Remarks on the ill-posedness of the Prandtl equation

TL;DR

This paper investigates the ill-posedness of the Prandtl equation, demonstrating non-existence and instability of solutions under various conditions, highlighting fundamental issues in its mathematical formulation.

Contribution

It extends previous work by providing new ill-posedness results for linear and nonlinear Prandtl equations, including in Sobolev and $C^ abla$ settings.

Findings

No local $C^ abla$ solutions for some linearized Prandtl equations with smooth initial data.

Non-existence of Lipschitz continuous solutions in the nonlinear Sobolev setting.

Ill-posedness in space related to Oleinik's results for monotonic data.

Abstract

In the lines of a recent paper by Gerard-Varet and Dormy, we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some $C^\infty$ initial data, local in time $C^\infty$ solutions do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.