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**Systems of conservation laws. I: Hyperbolicity, entropy, shock waves.
(Systèmes de lois de conservation. I: Hyperbolicité, entropies, ondes de choc.)**
*(French)*
Zbl 0930.35002

Fondations. Paris: Diderot Editeur. xii, 298 p. (1996).

This book is the first volume of a comprehensive work on the theory of the systems of hyperbolic and parabolic conservation laws (for Vol. II see Zbl 0930.35003 below). In recent years there was an enormous progress in this field, thus it was really needed to present both recent and more classical topics in a unified style, trying to update the classical textbooks on the subject (see for instance J. Smoller [Shock waves and reaction-diffusion equations. 2nd ed., Springer-Verlag (1994; Zbl 0807.35002)] and A. Majda [Compressible fluid flow and systems of conservation laws in several space variables. Springer-Verlag (1984; Zbl 0537.76001)]).

This first volume, mainly devoted to the basic aspect of the theory, contains some fundamental aspects of hyperbolic and parabolic systems in conservative form, considering both weak and classical solutions. Quite a few exercises are proposed to the reader.

We indicate briefly the content of each chapter. In Chapter 1 various examples of nonlinear hyperbolic systems are discussed, including e.g. gas dynamics, electromagnetism, hyperelasticity. Chapter 2 is devoted to the Kruzhkov theory of one-dimensional scalar conservation laws. Chapter 3 presents in some detail the theory of linear and quasilinear hyperbolic systems, introducing the concept of weak solutions and entropy conditions. The Riemann problem for genuinely nonlinear systems is explained in Chapter 4 for general systems, following the classical ideas of Lax; several examples are presented.

In Chapter 5 there is a detailed presentation of the classical Glimm’s proof of existence of entropy solutions for general systems of conservation laws. Finally, Chapters 6 and 7 are devoted to second-order perturbations and to shock profiles. The existence of smooth solutions that are globally defined in time is obtained for strictly dissipative viscosity matrices. The author presents several results concerning the existence and the nonlinear stability of shock profiles associated with shock waves satisfying Lax’s shock admissibility inequalities.

The book is indicated as a very useful textbook for graduated courses on the topics.

This first volume, mainly devoted to the basic aspect of the theory, contains some fundamental aspects of hyperbolic and parabolic systems in conservative form, considering both weak and classical solutions. Quite a few exercises are proposed to the reader.

We indicate briefly the content of each chapter. In Chapter 1 various examples of nonlinear hyperbolic systems are discussed, including e.g. gas dynamics, electromagnetism, hyperelasticity. Chapter 2 is devoted to the Kruzhkov theory of one-dimensional scalar conservation laws. Chapter 3 presents in some detail the theory of linear and quasilinear hyperbolic systems, introducing the concept of weak solutions and entropy conditions. The Riemann problem for genuinely nonlinear systems is explained in Chapter 4 for general systems, following the classical ideas of Lax; several examples are presented.

In Chapter 5 there is a detailed presentation of the classical Glimm’s proof of existence of entropy solutions for general systems of conservation laws. Finally, Chapters 6 and 7 are devoted to second-order perturbations and to shock profiles. The existence of smooth solutions that are globally defined in time is obtained for strictly dissipative viscosity matrices. The author presents several results concerning the existence and the nonlinear stability of shock profiles associated with shock waves satisfying Lax’s shock admissibility inequalities.

The book is indicated as a very useful textbook for graduated courses on the topics.

Reviewer: Alberto Tesei (Roma)

### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

35L65 | Hyperbolic conservation laws |

35L67 | Shocks and singularities for hyperbolic equations |

35L40 | First-order hyperbolic systems |