Mathematical Geophysics An Introduction to

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.

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

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

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

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