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**Compressible fluid flow and systems of conservation laws in several space variables.**
*(English)*
Zbl 0537.76001

Applied Mathematical Sciences, 53. New York etc.: Springer-Verlag. VIII, 159 p. DM 48.00; $ 17.50 (1984).

The book consists of 4 chapters, and an index. There is a bibliography at the end of each chapter. The book deals with quasilinear hyperbolic systems in one and several space variables. These systems are important in fluid mechanics, combustion theory and elsewhere. In applications these systems can be often written as systems of conservation laws.

In chapter 1, introduction, some physical examples are given (gas dynamics, combustion theory, nonlinear wave equation, etc.), and weakly nonlinear asymptotics of solutions of a class of systems of conservation laws are discussed. In chapter 2, smooth solutions and the equations of incompressible fluid flow and the existence of smooth solutions for a general system of conservation laws with smooth initial data is discussed. The smooth solution may not exist because of the blow-up or because of the formation of shock waves. Compressible and incompressible fluid flows and combustion equations at low Mach’s number are studied.

In chapter 3, the formation of shock waves in smooth solutions and a number of the results concerning the breakdown of smooth solutions are discussed (shock formation for scalar conservation laws in several space variables, for 2\(\times 2\) strictly hyperbolic systems, for some \(m\times m\) systems, and for a quasilinear wave equation). In chapter 4, the existence and stability of shock fronts in several space variables, the discontinuous weak solutions of systems of conservation laws in several space variables are discussed.

In chapter 1, introduction, some physical examples are given (gas dynamics, combustion theory, nonlinear wave equation, etc.), and weakly nonlinear asymptotics of solutions of a class of systems of conservation laws are discussed. In chapter 2, smooth solutions and the equations of incompressible fluid flow and the existence of smooth solutions for a general system of conservation laws with smooth initial data is discussed. The smooth solution may not exist because of the blow-up or because of the formation of shock waves. Compressible and incompressible fluid flows and combustion equations at low Mach’s number are studied.

In chapter 3, the formation of shock waves in smooth solutions and a number of the results concerning the breakdown of smooth solutions are discussed (shock formation for scalar conservation laws in several space variables, for 2\(\times 2\) strictly hyperbolic systems, for some \(m\times m\) systems, and for a quasilinear wave equation). In chapter 4, the existence and stability of shock fronts in several space variables, the discontinuous weak solutions of systems of conservation laws in several space variables are discussed.

Reviewer: A.G.Ramm

### MSC:

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

80A25 | Combustion |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

76V05 | Reaction effects in flows |

76N15 | Gas dynamics (general theory) |

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