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 posseses a unique maximal strong solution, which we apply to a three dimensional Stochastic Navier-Stokes Equation. Inspired by the work of [Kato and Lai,1984] in the deterministic setting, we provide 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,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 [Goodair et al, 2022] which give the full details for the abstract well-posedness arguments and application to the Navier-Stokes Equation.

18 pages, conference proceedings from Stochastic Transport in Upper Ocean Dynamics 2021

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