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Fourier analysis and nonlinear partial differential equations. (English) Zbl 1227.35004
Grundlehren der Mathematischen Wissenschaften 343. Berlin: Heidelberg (ISBN 978-3-642-16829-1/hbk; 978-3-642-16830-7/ebook). xvi, 523 p. EUR 106.95 (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.


MSC:
35-02Research monographs (partial differential equations)
35Q30Stokes and Navier-Stokes equations
35Q35PDEs in connection with fluid mechanics
35Q41Time-dependent Schrödinger equations, Dirac equations
35Q55NLS-like (nonlinear Schrödinger) equations
42B25Maximal functions, Littlewood-Paley theory
42B37Harmonic analysis and PDE
76B03Existence, uniqueness, and regularity theory (fluid mechanics)
76D03Existence, uniqueness, and regularity theory
76N10Compressible fluids, general
42-02Research monographs (Fourier analysis)
35L60Nonlinear first-order hyperbolic equations