The Zero Viscosity Limit of Stochastic Navier-Stokes Flows

TL;DR

This paper extends Kato's Criterion to stochastic Navier-Stokes flows, analyzing their convergence to Euler solutions under various noise models and scalings, and establishes existence and uniqueness of solutions in different dimensions.

Contribution

It introduces a stochastic analogue of Kato's Criterion for inviscid limits, covering multiple noise types and scalings, and proves existence and uniqueness of solutions in 2D and 3D.

Findings

The criterion applies to stochastic flows with additive, multiplicative, and transport noise.

Optimality of the $ u^{1/2}$ scaling is established.

Existence of weak solutions in 3D and strong solutions in 2D is proven.

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.