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On the ill-posedness of the Prandtl equation

Author(s): David Gérard-Varet; Emmanuel Dormy
Journal: J. Amer. Math. Soc. 23 (2010), 591-609.
MSC (2010): Primary 35-XX; Secondary 76-XX
Posted: November 24, 2009
MathSciNet review: 2601044
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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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