On the ill-posedness of the Prandtl equationReport as inadecuate

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1 DMA - Département de Mathématiques et Applications 2 LPS - Laboratoire de Physique Statistique de l-ENS

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 non-degenerate critical points. Interestingly, the strong instability is due to vicosity, which is coherent with well-posedness results obtained for the inviscid version of the equation. A numerical study of this instability is also provided.

Author: David Gerard-Varet - Emmanuel Dormy -

Source: https://hal.archives-ouvertes.fr/


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