%0 Journal Article
%T On the ill-posedness of the Prandtl equation
%A David Gerard-Varet
%A Emmanuel Dormy
%J Mathematics
%D 2009
%I arXiv
%R 10.1090/S0894-0347-09-00652-3
%X 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.
%U http://arxiv.org/abs/0904.0434v2