A mathematical introduction to fluid mechanics.

*(English)*Zbl 0417.76002
Universitext. New York, Heidelberg, Berlin: Springer-Verlag. viii, 205 p. DM 29.00; $ 16.00 (1979).

The present book originated in a course in fluid mechanics taught in the Department of Mathematics of the University of California, Berkeley in Spring 1978.

Like all the other books and papers written by J. E. Marsden as author or co-author it is highly recommendable to read, especially to those who are Interested in a rigorous mathematical treatment of fluid mechanics. The book written by A. J. Chorin and J. E. Marsden provides an easy readable introduction to fluid mechanics although it covers most of the subject contained in ordinary physics books.

Chapter one deals with isentropic flows and the Euler equations. The usual conservation laws and vorticity theorems are treated and the Navier-Stokes equations for incompressible flows are introduced. The second chapter is concerned with potential and slightly viscous flows. Worth mentioning – because unusual for introductory books – is the part, where the connection between vorticitiy flows, Hamiltonian systems and Lie groups is pointed out. Then the Prandtl boundary layer equations are discussed. Finally the theory of vortex sheets is presented utilizing some concepts from probability theory.

In the last chapter the authors give a short survey of the fundamentals of compressible flows. This is according to the reviewer’s opinion the most interesting part of the book. It contains a very clarifying introduction to shock waves and weak solutions of partial differential equations, which in standard physics books is in general not treated in full mathematical rigor.

What makes the books so valuable to newcomers in the field of fluid mechanics is the successful combination of Physics and Mathematics. Showing this link maintains the authors’ main intention throughout the book.

Like all the other books and papers written by J. E. Marsden as author or co-author it is highly recommendable to read, especially to those who are Interested in a rigorous mathematical treatment of fluid mechanics. The book written by A. J. Chorin and J. E. Marsden provides an easy readable introduction to fluid mechanics although it covers most of the subject contained in ordinary physics books.

Chapter one deals with isentropic flows and the Euler equations. The usual conservation laws and vorticity theorems are treated and the Navier-Stokes equations for incompressible flows are introduced. The second chapter is concerned with potential and slightly viscous flows. Worth mentioning – because unusual for introductory books – is the part, where the connection between vorticitiy flows, Hamiltonian systems and Lie groups is pointed out. Then the Prandtl boundary layer equations are discussed. Finally the theory of vortex sheets is presented utilizing some concepts from probability theory.

In the last chapter the authors give a short survey of the fundamentals of compressible flows. This is according to the reviewer’s opinion the most interesting part of the book. It contains a very clarifying introduction to shock waves and weak solutions of partial differential equations, which in standard physics books is in general not treated in full mathematical rigor.

What makes the books so valuable to newcomers in the field of fluid mechanics is the successful combination of Physics and Mathematics. Showing this link maintains the authors’ main intention throughout the book.

Reviewer: D. Krausser (Berlin)

##### MSC:

76-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to fluid mechanics |

76Bxx | Incompressible inviscid fluids |

76Dxx | Incompressible viscous fluids |

35Q30 | Navier-Stokes equations |

35Q31 | Euler equations |

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

35L65 | Hyperbolic conservation laws |