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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Well-posedness of the Prandtl equation in Sobolev spaces
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by R. Alexandre, Y.-G. Wang, C.-J. Xu and T. Yang PDF
J. Amer. Math. Soc. 28 (2015), 745-784 Request permission

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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Additional Information
  • 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
  • MR Author ID: 291072
  • 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
  • MR Author ID: 303932
  • Email: matyang@cityu.edu.hk
  • Received by editor(s): March 7, 2012
  • Received by editor(s) in revised form: April 10, 2014
  • Published electronically: June 6, 2014
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3327535