Mathematical Geophysics An Introduction to Rotating Fluids and the Navier-Stokes Equations

by Chemin, Jean-Yves; Desjardins, Benoit; Gallagher, Isabelle; Grenier, Emmanuel

• ISBN13: 9780198571339
• ISBN10: 019857133X
• Format: Hardcover
• Copyright: 2006-06-15
• Publisher: Clarendon Press

Summary

Aimed at graduate students, researchers and academics in mathematics, engineering, oceanography, meteorology, and mechanics, this text provides a detailed introduction to the physical theory of rotating fluids, a significant part of geophysical fluid dynamics. The text is divided into four parts, with the first part providing the physical background of the geophysical models to be analyzed. Part two is devoted to a self contained proof of the existence of weak (or strong) solutions to the imcompressible Navier-Stokes equations. Part three deals with the rapidly rotating Navier-Stokes equations, first in the whole space, where dispersion effects are considered. The case where the domain has periodic boundary conditions is then analyzed, and finally rotating Navier-Stokes equations between two plates are studied, both in the case of periodic horizontal coordinated and those in R2. In Part IV, the stability of Ekman boundary layers and boundary layer effects in magnetohydrodynamics and quasigeostrophic equations are discussed. The boundary layers which appear near vertical walls are presented and formally linked with the classical Prandlt equations. Finally spherical layers are introduced, whose study is completely open.

Author Biography

Jean-Yves Chemin is a Professor at the University of Paris VI Benoit Desjardins is based at the Centre of Atomic Studies centre de Bruyers le Chatel Isabelle Gallagher is a Professor at the Institut de Mathematiques de Jussieu
Emmanuel Greiner is based at the Ecole Normale Superiore de Lyon

Table of Contents

Preface	ix
Acknowledgements	xi
PART I GENERAL INTRODUCTION	1	(14)
PART II ON THE NAVIER—STOKES EQUATIONS	15	(70)
1 Some elements of functional analysis	17	(16)
1.1 Function spaces	17	(2)
1.2 The Stokes problem	19	(3)
1.3 A brief overview of Sobolev spaces	22	(6)
1.3.1 Definition in the case of the whole space Rd	22	(1)
1.3.2 Sobolev embeddings	23	(4)
1.3.3 A compactness result	27	(1)
1.4 Proof of the regularity of the pressure	28	(5)
2 Weak solutions of the Navier–Stokes equations	33	(20)
2.1 Spectral properties of the Stokes operator	33	(9)
2.1.1 The case of bounded domains	33	(3)
2.1.2 The general case	36	(6)
2.2 The Leray theorem	42	(11)
2.2.1 Construction of approximate solutions	44	(2)
2.2.2 A priori bounds	46	(1)
2.2.3 Compactness properties	47	(1)
2.2.4 End of the proof of the Leray theorem	48	(5)
3 Stability of Navier–Stokes equations	53	(30)
3.1 The time-dependent Stokes problem	53	(3)
3.2 Stability in two dimensions	56	(2)
3.3 Stability in three dimensions	58	(6)
3.4 Stable solutions in a bounded domain	64	(8)
3.4.1 Intermediate spaces	64	(2)
3.4.2 The well-posedness result	66	(5)
3.4.3 Some remarks about stable solutions	71	(1)
3.5 Stable solutions in a domain without boundary	72	(5)
3.6 Blow-up condition and propagation of regularity	77	(10)
3.6.1 Blow-up condition	77	(3)
3.6.2 Propagation of regularity	80	(3)
4 References and remarks on the Navier–Stokes equations	83	(2)
PART III ROTATING FLUIDS	85	(134)
5 Dispersive cases	87	(30)
5.1 A brief overview of dispersive phenomena	87	(6)
5.1.1 Strichartz-type estimates	89	(2)
5.1.2 Illustration of the wave equation	91	(2)
5.2 The particular case of the Rossby operator in R³	93	(7)
5.3 Application to rotating fluids in R³	100	(17)
5.3.1 Study of the limit system	101	(1)
5.3.2 Existence and convergence of solutions to the rotating-fluid equations	102	(6)
5.3.3 Global well-posedness	108	(9)
6 The periodic case	117	(38)
6.1 Setting of the problem, and statement of the main result	117	(4)
6.2 Derivation of the limit system in the energy space	121	(4)
6.3 Properties of the limit quadratic form Q	125	(7)
6.4 Global existence and stability for the limit system	132	(8)
6.5 Construction of an approximate solution	140	(4)
6.6 Study of the limit system with anisotropic viscosity	144	(11)
7 Ekman boundary layers for rotating fluids	155	(62)
7.1 The well-prepared linear problem	159	(9)
7.2 Non-linear estimates in the well-prepared case	168	(4)
7.3 The convergence theorem in the well-prepared case	172	(4)
7.4 The ill-prepared linear problem	176	(23)
7.5 Non-linear estimates in the ill-prepared case	199	(3)
7.6 The convergence theorem in the whole space	202	(8)
7.7 The convergence theorem in the periodic case	210	(11)
7.7.1 Proof of the theorem	210	(7)
8 References and remarks on rotating fluids	217	(2)
PART IV PERSPECTIVES	219	(20)
9 Stability of horizontal boundary layers	221	(6)
9.1 Critical Reynolds number	221	(3)
9.2 Energy of a small perturbation	224	(1)
9.3 Rolls and turbulence	225	(2)
10 Other systems	227	(4)
10.1 Large magnetic fields	227	(1)
10.2 A rotating MHD system	228	(1)
10.3 Quasigeostrophic limit	228	(3)
11 Vertical layers	231	(6)
11.1 Introduction	231	(1)
11.2 E1/3 layer	232	(1)
11.3 E1/4 layer	232	(3)
11.4 Mathematical problems	235	(2)
12 Other layers	237	(2)
12.1 Sphere	237	(1)
12.2 Spherical shell	237	(1)
12.3 Layer between two differentially rotating spheres	238	(1)
References	239	(8)
List of Notations	247	(2)
Index	249