The Zero Viscosity Limit of Stochastic Navier-Stokes Flows

Abstract: We introduce an analogue to Kato's Criterion regarding the inviscid convergence of stochastic Navier-Stokes flows 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. To enable the analysis we prove the existence of probabilistically weak, analytically weak solutions to a general stochastic Navier-Stokes Equation on a bounded domain with no-slip boundary condition in three spatial dimensions, as well as the existence and uniqueness of probabilistically strong, analytically weak solutions in two dimensions. The criterion applies for these solutions in both two and three dimensions, with some technical simplifications in the 2D case.