The interaction between rough vortex patch and boundary layer

TL;DR

This paper studies the asymptotic behavior of Navier-Stokes solutions with rough initial vorticity near boundaries, establishing convergence to Euler solutions and analyzing boundary layer interactions at high Reynolds numbers.

Contribution

It introduces a new functional framework to handle rough vortex patches and boundary layer effects, proving convergence in the Yudovich class under high Reynolds number conditions.

Findings

Established $L^p$ convergence of Navier-Stokes to Euler solutions

Developed a Kato-type criterion for rough initial data

Controlled boundary layer interactions with a novel functional approach

Abstract

In this paper, we investigate the asymptotic behavior of solutions to the Navier-Stokes equations in the half-plane under high Reynolds number conditions, where the initial vorticity belongs to the Yudovich class and is supported away from the boundary. We establish the $L^p$ ($2\leq p< \infty$) convergence of solutions from the Navier-Stokes equations to those of the Euler equations. One of the main difficulties stems from the limited regularity of the initial data, which hinders the derivation of an asymptotic expansion. To overcome this challenge, we first prove a Kato-type criterion adapted to the Yudovich class setting. We then obtain uniform estimates for the Navier-Stokes equations -- a non-trivial task due to the strong boundary layer effects. A key component of our approach is the introduction of a suitable functional framework, which enables us to control the interaction between the rough vortex patch and the boundary layer.