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**Incompressible perfect fluids.
(Fluides parfaits incompressibles.)**
*(French)*
Zbl 0829.76003

Astérisque. 230. Paris: Société Mathématique de France, 177 p. (1995).

This book is based on the three lecture courses given to students of Nankai Institute in Tian-Jin (1991), l’Ecole Polytechnique (1992), and l’Ecole Normale superieure de Cachan (1992-1993).

The book is divided into 9 chapters. The first chapter is introductory and aims to review some basic concepts of ideal fluids. Chapter 2, on Littlewood-Paley theory, starts by proving some elementary inequalities for the functions whose Fourier transforms have compact support. Then the Sobolev and Hölder spaces are characterised by a partition of unity associated with a cover by dyadic wreaths. Chapter 3 investigates the action of the operators of type \(\partial_i \partial_j \Delta^{- 1}\) on spaces of functions. The fourth chapter is devoted to the Euler equations with regular initial conditions. Chapter 5 investigates the solutions of the Euler equations for incompressible fluid, in two- dimensional space with bounded vorticity, and then discusses the evolution of regularity of vortex patches. The problem of seeking for a solutions of Euler equations for incompressible fluids in two dimensions, when the initial vorticity is the measure of the length of a compact curve, is investigated in chapter 6. Chapter 7 deals with the notion of wave front of distributions. Some properties of high regularity in time of the solutions of Euler equations are discussed in chapter 8. The problem of vortex patches from chapter 5, when the singularities are present on the border of vortex patches, is covered in chapter 9.

The book is excellently written and produced. It is highly useful for students and research workers.

The book is divided into 9 chapters. The first chapter is introductory and aims to review some basic concepts of ideal fluids. Chapter 2, on Littlewood-Paley theory, starts by proving some elementary inequalities for the functions whose Fourier transforms have compact support. Then the Sobolev and Hölder spaces are characterised by a partition of unity associated with a cover by dyadic wreaths. Chapter 3 investigates the action of the operators of type \(\partial_i \partial_j \Delta^{- 1}\) on spaces of functions. The fourth chapter is devoted to the Euler equations with regular initial conditions. Chapter 5 investigates the solutions of the Euler equations for incompressible fluid, in two- dimensional space with bounded vorticity, and then discusses the evolution of regularity of vortex patches. The problem of seeking for a solutions of Euler equations for incompressible fluids in two dimensions, when the initial vorticity is the measure of the length of a compact curve, is investigated in chapter 6. Chapter 7 deals with the notion of wave front of distributions. Some properties of high regularity in time of the solutions of Euler equations are discussed in chapter 8. The problem of vortex patches from chapter 5, when the singularities are present on the border of vortex patches, is covered in chapter 9.

The book is excellently written and produced. It is highly useful for students and research workers.

Reviewer: V.A.Sava (Iaşi)

### MSC:

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

76Bxx | Incompressible inviscid fluids |

35Q35 | PDEs in connection with fluid mechanics |