On the ill-posedness of the Prandtl equation
David Gérard-Varet1
and Emmanuel Dormy2
1DMA/CNRS, Ecole Normale Supérieure, 45 rue d'Ulm,75005 Paris, France.
2ENS/IPGP/CNRS, Ecole Normale Supérieure, 29 rue Lhomond,
75005 Paris, France.
J. Amer. Math. Soc. 23 (2010), 591-609.
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.