##
**Multi-dimensional hyperbolic partial differential equations. First-order systems and applications.**
*(English)*
Zbl 1113.35001

Oxford Mathematical Monographs; Oxford Science Publications. Oxford: Oxford University Press (ISBN 0-19-921123-X/hbk). xxv, 508 p. (2007).

The book is devoted to hyperbolic partial differential equations in \(\mathbb R^n\), in particular to multidimensional first-order systems of conservation laws. The authors concentrate mostly on first-order systems, but higher-order equations are referred to from place to place.

The book consists of four parts and a big appendix. The first part is devoted to linear Cauchy problems with constant coefficients (chapter 1) and variable coefficients (chapter 2). Many important definitions are given here. The second chapter presents the symmterizers technique, applying the pseudo- and paradifferential calculus.

The second part deals with the initial boundary value problems: symmetric dissipative, constant-coefficient in a half space, with characteristic boundaries. The theory of Kreiss’ symmetrizers and Lopatinskiĭ stability condition is used. A chapter is devoted to variable-coefficient initial boundary value problems; here pseudo- and paradifferential calculus is applied to extend well-posedness results to more general situations. The classification of the linear problems is presented.

In the third part the nonlinear problems are studied. The first chapter reviews Cauchy problems for quasilinear systems. Well-posedness is understood in the Sobolev spaces of high index, so the solutions are smooth enough (and thus classical). Then the mixed problems for quasilinear systems are studied. The last chapter is devoted to multidimensional shocks.

The fourth part is entirely devoted to gas dynamics. The pressure law is more general than the famous \(\gamma\)-law. The first chapter is called “The Euler Equations for Real Fluids”; there some basic questions regarding hyperbolicity and symmetrizability are discussed. The second chapter is devoted to boundary conditions for real fluids. In the last chapter the shock waves in real fluids are studied.

The fifth part is an appendix and contains basic calculus results, information on Laplace and Fourier transform, Paley-Wiener theory; much space is devoted to pseudo- and paradifferential calculus (a powerful tool used throughout the book).

Most chapters of the book are self-contained.

The book consists of four parts and a big appendix. The first part is devoted to linear Cauchy problems with constant coefficients (chapter 1) and variable coefficients (chapter 2). Many important definitions are given here. The second chapter presents the symmterizers technique, applying the pseudo- and paradifferential calculus.

The second part deals with the initial boundary value problems: symmetric dissipative, constant-coefficient in a half space, with characteristic boundaries. The theory of Kreiss’ symmetrizers and Lopatinskiĭ stability condition is used. A chapter is devoted to variable-coefficient initial boundary value problems; here pseudo- and paradifferential calculus is applied to extend well-posedness results to more general situations. The classification of the linear problems is presented.

In the third part the nonlinear problems are studied. The first chapter reviews Cauchy problems for quasilinear systems. Well-posedness is understood in the Sobolev spaces of high index, so the solutions are smooth enough (and thus classical). Then the mixed problems for quasilinear systems are studied. The last chapter is devoted to multidimensional shocks.

The fourth part is entirely devoted to gas dynamics. The pressure law is more general than the famous \(\gamma\)-law. The first chapter is called “The Euler Equations for Real Fluids”; there some basic questions regarding hyperbolicity and symmetrizability are discussed. The second chapter is devoted to boundary conditions for real fluids. In the last chapter the shock waves in real fluids are studied.

The fifth part is an appendix and contains basic calculus results, information on Laplace and Fourier transform, Paley-Wiener theory; much space is devoted to pseudo- and paradifferential calculus (a powerful tool used throughout the book).

Most chapters of the book are self-contained.

Reviewer: Ilya A. Chernov (Petrozavodsk)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35Lxx | Hyperbolic equations and hyperbolic systems |

76L05 | Shock waves and blast waves in fluid mechanics |

35L65 | Hyperbolic conservation laws |