Conformal and Potential Analysis in Hele-Shaw Cells (eBook)

Contents 5
Preface 8
1 Introduction and Background 10
1.1 Newtonian .uids 10
1.2 The Navier-Stokes equations 11
1.3 Riemann map and Carath ´ eodory kernel convergence 18
1.4 Hele-Shaw .ows 20
1.5 Complex moments 31
1.6 Further remarks on the Polubarinova-Galin equation 32
1.7 The Schwarz function 33
2 Explicit Strong Solutions 35
2.1 Classical solutions 35
2.2 Corner .ows 45
3 Weak Solutions and Balayage 60
3.1 De.nition of weak solution 60
3.2 Existence and uniqueness of weak solutions 63
3.3 General properties of weak solutions 65
3.4 Regularity of the boundary 67
3.5 Balayage point of view 69
3.6 Existence and non-branching backward 72
of weak solutions 72
3.7 Hele-Shaw .ow and quadrature domains 76
3.8 Multidimensional Hele-Shaw .ow and other 80
generalizations 80
4 Geometric Properties 82
4.1 Distance to the boundary 82
4.2 Special classes of univalent functions 83
4.3 Hereditary shape of phase domains 85
4.4 Infinite life-span of starlike dynamics 106
4.5 Solidfication and melting in potential .ows 108
4.6 Geometry of weak solutions 111
5 Capacities and Isoperimetric Inequalities 119
5.1 Conformal invariants and capacities 120
5.2 Hele-Shaw cells with obstacles 127
5.3 Isoperimetric inequality for a corner .ow 133
5.4 Melting of a bounded crystal 137
6 General Evolution Equations 140
6.1 The Löwner-Kufarev equation 142
6.2 Quasiconformal maps and Teichmüller spaces 143
6.3 Di. S1/Rot S1 embedded into T 149
6.4 Infinitesimal descriptions of semi-.ows 153
6.5 Parametric representation of univalent maps with quasiconformal extensions 156
6.6 Fractal growth 171
7 Hele-Shaw Evolution and Strings 176
7.1 Bosonic strings 178
7.2 Correlator 184
7.3 Virasoro algebra 187
7.4 Hele-Shaw worldsheet 193
Bibliography 209
List of Symbols 231
Index 233