The viscous variational wave equation with transport noise

Abstract

This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal continuity of solutions to this system in $L^2_x$. Martingale solutions are extracted from a two-level Galerkin approximation via the Skorokhod--Jakubowski theorem. We use the apparatus of Dudley maps to streamline this stochastic compactness method, bypassing the usual martingale identification argument. Pathwise uniqueness for the system is established through a renormalisation procedure that involves double commutator estimates and a delicate handling of noise and nonlinear terms. New model-specific commutator estimates are proven.