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

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 $d=2$.