On the vanishing viscosity limit for 2D incompressible flows with unbounded vorticity

TL;DR

This paper proves strong convergence of vorticities in 2D incompressible flows with unbounded initial vorticity during the vanishing viscosity limit, extending previous results to less restrictive initial conditions.

Contribution

It extends the vanishing viscosity limit results to initial vorticities in L^p spaces with p>1, using a simplified DiPerna-Lions renormalization approach.

Findings

Strong convergence of vorticities established for p>1

Extension of previous results from bounded to unbounded vorticity cases

Simplified proof using classical renormalization theory

Abstract

We show strong convergence of the vorticities in the vanishing viscosity limit for the incompressible Navier-Stokes equations on the two-dimensional torus, assuming only that the initial vorticity of the limiting Euler equations is in $L^p$ for some $p>1$. This substantially extends a recent result of Constantin, Drivas and Elgindi, who proved strong convergence in the case $p=\infty$. Our proof, which relies on the classical renormalization theory of DiPerna-Lions, is surprisingly simple.