The interaction between rough vortex patch and boundary layer

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.