Remarks on high Reynolds numbers hydrodynamics and the inviscid limit

TL;DR

This paper establishes conditions under which weak limits of Navier-Stokes solutions at high Reynolds numbers converge to solutions of the Euler equations, focusing on bounded domains and local enstrophy bounds.

Contribution

It proves that under certain boundedness and scaling conditions, inviscid limits of Navier-Stokes solutions satisfy the Euler equations in bounded domains.

Findings

Weak $L^2$ limits satisfy Euler equations if local enstrophies are bounded.

In 3D, $t$-a.e. weak limits satisfy Euler equations under a scaling property.

Conditions are imposed away from boundaries, allowing for wild solutions.

Abstract

We prove that any weak space-time $L^2$ vanishing viscosity limit of a sequence of strong solutions of Navier-Stokes equations in a bounded domain of ${\mathbb{R}}^2$ satisfies the Euler equation if the solutions' local enstrophies are uniformly bounded. We also prove that $t-a.e.$ weak $L^2$ inviscid limits of solutions of 3D Navier-Stokes equations in bounded domains are weak solutions of the Euler equation if they locally satisfy a scaling property of their second order structure function. The conditions imposed are far away from boundaries, and wild solutions of Euler equations are not a priori excluded in the limit.