Perfect Incompressible Fluids

by Chemin, Jean-Yves; Gallagher, Isabelle; Iftimie, Dragos

• ISBN13: 9780198503972
• ISBN10: 0198503970
• Format: Hardcover
• Copyright: 1998-12-10
• Publisher: Clarendon Press

Summary

The aim of this book is to offer a direct and self-contained access to some of the new or recent results in fluid mechanics. It gives an authoritative account on the theory of the Euler equations describing a perfect incompressible fluid. First of all, the text derives the Euler equationsfrom a variational principle, and recalls the relations on vorticity and pressure. Various weak formulations are proposed. The book then presents the tools of analysis necessary for their study: Littlewood-Paley theory, action of Fourier multipliers on L spaces, and partial differentialcalculus. These techniques are then used to prove various recent results concerning vortext patches or sheets, essentially the persistence of the smoothness of the boundary of a vortex patch, even if that smoothness allows singular points, as well as the existence of weak solutions of thevorticity sheet type. The text also presents properties of microlocal (analytic or Gevrey) regularity of the solutions of Euler equations, and provides links of such properties to the smoothness in time of the flow of the solution vector field.

Table of Contents

1 Presentation of the equations	1	(15)
1.1 What is a perfect fluid?	1	(3)
1.2 From Lagrange to Euler	4	(4)
1.3 Vorticity, pressure and dimension 2	8	(7)
1.4 References and remarks	15	(1)
2 Littlewood-Paley theory	16	(28)
2.1 Dyadic decomposition	16	(4)
2.2 Sobolev spaces	20	(6)
2.3 Holder spaces	26	(6)
2.4 Paradifferential calculus	32	(5)
2.5 The pressure and its gradient field	37	(6)
2.6 References and remarks	43	(1)
3 Concerning Biot-Savart's law	44	(21)
3.1 L(p) estimates	44	(3)
3.2 L(XXX) estimates: some examples	47	(5)
3.3 Riesz operators and bounded functions	52	(11)
3.4 References and remarks	63	(2)
4 The case of smooth initial data	65	(20)
4.1 Resolution of a model problem	65	(11)
4.2 Application to Euler's equation	76	(7)
4.3 References and remarks	83	(2)
5 The case of bounded vorticity	85	(24)
5.1 Yudovich's theorem	85	(5)
5.2 On ordinary differential equations	90	(4)
5.3 An example	94	(2)
5.4 The vortex patch problem	96	(1)
5.5 Proof of the persistence	97	(9)
5.6 References and remarks	106	(3)
6 Vortex sheets	109	(10)
6.1 Presentation of the problem	109	(2)
6.2 The study of the function G	111	(1)
6.3 The limit	112	(5)
6.4 References and remarks	117	(2)
7 The wave front and the product	119	(21)
7.1 Presentation of the wave front	119	(6)
7.2 When can the product be defined?	125	(5)
7.3 Analytic and Gevrey wave-front sets	130	(9)
7.4 References and remarks	139	(1)
8 Analyticity and Gevrey regularity	140	(17)
8.1 Statement of the theorems	140	(1)
8.2 Multilinear operators	141	(9)
8.3 Regularity along the flow lines	150	(6)
8.4 References and remarks	156	(1)
9 Singular vortex patches	157	(24)
9.1 Presentation of the problem	157	(4)
9.2 Local Littlewood-Paley theory	161	(8)
9.3 Dynamics of singular patches	169	(11)
9.4 References and remarks	180	(1)
References	181	(5)
Index	186