The zero viscosity limit of stochastic Navier–Stokes flows

Abstract

We introduce an analogue to Kato’s Criterion regarding the inviscid convergence of weak solutions of the stochastic Navier–Stokes equations to the strong solution of the deterministic Euler equation. Our assumptions cover additive, multiplicative and transport type noise models. This is achieved firstly for the typical noise scaling of ${\nu }^{\frac{1}{2}}$, before considering a new parameter which approaches zero with viscosity but at a potentially different rate. We determine the implications of this for our criterion and clarify a sense in which the scaling by ${\nu }^{\frac{1}{2}}$ is optimal. The criterion applies in both two and three dimensions, with some technical simplifications in the 2D case.