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 Stochastic Navier-Stokes Equation on a boundeded domain of R 3 . Inspired by the work of Kato and Lai [3] 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 [4]. Our application to the Incompressible Navier-Stokes equation on a bounded domain in R 3 matches the deterministic theory and represents the first well-posedness result for an SPDE with SALT noise on a bounded domain. This short work summarises the results and announces two papers [1], [2] which give the full details for the abstract well-posedness arguments and application to the Navier-Stokes Equation respectively.

Stochastic Calculus in Infinite Dimensions and SPDEs

“…Another such way that we look to simplify the equations in this framework to apply the standard theory is through taking finite dimensional approximations (which also motivated the previous result!). The scheme of application for this is referred to as a Galerkin Scheme, used traditionally in the analysis for highly non-trivial PDEs such as the Euler and Navier-Stokes Equations (see [11,12]) and indeed is used in the abstract stochastic solution method [1].…”

“…We have alluded quite heavily to the application of this framework for SALT [4] derived SPDEs, which is done for now in [1] and to be expanded upon in [2,3]. In [1] we establish an abstract solution method in the context of the Hilbert Spaces…”

“…Whilst this theory is adequate in such applications, mathematical models for physical phenomena far exceed those for the position of a particle; it is natural to ask the question of how to stochastically perturb an equation modelling (for example) the velocity or temperature of a particle, as a function of space and time. The ways one might do this are vast [7,8] and to this end we address the problem of how to define the stochastic integral t 0 Ψ s dW s (1) for Ψ : Ω × [0, ∞) × R n → R d and W a standard real valued Brownian Motion. The idea is to regard Ψ not as a pointwise defined function but rather an element of a function space, which is our motivating context for stochastic integration of Hilbert Space valued processes.…”

“…Such models are therefore good motivation for us to establish a very general framework for SPDEs, which aren't covered by the standard monographs in the field. In this framework we make the conversion between Stratonovich and Itô equations rigorous, and this framework is returned to for a very strong existence and uniqueness result in [1].…”

“…The framework that we establish facilitates an exceedingly broad class of SPDEs and noise structures, in which we build upon standard SDE theory and rigorously deduce a conversion between the Stratonovich and Itô Forms. The study of Stratonovich equations is largely motivated by the stochastic variational principle of SALT [4] for fluid dynamics, and we discuss an application of the framework to a Navier-Stokes Equation with Stochastic Lie Transport as seen in [1]. Moreover we prove a fundamental existence and uniqueness result (which to the best of our knowledge, is not present in the literature) for SPDEs evolving in a finite dimensional Hilbert Space driven by Cylindrical Brownian Motion, assuming the analogy to the Lipschitz and linear growth conditions for the standard existence and uniqueness theory for SDEs.…”