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. (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.

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.

Reviewer: Lijing Sun (Milwaukee)

##### MSC:

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 |