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Well-posedness of the Prandtl equation in Sobolev spaces


Authors: R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang
Journal: J. Amer. Math. Soc. 28 (2015), 745-784
MSC (2010): Primary 35M13, 35Q35, 76D10, 76D03, 76N20
DOI: https://doi.org/10.1090/S0894-0347-2014-00813-4
Published electronically: June 6, 2014
MathSciNet review: 3327535
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Abstract: We develop a new approach to study the well-posedness theory of the Prandtl equation in Sobolev spaces by using a direct energy method under a monotonicity condition on the tangential velocity field instead of using the Crocco transformation. Precisely, we firstly investigate the linearized Prandtl equation in some weighted Sobolev spaces when the tangential velocity of the background state is monotonic in the normal variable. Then to cope with the loss of regularity of the perturbation with respect to the background state due to the degeneracy of the equation, we apply the Nash-Moser-Hörmander iteration to obtain a well-posedness theory of classical solutions to the nonlinear Prandtl equation when the initial data is a small perturbation of a monotonic shear flow.


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R. Alexandre
Affiliation: Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China and Arts et Métiers ParisTech, Paris 75013, France
Email: radjesvarane.alexandre@paristech.fr

Y.-G. Wang
Affiliation: Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai, 200240, P. R. China
Email: ygwang@sjtu.edu.cn

C.-J. Xu
Affiliation: School of Mathematics, Wuhan University 430072, Wuhan, P. R. China, and Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France
Email: Chao-Jiang.Xu@univ-rouen.fr

T. Yang
Affiliation: Department of Mathematics, City University of Hong Kong, Hong Kong, P. R. China
Email: matyang@cityu.edu.hk

DOI: https://doi.org/10.1090/S0894-0347-2014-00813-4
Keywords: Prandtl equation, well-posedness theory, Sobolev spaces, energy method, monotonic velocity field, Nash-Moser-H\"ormander iteration.
Received by editor(s): March 7, 2012
Received by editor(s) in revised form: April 10, 2014
Published electronically: June 6, 2014
Article copyright: © Copyright 2014 American Mathematical Society