Remarks on the ill-posedness of the Prandtl equation

Details Remarks on the ill-posedness of the Prandtl equation Remarks on the ill-posedness of the Prandtl equation

2010-08-03

arxiv.org/abs/1008.0532v1

Mathematics

Analysis of PDEs

Scientific paper

In the lines of a recent paper by Gerard-Varet and Dormy, we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some $C^\infty$ initial data, local in time $C^\infty$ solutions do not exist. At the nonlinear level, we prove that if a flow exists in the Sobolev setting, it cannot be Lipschitz continuous. Besides ill-posedness in time, we also establish some ill-posedness in space, that casts some light on the results obtained by Oleinik for monotonic data.

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