Lp-Theory for Incompressible Newtonian Flows: Energy Preserving Boundary Conditions, Weakly Singular Domains

Cover......Page 1
Acknowledgements......Page 6
Contents......Page 7
Introduction......Page 10
References......Page 15
Abstract and Notation......Page 16
Bounded Weakly Singular Domains......Page 0
1.1 The Principles of Continuum Mechanics......Page 18
1.3 Conservation of Momentum......Page 21
1.4 Constitutive Equations......Page 23
References......Page 25
Energy Preserving Boundary Conditions......Page 27
2.1 Generic Transmission Conditions......Page 28
2.2 Impermeable Walls......Page 29
2.3 Free Surfaces and Interfaces......Page 30
2.5 Energy Preserving Boundary Conditions......Page 31
2.6 Vorticity and Pressure Boundary Conditions......Page 35
Remarks......Page 36
References......Page 37
Lp- Theory for Incompressible Newtonian Flows......Page 39
3.1 Necessary Regularity Conditions......Page 42
3.2 Additional Regularity of the Pressure Trace......Page 47
3.3 Necessary Compatibility Conditions......Page 49
3.4 Maximal Lp-Regularity of the Stokes Equations......Page 50
3.5 Associated Stokes Operators......Page 52
3.6 Well-Posedness of the Navier-Stokes Equations......Page 56
Remarks......Page 60
References......Page 66
Tools and Methods......Page 70
4.1 Lp-Theory for Elliptic Problems......Page 71
4.2 Lp-Theory for Parabolic Problems......Page 77
4.3 Special Solutions in a Halfspace......Page 80
4.4 Sectorial Operators and Functional Calculus......Page 86
4.5 A Splitting Scheme for the Stokes Equations......Page 93
4.6 Uniform Sobolev Type Embeddings......Page 97
Remarks......Page 100
References......Page 102
Maximal Lp-Regularity in a Halfspace......Page 104
5.1 Strategy......Page 106
5.2 The Transformed Stokes Equations......Page 109
5.3 Boundary Conditions Involving the Normal Velocity......Page 111
5.4 Boundary Conditions Involving the Pressure......Page 114
References......Page 117
Maximal Lp-Regularity\rin a Bent Halfspace......Page 118
6.1 Strategy......Page 120
6.2 Transformation to a Halfspace......Page 123
6.3 Boundary Conditions Involving the Normal Velocity......Page 127
References......Page 129
Maximal Lp-Regularity\rin a Bounded Smooth Domain......Page 130
7.1 Strategy......Page 132
7.2 Partition of the Domain......Page 134
7.3 The Regularity of the Pressure......Page 138
7.4 Boundary Conditions Involving the Normal Velocity......Page 139
7.5 Boundary Conditions Involving the Pressure......Page 146
References......Page 153
Lp-Theory in\rWeakly Singular Domains......Page 154
8.1 Weakly Singular Domains......Page 155
8.2 Necessary Regularity Conditions......Page 159
8.3 Necessary Compatibility Conditions......Page 161
8.4 Maximal Lp-Regularity of the Stokes Equations......Page 162
8.5 Lp-Theory for Elliptic Problems......Page 163
8.7 The Reflection Method......Page 165
Remarks......Page 167
References......Page 168
Bibliography......Page 169
Basic Operations......Page 175
Boundary Conditions......Page 176
Equations......Page 177
Function Spaces......Page 178
Maximal Regularity Spaces......Page 179
Index......Page 180