Strong Convergence of Vorticities in the 2D Viscosity Limit on a Bounded Domain

TL;DR

This paper proves strong convergence of vorticities in the 2D viscosity limit on bounded domains, extending previous results by analyzing the evolution of weak convergence defects under bounded p-enstrophies.

Contribution

It advances the understanding of the vanishing viscosity limit by establishing local strong convergence of vorticities for p-enstrophies with p>2, using a novel analysis of weak convergence defects.

Findings

Established local strong convergence of vorticities for p-enstrophies with p>2.

Extended previous interior convergence results to include boundary effects.

Provided a new method analyzing the evolution of weak convergence defects.

Abstract

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the $p$-enstrophies, $p>2$. The key novel idea is the analysis of the evolution of the weak convergence defect.