Stochastic constrained Navier–Stokes equations on [formula omitted

Abstract

We study two dimensional Navier–Stokes equations driven by a multiplicative Gaussian noise in the Stratonovich form along with a constraint on the L2-norm of the solution. In the deterministic setting [5], it was shown that the global solution exists only on a two dimensional torus and hence we focus on such a case here. The existence of a martingale solution is shown. Moreover, the pathwise uniqueness of the solution is proved by using Schmalfuss idea [24], concluding the existence of a strong solution via a Yamada–Watanabe type result from Ondreját [21].