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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the ill-posedness of the Prandtl equation
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by David Gérard-Varet and Emmanuel Dormy PDF
J. Amer. Math. Soc. 23 (2010), 591-609 Request permission

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
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 23 (2010), 591-609
  • MSC (2010): Primary 35-XX; Secondary 76-XX
  • DOI: https://doi.org/10.1090/S0894-0347-09-00652-3
  • MathSciNet review: 2601044