Strong convergence of the vorticity for the 2D Euler Equations in the inviscid limit

TL;DR

This paper establishes uniform-in-time convergence of Navier-Stokes vorticity solutions to Euler solutions in the inviscid limit, providing rates of convergence and energy conservation results.

Contribution

It proves strong $L^p$ convergence of vorticity in the inviscid limit and derives convergence rates for solutions with bounded vorticity.

Findings

Uniform-in-time $L^p$ convergence of vorticity

Convergence rates in $L^p$ for bounded vorticity solutions

Energy conservation in the inviscid limit

Abstract

In this paper we prove the uniform-in-time $L^p$ convergence in the inviscid limit of a family $\omega^\nu$ of solutions of the $2D$ Navier-Stokes equations towards a renormalized/Lagrangian solution $\omega$ of the Euler equations. We also prove that, in the class of solutions with bounded vorticity, it is possible to obtain a rate for the convergence of $\omega^\nu$ to $\omega$ in $L^p$. Finally, we show that solutions of the Euler equations with $L^p$ vorticity, obtained in the vanishing viscosity limit, conserve the kinetic energy. The proofs are given by using both a (stochastic) Lagrangian approach and an Eulerian approach.