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 Ito 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 \(d=2\).