On the 3D Navier-Stokes Equations with Stochastic Lie Transport

We prove the existence and uniqueness of maximal solutions to the 3D SALT (Stochastic Advection by Lie Transport) Navier-Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively. In particular we demonstrate the efficacy of Goodair et al. (Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations, 2023. Stochastics and Partial Differential Equations: Analysis and Computations, pp.1-64) in showing the well-posedness for both the velocity and vorticity form of the equation, as well as obtaining the first analytically strong existence result for a fluid equation perturbed by Lie transport noise on a bounded domain.