Existence and Uniqueness of Maximal Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport

Abstract

We present here a criterion to conclude that an abstract SPDE possesses a unique maximal strong solution, which we apply to a three dimensional Stochastic Navier-Stokes Equation. Motivated by the work of Kato and Lai we ask that there is a comparable result here in the stochastic case whilst facilitating a variety of noise structures such as additive, multiplicative and transport. In particular our criterion is designed to fit viscous fluid dynamics models with Stochastic Advection by Lie Transport (SALT) as introduced in Holm (Proc R Soc A: Math Phys Eng Sci 471(2176):20140963, 2015). Our application to the Incompressible Navier-Stokes equation matches the existence and uniqueness result of the deterministic theory. This short work summarises the results and announces two papers (Crisan et al., Existence and uniqueness of maximal strong solutions to nonlinear SPDEs with applications to viscous fluid models, in preparation; Crisan and Goodair, Analytical properties of a 3D stochastic Navier-Stokes equation, 2022, in preparation) which give the full details for the abstract well-posedness arguments and application to the Navier-Stokes Equation respectively.