Abstract
We show the local in time well-posedness of the Prandtl equations for data with Gevrey 2 regularity in x and Sobolev regularity in y. The main novelty of our result is that we do not make any assumption on the structure of the initial data: no monotonicity or hypothesis on the critical points. Moreover, our general result is optimal in terms of regularity, in view of the ill-posedness result of Gérard-Varet and Dormy (J Am Math Soc 23(2):591–609, 2010).
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Acknowledgements
The authors thank Weiren Zhao for useful remarks on the first version of this manuscript. They acknowledge the support of the Université Sorbonne Paris Cité, through the funding “Investissements d’Avenir”, convention ANR-11-IDEX-0005. This work was also supported by the SingFlows Project, Grant ANR-18-CE40-0027 of the French National Research Agency (ANR). H.D. is grateful to the People Programme (Marie Curie Actions) of the European Unions Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. PCOFUND-GA-2013-609102, through the PRESTIGE programme coordinated by Campus France. D.G.-V. acknowledges the support of the Institut Universitaire de France.
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Dietert, H., Gérard-Varet, D. Well-Posedness of the Prandtl Equations Without Any Structural Assumption. Ann. PDE 5, 8 (2019). https://doi.org/10.1007/s40818-019-0063-6
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DOI: https://doi.org/10.1007/s40818-019-0063-6