Stochastic Camassa-Holm equation with convection type noise

TL;DR

This paper studies a stochastic version of the Camassa-Holm equation with convection noise, proving local strong solutions using operator theory, which could impact the analysis of other nonlinear stochastic PDEs.

Contribution

It introduces a stochastic Camassa-Holm equation with convection noise and establishes existence and uniqueness of solutions via Kato's operator theory.

Findings

Proved local strong solutions exist and are unique.

Transformed the stochastic PDE into a random quasi-linear PDE.

Potential applications to other nonlinear stochastic PDEs.

Abstract

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.