Remarks on the ill-posedness of the Prandtl equation

Abstract

In the lines of the recent paper (J. Amer. Math. Soc. 23(2) (2010), 591-609), we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary non-monotonic shear flows, we show that for some C ∞ initial data, local in time H 1 solutions of the linearized Prandtl equation 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.