On uniqueness of dissipative solutions of the Camassa-Holm equation

TL;DR

This paper proves the uniqueness of dissipative weak solutions for the Camassa-Holm equation, confirming the model's mathematical consistency and physical relevance for shallow water wave phenomena including wave-breaking.

Contribution

It establishes the uniqueness of dissipative solutions, completing the global well-posedness theory for the Camassa-Holm equation.

Findings

Dissipative weak solutions are unique.

Supports the physical relevance of the Camassa-Holm model.

Provides a mathematical foundation for wave-breaking analysis.

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.