Local Martingale and Pathwise Solutions for an Abstract Fluids Model

TL;DR

This paper develops an abstract framework for establishing local martingale and pathwise solutions to nonlinear stochastic evolution systems, with applications to 3D primitive equations of oceans and atmosphere under white noise forcing.

Contribution

It introduces a general abstract approach to prove existence and uniqueness of solutions for stochastic fluid models, extending to various related equations and approximations.

Findings

Established local existence and uniqueness of solutions

Applied framework to 3D primitive equations on the {eta}-plane

Framework covers multiple fluid dynamics models

Abstract

We establish the existence and uniqueness of both local martingale and local pathwise solutions of an abstract nonlinear stochastic evolution system. The primary application of this abstract framework is to infer the local existence of strong, pathwise solutions to the 3D primitive equations of the oceans and atmosphere forced by a nonlinear multiplicative white noise. Instead of developing our results specifically for the 3D primitive equations we choose to develop them in a slightly abstract framework which covers many related forms of these equations (atmosphere, oceans, coupled atmosphere-ocean, on the sphere, on the {\beta}-plane approximation etc and the incompressible Navier-Stokes equations). In applications, all of the details are given for the {\beta}-plane approximation of the oceans equations.