Affiliations: Equipe d'analyse fonctionnelle, Université Denis Diderot, Institut de Mathématiques de Jussieu, Paris, France. E-mail: [email protected] | Equipe d'analyse fonctionnelle, Université Pierre et Marie Curie, Institut de Mathématiques de Jussieu, Paris, France. E-mail: [email protected]
Note: [] Corresponding author: D. Gérard-Varet, Equipe d'analyse fonctionnelle, Université Denis Diderot (Paris 7), Institut de Mathématiques de Jussieu, UMR CNRS 7586, Paris, France. E-mail: [email protected].
Note: [] Current address: Division of Applied Mathematics, Brown University, Providence, RI, USA.
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 H1 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.
