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

TL;DR

This paper establishes a criterion for the existence and uniqueness of maximal solutions to 3D stochastic Navier-Stokes equations with various noise structures, extending deterministic results to stochastic fluid dynamics models.

Contribution

It introduces a new criterion for well-posedness of stochastic PDEs, specifically applied to Navier-Stokes equations with stochastic Lie transport, covering multiple noise types.

Findings

Proves existence and uniqueness of maximal solutions for stochastic Navier-Stokes.

Extends deterministic Navier-Stokes results to stochastic setting with SALT.

Provides a framework applicable to various noise structures in fluid models.

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.