A note on the Prandtl boundary layers

TL;DR

This paper investigates the ill-posedness of the nonlinear Prandtl equation, demonstrating the invalidity of boundary-layer expansions for non-monotonic flows and establishing well-posedness only for monotonic shear flows.

Contribution

It reveals the nonlinear ill-posedness of the Prandtl equation near non-monotonic shear flows and introduces a weak notion of well-posedness, clarifying conditions for validity.

Findings

Boundary-layer expansions are invalid for non-monotonic shear flows in Sobolev spaces.

The nonlinear Prandtl equation is ill-posed near non-stationary, non-monotonic shear flows.

Oleinik's monotonic solutions are well-posed.

Abstract

This note concerns a nonlinear ill-posedness of the Prandtl equation and an invalidity of asymptotic boundary-layer expansions of incompressible fluid flows near a solid boundary. Our analysis is built upon recent remarkable linear ill-posedness results established by G\'erard-Varet and Dormy [2], and an analysis in Guo and Tice [5]. We show that the asymptotic boundary-layer expansion is not valid for non-monotonic shear layer flows in Sobolev spaces. We also introduce a notion of weak well-posedness and prove that the nonlinear Prandtl equation is not well-posed in this sense near non-stationary and non-monotonic shear flows. On the other hand, we are able to verify that Oleinik's monotonic solutions are well-posed.