Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit

TL;DR

This paper demonstrates that under certain conditions on the inertial range structure functions, weak limits of Navier-Stokes solutions as viscosity vanishes are solutions of the Euler equations, allowing for non-uniqueness and dissipation.

Contribution

It establishes conditions under which weak solutions of Navier-Stokes converge to Euler solutions in the vanishing viscosity limit, including cases with slip boundary conditions.

Findings

Weak limits of Navier-Stokes solutions are Euler solutions under positive structure function exponents.

The result applies to both no-slip and slip boundary conditions.

Non-unique, possibly dissipative Euler solutions can emerge as limits.

Abstract

We prove that if the local second-order structure function exponents in the inertial range remain positive uniformly in viscosity, then any spacetime $L^2$ weak limit of Leray--Hopf weak solutions of the Navier-Stokes equations on any bounded domain $\Omega\subset \mathbb{R}^d$, $d= 2,3$ is a weak solution of the Euler equations. This holds for both no-slip and Navier-friction conditions with viscosity-dependent slip length. The result allows for the emergence of non-unique, possibly dissipative, limiting weak solutions of the Euler equations.