A note on stochastic Navier-Stokes equations with not regular multiplicative noise

TL;DR

This paper investigates stochastic Navier-Stokes equations with irregular multiplicative noise, establishing the existence of weak solutions in two and three dimensions and proving pathwise uniqueness in two dimensions.

Contribution

It introduces methods to handle stochastic Navier-Stokes equations with non-regular noise where Itô calculus is not applicable, proving existence and uniqueness results.

Findings

Existence of weak solutions in 2D and 3D

Pathwise uniqueness in 2D

Handling irregular noise without Itô calculus

Abstract

We consider the Navier-Stokes equations in $\mathbb R^d$ ($d=2,3$) with a stochastic forcing term which is white noise in time and coloured in space; the spatial covariance of the noise is not too regular, so It\^o calculus cannot be applied in the space of finite energy vector fields. We prove existence of weak solutions for $d=2,3$ and pathwise uniqueness for $d=2$.