Fourier analysis and nonlinear partial differential equations. (English) Zbl 1227.35004

Grundlehren der Mathematischen Wissenschaften 343. Berlin: Springer (ISBN 978-3-642-16829-1/hbk; 978-3-642-16830-7/ebook). xvi, 523 p. (2011).
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.
Chapter 1 is a review of some elementary harmonic analysis theory: the atomic decomposition, interpolation theorem, Hardy-Littlewood maximal function, Fourier transform, Sobolev spaces etc. Chapter 2 is an exposition of the Littlewood-Paley theory. A complete theory of strong solutions for transport and transport-diffusion equations is given in Chapter 3. Linear and quasilinear symmetric systems are investigated in Chapter 4. Chapters 5, 6 and 10 deal with the incompressible Navier-Stokes system and the compressible Navier-Stokes system respectively. Chapter 7 studies the Euler system for inviscid incompressible fluids. Chapter 8 is devoted to Strichartz estimates for the Schrödinger and wave equations. The quasilinear Strichartz estimate for a class of quasilinear wave equations is studied in Chapter 9.
The lack of exercises seems to be a minor drawback. Some proofs can be relegated as exercises; consequently the reader can be involved in a more active way. However, the references, historical background, and discussion of possible future developments at the end of each chapter are very convenient for the readers.

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35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35Q30 Navier-Stokes equations
35Q35 PDEs in connection with fluid mechanics
35Q41 Time-dependent Schrödinger equations and Dirac equations
35Q55 NLS equations (nonlinear Schrödinger equations)
42B25 Maximal functions, Littlewood-Paley theory
42B37 Harmonic analysis and PDEs
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
35L60 First-order nonlinear hyperbolic equations
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