On the ill-posedness of the Prandtl equation

David Gerard-Varet (1) , Emmanuel Dormy (2)

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

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.

Domaines: Equations aux dérivées partielles [math.AP]

David Gerard-Varet, Emmanuel Dormy. On the ill-posedness of the Prandtl equation. 2009. ⟨hal-00372942v3⟩