Conformal And Potential Analysis in Hele-shaw

This monograph aims at giving a presentation of recent and new ideas that arise from the problems of planar fluid dynamics and which are interesting from the point of view of geometric function theory and potential theory. In particular, this book is concerned with geometric problems for Hele-Shaw flows. Also Hele-Shaw flows on parameter spaces (e.g., the Teichm??ller space) are treated and connections with string theory are revealed. Ultimately, the interaction between several branches of complex and potential analysis, and planar fluid mechanics is discussed. For most parts of this book the background provided by graduate courses in real and complex analysis, in particular, the theory of conformal mappings and in fluid mechanics is assumed. There are some historical remarks concerning the people that have contributed to the topic. The book is as self-contained as possible.

Preface

ix

1 Introduction and Background

1

(26)

1.1 Newtonian fluids

1

(1)

1.2 The Navier-Stokes equations

2

(7)

1.2.1 The continuity equation

3

(1)

1.2.2 The Euler equation

4

(1)

1.2.3 The Navier-Stokes equation

5

(1)

1.2.4 Dynamical similarity and the Reynolds number

5

(2)

1.2.5 Vorticity, two-dimensional flows

7

(2)

1.3 Riemann map and Caratheodory kernel convergence

9

(2)

1.4 Hele-Shaw flows

11

(11)

1.4.1 The Stokes-Leibenzon model

13

(2)

1.4.2 The Polubarinova-Galin equation

15

(4)

1.4.3 Local existence and ill/well-posedness

19

(1)

1.4.4 Regularizations

20

(1)

1.4.5 Numerical treatment

21

(1)

1.5 Complex moments

22

(1)

1.6 Further remarks on the Polubarinova-Galin equation

23

(1)

1.7 The Schwarz function

24

(3)

2 Explicit Strong Solutions

27

(26)

2.1 Classical solutions

27

(10)

2.1.1 Polubarinova and Galin's cardioid

27

(2)

2.1.2 Rational solutions of the Polubarinova-Galin equation

29

(6)

2.1.3 Saffman-Taylor fingers

35

(2)

2.2 Corner flows

37

(16)

2.2.1 Mathematical model

38

(3)

2.2.2 Logarithmic perturbations of the trivial solution

41

(3)

2.2.3 Self-similar bubbles

44

(9)

3 Weak Solutions and Balayage

53

(22)

3.1 Definition of weak solution

53

(3)

3.2 Existence and uniqueness of weak solutions

56

(2)

3.3 General properties of weak solutions

58

(2)

3.4 Regularity of the boundary

60

(2)

3.5 Balayage point of view

62

(3)

3.6 Existence and non-branching backward of weak solutions

65

(4)

3.7 Hele-Shaw flow and quadrature domains

69

(4)

3.8 Multidimensional Hele-Shaw flow and other generalizations

73

(2)

4 Geometric Properties

75

(38)

4.1 Distance to the boundary

75

(1)

4.2 Special classes of univalent functions

76

(2)

4.3 Hereditary shape of phase domains

78

(21)

4.3.1 Bounded dynamics

78

(9)

4.3.2 Dynamics with small surface tension

87

(2)

4.3.3 Geometric properties in the presence of surface tension

89

(3)

4.3.4 Unbounded regions with bounded complement

92

(4)

4.3.5 Unbounded regions with the boundary extending to infinity

96

(3)

4.4 Infinite life-span of starlike dynamics

99

(2)

4.5 Solidification and melting in potential flows

101

(3)

4.5.1 Close-to-parabolic semi-infinite crystal

102

(2)

4.6 Geometry of weak solutions

104

(9)

4.6.1 Starlikeness of the weak solution

104

(1)

4.6.2 The inner normal theorem

105

(4)

4.6.3 Distance to the boundary (revisited)

109

(4)

5 Capacities and Isoperimetric Inequalities

113

(22)

5.1 Conformal invariants and capacities

114

(7)

5.1.1 Modulus of a family of curves

114

(1)

5.1.2 Reduced modulus and capacity

115

(3)

5.1.3 Integral means and the radius-area problem

118

(3)

5.2 Hele-Shaw cells with obstacles

121

(6)

5.2.1 Robin's capacity and Robin's reduced modulus

121

(3)

5.2.2 A problem with an obstacle

124

(3)

5.3 Isoperimetric inequality for a corner flow

127

(4)

5.4 Melting of a bounded crystal

131

(4)

6 General Evolution Equations

135

(36)

6.1 The Löwner-Kufarev equation

137

(1)

6.2 Quasiconformal maps and Teichmüller spaces

138

(6)

6.2.1 Quasiconformal maps

139

(1)

6.2.2 The universal Teichmüller space

140

(4)

6.3 Diff S¹-/Rot S¹- embedded into T

144

(4)

6.3.1 Homogeneous manifold Diff S¹ /Rot S¹

144

(2)

6.3.2 Douady-Earle extension

146

(1)

6.3.3 Semi-flows on T and M

147

(1)

6.4 Infinitesimal descriptions of semi-flows

148

(3)

6.5 Parametric representation of univalent maps

151

(15)

6.5.1 Semigroups of conformal maps

151

(3)

6.5.2 Evolution families and differential equations

154

(7)

6.5.3 The Löwner-Kufarev ordinary differential equation

161

(3)

6.5.4 Univalent functions smooth on the boundary

164

(1)

6.5.5 An application to Hele-Shaw flows

165

(1)

6.6 Fractal growth

166

(5)

7 Hele-Shaw Evolution and Strings

171

(33)

7.1 Bosonic strings

173

(6)

7.1.1 Initial definitions

173

(2)

7.1.2 Metrics

175

(1)

7.1.3 Dynamics

176

(1)

7.1.4 Symmetries

177

(2)

7.2 Correlator

179

(3)

7.2.1 Correlator and partition function

179

(1)

7.2.2 Conformal anomaly. Liouville theory

180

(2)

7.2.3 Superstrings

182

(1)

7.3 Virasoro algebra

182

(6)

7.3.1 Dirac and Kronecker δ-functions

182

(1)

7.3.2 Energy-momentum tensor

183

(3)

7.3.3 Central extension of the Witt algebra

186

(2)

7.4 Hele-Shaw worldsheet

188

(16)

7.4.1 Integrable hierarchies

188

(1)

7.4.2 Integrable structure of the Dirichlet problem

189

(3)

7.4.3 Hele-Shaw worldsheet

192

(4)

7.4.4 Variation of the logarithmic action

196

(5)

7.4.5 Virasoro generators and Neretin polynomials

201

(3)

References

204

(23)

List of symbols

227

(2)

Index

229

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Conformal And Potential Analysis in Hele-shaw Cells

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Table of Contents

Supplemental Materials