Fourier Analysis and Nonlinear Partial Differential Equations
In recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of
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Series editors M. Berger P. de la Harpe F. Hirzebruch N.J. Hitchin L. Hörmander A. Kupiainen G. Lebeau F.-H. Lin B.C. Ngô M. Ratner D. Serre Ya.G. Sinai N.J.A. Sloane A.M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan
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Hajer Bahouri Jean-Yves Chemin Raphaël Danchin
Fourier Analysis and Nonlinear Partial Differential Equations
Hajer Bahouri Départment de Mathématiques Faculté des Sciences de Tunis Campus Universitaire Université de Tunis El Manar 2092 Tunis Tunisia [email protected]
Raphaël Danchin Centre de Mathématiques Faculté de Sciences et Technologie Université Paris XII-Val de Marne 61, avenue du Général de Gaulle 94 010 Créteil Cedex France [email protected]
Jean-Yves Chemin Laboratoire Jacques-Louis Lions Université Pierre et Marie Curie Boîte courrier 187 75252 Paris Cedex 05 France [email protected]
ISSN 0072-7830 ISBN 978-3-642-16829-1 e-ISBN 978-3-642-16830-7 DOI 10.1007/978-3-642-16830-7 Springer Heidelberg Dordrecht London New York Mathematics Subject Classification: 35Q35, 76N10, 76D05, 35Q31, 35Q30 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: VTEX, Vilnius Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
A la m´emoire de Noomann Bassou
Preface
Since the 1980s, Fourier analysis methods have become of ever greater interest in the study of linear and nonlinear partial differential equations. In particular, techniques based on Littlewood–Paley decomposition have proven to be very efficient in the study of evolution equations. Littlewood–Paley decomposition originates with Littlewood and Paley’s works in the early 1930s and provides an elementary device for splitting a (possibly rough) function into a sequence of spectrally well localized smooth functions. In particular, differentiation acts almost as a multiplication on each term of the sequence. However, its systematic use for nonlinear partial differential equations is rather recent. In this context, the main breakthrough was achieved after J.-M. Bony introduced the paradifferential calculus in his pioneering 1981 paper (see [39]) and
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