Perfect incompressible fluids / Jean-Yves Chemin ; translated by Isabelle Gallagher and Dragos Iftimie.

Format:

Book

Author/Creator:

Chemin, Jean-Yves.

Series:

Oxford lecture series in mathematics and its applications ; 14.

Oxford science publications

Oxford lecture series in mathematics and its applications ; 14

Standardized Title:

Fluides parfaits incompressibles. English

Language:

English

French

Subjects (All):

Non-Newtonian fluids.

Physical Description:

x, 187 pages : illustrations ; 24 cm.

Place of Publication:

Oxford : Clarendon Press ; New York : Oxford University Press, 1998.

Summary:

An accessible and self-contained introduction to recent advances in fluid dynamics, this book covers the Euler equations for a perfect incompressible fluid. It develops the key toots for analysis, including the Littlewood-Paley theory, Fourier multipliers on L spaces, and partial differential calculus, and then proves recent results for vortex patches and sheets.

Contents:

1 Presentation of the equations 1

1.1 What is a perfect fluid? 1

1.2 Prom Lagrange to Euler 4

1.3 Vorticity, pressure and dimension 2 8

1.4 References and remarks 15

2 Littlewood-Paley theory 16

2.1 Dyadic decomposition 16

2.2 Sobolev spaces 20

2.3 Hölder spaces 26

2.4 Paradifferential calculus 32

2.5 The pressure and its gradient field 37

2.6 References and remarks 43

3 Concerning Biot-Savart's law 44

3.1 L^p estimates 44

3.2 L^∞ estimates: some examples 47

3.3 Riesz operators and bounded functions 52

3.4 References and remarks 63

4 The case of smooth initial data 65

4.1 Resolution of a model problem 65

4.2 Application to Euler's equation 76

4.3 References and remarks 83

5 The case of bounded vorticity 85

5.1 Yudovich's theorem 85

5.2 On ordinary differential equations 90

5.3 An example 94

5.4 The vortex patch problem 96

5.5 Proof of the persistence 97

5.6 References and remarks 106

6 Vortex sheets 109

6.1 Presentation of the problem 109

6.2 The study of the function G 111

6.3 The limit 112

6.4 References and remarks 117

7 The wave front and the product 119

7.1 Presentation of the wave front 119

7.2 When can the product be defined? 125

7.3 Analytic and Gevrey wave-front sets 130

7.4 References and remarks 139

8 Analyticity and Gevrey regularity 140

8.1 Statement of the theorems 140

8.2 Multilinear operators 141

8.3 Regularity along the flow lines 150

8.4 References and remarks 156

9 Singular vortex patches 157

9.1 Presentation of the problem 157

9.2 Local Littlewood-Paley theory 161

9.3 Dynamics of singular patches 169

9.4 References and remarks 180.

Notes:

Includes bibliographical references (pages [181]-185) and index.

ISBN:

0198503970

OCLC:

39380126