On the ill-posedness of the Prandtl equation
Analysis of PDEs
2015-05-13 v2
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.
Keywords
Cite
@article{arxiv.0904.0434,
title = {On the ill-posedness of the Prandtl equation},
author = {David Gerard-Varet and Emmanuel Dormy},
journal= {arXiv preprint arXiv:0904.0434},
year = {2015}
}
Comments
A few more typos corrected. Definition of L_eps added on page 12. Confusing semigroup notation corrected