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**Mathematical theory of incompressible nonviscous fluids.**
*(English)*
Zbl 0789.76002

Applied Mathematical Sciences. 96. New York, NY: Springer-Verlag. xi, 283 p. (1994).

Although being more than two centuries old, the theory of incompressible nonviscous flows (so-called perfect flows) is far from being complete and generates a permanent interest within mathematical community. Many basic questions in this theory remain still open; e.g., one may recall such famous challenging problems as development of singularities in a finite time or the long-time behaviour of smooth solutions.

Among many books devoted to the subject, no recent one, in authors’ opinion, is oriented toward mathematical physics. Thus, the purpose of the authors was to fill this gap and write a mathematically rigorous book presenting, on the other hand, the underlying physical ideas in a clear and natural manner.

The text consists of seven chapters. The introductory chapter 1 starts out with the derivation of the equations of motion of incompressible nonviscous fluids (the Euler equations) and discusses such basic notions as potential and vorticity flows, stream function, conservation laws. Chapter 2 deals with existence, uniqueness, and regularity of the solution to the initial value problem for the Euler equation. In chapter 3, the concept of stability is introduced and the results obtained are applied to stationary solutions of the Euler equation. The vortex model is explained in chapter 4 both in two and three dimensions. In chapter 5, the spectral and vortex methods are considered as finite-dimensional approximation schemes for the Euler equation, and their convergence is investigated. Chapter 6 is devoted to the study of discontinuities arising in the perfect fluids. As examples, the evolution of vortex sheets and the evolution of density discontinuities (water waves) are presented.

The remainder of the book, chapter 7, is essentially different from that of all the others. Namely, the Euler equations are replaced here by the Navier-Stokes equations in order to take into account dissipative effects omitted up to now. On this basis, the authors provide the description of turbulent flows and illustrate the statistical theories from both a phenomenological and mathematical point of view. In particular, the statistical mechanics of vortex systems and its connection with invariant measures are thoroughly investigated.

The final section of each chapter contains discussion of the existing literature and further developments, as well as technical appendices and some exercises with a varying degree of difficulty. The authors hope that solving them is the best test to check that the objectives have been reached.

To summarize, we can say that the book is very well organized and carefully written. It is essentially addressed to mathematicians working on nonlinear problems and to postgraduated students who want to enter the current research in fluid dynamics areas.

Among many books devoted to the subject, no recent one, in authors’ opinion, is oriented toward mathematical physics. Thus, the purpose of the authors was to fill this gap and write a mathematically rigorous book presenting, on the other hand, the underlying physical ideas in a clear and natural manner.

The text consists of seven chapters. The introductory chapter 1 starts out with the derivation of the equations of motion of incompressible nonviscous fluids (the Euler equations) and discusses such basic notions as potential and vorticity flows, stream function, conservation laws. Chapter 2 deals with existence, uniqueness, and regularity of the solution to the initial value problem for the Euler equation. In chapter 3, the concept of stability is introduced and the results obtained are applied to stationary solutions of the Euler equation. The vortex model is explained in chapter 4 both in two and three dimensions. In chapter 5, the spectral and vortex methods are considered as finite-dimensional approximation schemes for the Euler equation, and their convergence is investigated. Chapter 6 is devoted to the study of discontinuities arising in the perfect fluids. As examples, the evolution of vortex sheets and the evolution of density discontinuities (water waves) are presented.

The remainder of the book, chapter 7, is essentially different from that of all the others. Namely, the Euler equations are replaced here by the Navier-Stokes equations in order to take into account dissipative effects omitted up to now. On this basis, the authors provide the description of turbulent flows and illustrate the statistical theories from both a phenomenological and mathematical point of view. In particular, the statistical mechanics of vortex systems and its connection with invariant measures are thoroughly investigated.

The final section of each chapter contains discussion of the existing literature and further developments, as well as technical appendices and some exercises with a varying degree of difficulty. The authors hope that solving them is the best test to check that the objectives have been reached.

To summarize, we can say that the book is very well organized and carefully written. It is essentially addressed to mathematicians working on nonlinear problems and to postgraduated students who want to enter the current research in fluid dynamics areas.

Reviewer: O.Titow (Berlin)

### MSC:

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

76Bxx | Incompressible inviscid fluids |

35Q35 | PDEs in connection with fluid mechanics |

76F20 | Dynamical systems approach to turbulence |