Damping-Diffusion-Noise Interactions in the Stochastic Camassa-Holm Equation

2024/10/10 by Diego Alonso‐Orán, Diego Alonso-Orán, Peter H. C. Pang +4
Physics and Astronomy · Computer Science · #math.AP

paper · pdf · doi:10.48550/arxiv.2410.07649

Abstract

In this paper we investigate the effects of the interaction between time-inhomogeneous damping, non-local diffusion, and noise on classical solutions to the Camassa-Holm equation incorporating these features. First, a local-in-time theory is established, covering the existence, uniqueness, and a blow-up criterion under relatively general conditions for these interacting mechanisms. Subsequently, we identify different conditions on the interactions between damping, diffusion, and noise that guarantee the global regularity and long-time behaviour of classical solutions. Notably, we demonstrate the existence of an evolution system of measures, which generalizes the concept of invariant measures to the time-inhomogeneous setting.

Citations

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Related

• Global existence of dissipative solutions to the Camassa–Holm equation with transport noise
• On the stochastic Dullin-Gottwald-Holm equation: Global existence and wave-breaking phenomena
• Existence and Blow-up of solutions for Stochastic Modified Two-component Camassa-Holm System
• A comparative numerical study of stochastic Hamiltonian Camassa-Holm equations
• Modulation analysis of the stochastic Camassa-Holm equation with pure jump noise