Stochastic Constrained Navier-Stokes Equations on $\mathbb{T}^2$

TL;DR

This paper investigates stochastic constrained Navier-Stokes equations on a 2D torus, proving existence, uniqueness, and strong solutions for the equations driven by multiplicative Gaussian noise.

Contribution

It extends deterministic results to stochastic settings, establishing existence and uniqueness of solutions for constrained Navier-Stokes equations on the 2D torus.

Findings

Existence of martingale solutions

Pathwise uniqueness of solutions

Existence of strong solutions

Abstract

We study constrained 2-dimensional Navier-Stokes Equations driven by a multiplicative Gaussian noise in the Stratonovich form. In the deterministic case [4] we showed the existence of global solutions only on a two dimensional torus and hence we concentrated on such a case here. We prove the existence of a martingale solution and later using Schmalfuss idea [20] we show the pathwise uniqueness of the solutions. We also establish the existence of a strong solution using a Yamada-Watanabe type result from Ondrej\'{a}t [17].