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 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.

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

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 andeasy 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)

• , 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

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