of the Camassa-Holm equation are unique. Thus we complete the global well-posedness theory of this celebrated model of shallow water, initiated by a general proof of existence in [Z. Xin, P. Zhang Comm. Pure Appl. Math. 53 (2000)]. As the dissipative weak solutions, being viscosity solutions, seem to constitute the most physically relevant class of solutions in the wave-breaking regime, the result provides mathematical rationale for the feasibility of the Camassa-Holm equation as a model of water waves encompassing both soliton interactions and wave-breaking.

Citations

• Global Conservative Solutions of the Camassa–Holm Equation
• Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation
• The Hydrodynamical Relevance of the Camassa–Holm and Degasperis–Procesi Equations

Related

• Uniqueness of Dissipative Solution for Camassa-Holm Equation with Peakon-Antipeakon Initial Data
• On the inconsistency of the Camassa-Holm model with the shallow water theory
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• Existence and uniqueness of the globally conservative solutions for a weakly dissipative Camassa-Holm equation in time weighted H¹(ℝ) space
• Existence and uniqueness of global weak solutions to a generalized Camassa-Holm equation