Boundary layers interactions in the plane parallel incompressible flows

TL;DR

This paper investigates the complex interactions between boundary and transition layers in inviscid limits of incompressible flows with boundaries and discontinuities, providing a rigorous mathematical analysis for plane parallel flows.

Contribution

It initiates a mathematical study of boundary and transition layer interactions in inviscid limits, proving strong convergence for plane parallel flows.

Findings

Established strong convergence in the inviscid limit.

Analyzed boundary and transition layer interactions.

Extended understanding of flow behavior near boundaries and discontinuities.

Abstract

We study the inviscid limit problem of the incompressible flows in the presence of both impermeable regular boundaries and a hypersurface transversal to the boundary across which the inviscid flow has a discontinuity jump. In the former case, boundary layers have been introduced by Prandtl as correctors near the boundary between the inviscid and viscous flows. In the latter case, the viscosity smoothes out the discontinuity jump by creating a transition layer which has the same amplitude and thickness as the Prandtl layer. In the neighborhood of the intersection of the impermeable boundary and of the hypersurface, interactions between the boundary and the transition layers must then be considered. In this paper, we initiate a mathematical study of this interaction and carry out a strong convergence in the inviscid limit for the case of the plane parallel flows introduced by Di Perna and Majda in \cite{DM}.