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

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