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arXiv · 1008.0532

Remarks on the ill-posedness of the Prandtl equation

David Gerard-Varet·Toan Nguyen
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Abstract

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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David Gerard-Varet, Toan Nguyen. 2010-08-03. Remarks on the ill-posedness of the Prandtl equation. https://arxiv.org/abs/1008.0532

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