Preface ......................................................... v
1 Preliminaries ................................................ 1
1.1 Notation ................................................ 1
1.2 Newtonian Potential ..................................... 5
1.3 Equation div u = b ...................................... 7
1.4 Necas Imbedding Theorem ................................ 16
1.5 Spaces of Solenoidal Vector Fields ..................... 21
1.6 Linear Functionals Vanishing on Divergence Free
Vector Fields .......................................... 22
1.7 Helmholtz-Weyl Decomposition ........................... 26
1.8 Comments ............................................... 29
2 Linear Stationary Problem ................................... 31
2.1 Existence and Uniqueness of Weak Solutions ............. 31
2.2 Coercive Estimates ..................................... 33
2.3 Local Regularity ....................................... 36
2.4 Further Local Regularity Results, n = 2,3 .............. 37
2.5 Stokes Operator in Bounded Domains ..................... 41
2.6 Comments ............................................... 45
3 Non-Linear Stationary Problem ............................... 47
3.1 Existence of Weak Solutions ............................ 47
3.2 Regularity of Weak Solutions ........................... 52
3.3 Comments ............................................... 60
4 Linear Non-Stationary Problem ............................... 61
4.1 Derivative in Time ..................................... 61
4.2 Explicit Solution ...................................... 64
4.3 Cauchy Problem ......................................... 75
4.4 Pressure Field. Regularity ............................. 76
4.5 Uniqueness Results ..................................... 80
4.6 Local Interior Regularity .............................. 84
4.7 Local Boundary Regularity .............................. 88
4.8 Comments ............................................... 90
5 Non-linear Non-Stationary Problem ........................... 91
5.1 Compactness Results for Non-Stationary Problems ........ 91
5.2 Auxiliary Problem ...................................... 94
5.3 Weak Leray-Hopf Solutions ............................. 101
5.4 Multiplicative Inequalities and Related Questions ..... 106
5.5 Uniqueness of Weak Leray-Hopf Solutions. 2D Case ...... 109
5.6 Further Properties of Weak Leray-Hopf Solutions ....... 114
5.7 Strong Solutions ...................................... 119
5.8 Comments .............................................. 132
6 Local Regularity Theory for Non-Stationary Navier-Stokes
Equations .................................................. 133
6.1 ε-Regularity Theory ................................... 133
6.2 Bounded Ancient Solutions ............................. 149
6.3 Mild Bounded Ancient Solutions ........................ 158
6.4 Liouville Type Theorems ............................... 166
6.4.1 LPS Quantities ................................. 166
6.4.2 2D case ........................................ 167
6.4.3 Axially Symmetric Case with No Swirl ........... 170
6.4.4 Axially Symmetric Case ......................... 173
6.5 Axially Symmetric Suitable Weak Solutions ............. 178
6.6 Backward Uniqueness for Navier-Stokes Equations ....... 184
6.7 Comments .............................................. 188
7 Behavior of L3-Norm ........................................ 189
7.1 Main Result ........................................... 189
7.2 Estimates of Scaled Solutions ......................... 191
7.3 Limiting Procedure .................................... 197
7.4 Comments .............................................. 204
Appendix A Backward Uniqueness and Unique Continuation ....... 205
A.l Carleman-Type Inequalities ............................ 205
A.2 Unique Continuation Across Spatial Boundaries ......... 210
A.3 Backward Uniqueness for Heat Operator in Half Space ... 214
A.4 Comments .............................................. 219
Appendix В Lemarie-Riesset Local Energy Solutions ............ 221
B.1 Introduction .......................................... 221
B.2 Proof of Theorem 1.6 .................................. 225
B.3 Regularized Problem ................................... 233
B.4 Passing to Limit and Proof of Proposition 1.8 ......... 237
B.5 Proof of Theorem 1.7 .................................. 243
B.6 Density ............................................... 249
B.7 Comments .............................................. 250
Bibliography .................................................. 251
Index ......................................................... 257
|
