Global well-posedness of the viscous Camassa--Holm equation with gradient noise

TL;DR

This paper proves the global existence and uniqueness of weak solutions for a stochastic viscous Camassa--Holm equation with spatially dependent noise, using Galerkin methods, energy estimates, and stochastic analysis techniques.

Contribution

It establishes the first rigorous existence and uniqueness results for this class of stochastic PDEs with gradient noise and viscosity, extending the mathematical understanding of stochastic shallow water models.

Findings

Existence of weak solutions in H^m spaces.

Pathwise uniqueness of solutions.

Handling of nonlinear noise terms and Stratonovich-to-Itô correction.

Abstract

We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa--Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in $H^m$ ($m\in\mathbb{N}$) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod--Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-It\^{o} correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional It\^{o} formula and a DiPerna--Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.