Existence and uniqueness of maximal solutions to SPDEs with applications to viscous fluid equations

Abstract

We present two criteria for the existence and uniqueness of a maximal strong solution for a general class of stochastic partial differential equations. Each criterion has its corresponding set of assumptions and can be applied to viscous fluid equations with additive, multiplicative or a general transport type noise. In particular, we apply these criteria to demonstrate well-posedness results for the 3D SALT [Stochastic Advection by Lie Transport, (Holm in Proc R Soc A Math Phys Eng Sci 471:20140963, 2015)] Navier–Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively.