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On the ill-posedness of the Prandtl equation

Authors: David Gérard-Varet and Emmanuel Dormy
Journal: J. Amer. Math. Soc. 23 (2010), 591-609
MSC (2010): Primary 35-XX; Secondary 76-XX
Published electronically: November 24, 2009
MathSciNet review: 2601044
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Abstract: The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data, or for data with monotonicity properties. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes for the linearization around solutions with nondegenerate critical points. Interestingly, the strong instability is due to viscosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

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Additional Information

David Gérard-Varet
Affiliation: DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm,75005 Paris, France

Emmanuel Dormy
Affiliation: ENS/IPGP/CNRS, Ecole Normale Supérieure, 29 rue Lhomond, 75005 Paris, France

Received by editor(s): April 2, 2009
Published electronically: November 24, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.