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  • Book
  • © 2011

Fourier Analysis and Nonlinear Partial Differential Equations

  • First accessible work giving an exhaustive and up-to-date presentation of how to use Fourier analysis to study PDEs.

  • Written by experts in the field of Fourier analysis, this work presents a self-contained state of the art of techniques with applications to different classes of PDEs.

  • Both accessible to anyone with a good undergraduate level in analysis, as well as to experts and researchers.

  • Includes supplementary material: sn.pub/extras

Part of the book series: Grundlehren der mathematischen Wissenschaften (GL, volume 343)

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  • ISBN: 978-3-642-16830-7
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  • Own it forever
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  • Tax calculation will be finalised during checkout
Softcover Book USD 119.99
Price excludes VAT (USA)
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Table of contents (10 chapters)

  1. Front Matter

    Pages I-XV
  2. Basic Analysis

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 1-50
  3. Littlewood–Paley Theory

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 51-121
  4. Transport and Transport-Diffusion Equations

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 123-167
  5. Quasilinear Symmetric Systems

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 169-202
  6. The Incompressible Navier–Stokes System

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 203-243
  7. Anisotropic Viscosity

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 245-289
  8. Euler System for Perfect Incompressible Fluids

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 291-333
  9. Strichartz Estimates and Applications to Semilinear Dispersive Equations

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 335-387
  10. Smoothing Effect in Quasilinear Wave Equations

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 389-428
  11. The Compressible Navier–Stokes System

    • Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin
    Pages 429-496
  12. Back Matter

    Pages 497-523

About this book

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 evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations.  It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity.

It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.

Keywords

  • 35Q35, 76N10, 76D05, 35Q31, 35Q30
  • Fourier analysis
  • Littlewood-Paley theory
  • estimates
  • partial differential equations

Reviews

From the reviews:

“The authors did make impressive contributions to a broad area of fluid dynamics. It is the first time that a coherent presentation of those research results is available, which will give easier access to the whole area to a broader audience. … It is a valuable contribution in the important area of the interest of the authors and will without question find its place in the mathematical libraries, and on the shelves of people working in those areas.” (Herbert Koch, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2014)

“The aim of the present monograph is to introduce methods from Fourier analysis, and in particular techniques based on the Littlewood–Paley decomposition, for the solution of nonlinear partial differential equations. … The presentation is fairly self-contained and only requires a solid background in measure theory and functional analysis. It will be of value to both graduate students and researchers interested in application of Fourier analysis to partial differential equations.” (G. Teschl, Monatshefte für Mathematik, Vol. 165 (3-4), March, 2012)

“This book is a well-written introduction to Fourier analysis, Littlewood-Paley theory and some of their applications to the theory of evolution equations. It is suitable for readers with a solid undergraduate background in analysis. A feature that distinguishes it from other books of this sort is its emphasis on using Littlewood-Paley decomposition to study nonlinear differential equations. … the references, historical background, and discussion of possible future developments at the end of each chapter are very convenient for its readers.” (Lijing Sun, Zentralblatt MATH, Vol. 1227, 2012)

“This book intends to prepare the reader how to apply tools from Fourier analysis to directly solve problems arising in the theory of non linear partial differential equations. … The presentation is well structured and easy to follow. … This is a textbook for advanced undergraduate or beginning graduate students with a good background in real and functional analysis. … even active researchers or mathematicians interested in the application of Fourier-analytic tools will find this book very useful.” (Peter R. Massopust, Mathematical Reviews, Issue 2011 m)

Authors and Affiliations

  • , Départment de Mathématiques, Université de Tunis El Manar, Tunis, Tunisia

    Hajer Bahouri

  • , Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris Cedex 05, France

    Jean-Yves Chemin

  • , Centre de Mathématiques, Université Paris XII - Val de Marne, Créteil Cedex, France

    Raphaël Danchin

Bibliographic Information

Buying options

eBook USD 89.00
Price excludes VAT (USA)
  • ISBN: 978-3-642-16830-7
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 119.99
Price excludes VAT (USA)
Hardcover Book USD 169.99
Price excludes VAT (USA)