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 $$L^2_x$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>L</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> . 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.