Inviscid Limit for Vortex Patches in A Bounded Domain

TL;DR

This paper proves that vortex patch solutions of the Navier-Stokes equations with Navier boundary conditions converge to Euler vortex patch solutions as viscosity approaches zero, with an almost optimal convergence rate in bounded domains.

Contribution

It establishes the inviscid limit for vortex patches in bounded domains with Navier boundary conditions, extending previous results from the whole space to bounded settings.

Findings

Convergence of Navier-Stokes vortex patches to Euler solutions as viscosity vanishes.

Global in time convergence in 2D and local in time in 3D.

Almost optimal convergence rate of $( u t)^{3/4- ext{small}}$ in $L^2$.

Abstract

In this paper, we consider the inviscid limit of the incompressible Navier-Stokes equations in a smooth, bounded and simply connected domain $\Omega \subset \mathbb{R}^d, d=2,3$. We prove that for a vortex patch initial data the weak Leray solutions of the incompressible Navier-Stokes equations with Navier boundary conditions will converge (locally in time for $d=3$ and globally in time for $d=2$) to a vortex patch solution of the incompressible Euler equation as the viscosity vanishes. In view of the results obtained in [1] and [19] which dealt with the case of the whole space, we derive an almost optimal convergence rate $(\nu t)^{\frac34-\varepsilon}$ for any small $\varepsilon>0$ in $L^2$.