| Preface |
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vii | |
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1 | (8) |
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1.1 The nonlinear stability problem for the Einstein equations |
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1 | (3) |
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1.2 Statement of the main result |
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4 | (5) |
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2 Overview of the Hyperboloidal Foliation Method |
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9 | (10) |
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2.1 The semi-hyperboloidal frame and the hyperboloidal frame |
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9 | (2) |
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2.2 Spacetime foliation and initial data set |
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11 | (3) |
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2.3 Coordinate formulation of the nonlinear stability property |
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14 | (2) |
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2.4 Bootstrap argument and construction of the initial data |
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16 | (1) |
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2.5 Outline of the Monograph |
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17 | (2) |
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3 Functional Analysis on Hyperboloids of Minkowski Spacetime |
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19 | (26) |
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3.1 Energy estimate on hyperboloids |
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19 | (6) |
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3.2 Sup-norm estimate based on curved characteristic integration |
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25 | (3) |
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3.3 Sup-norm estimate for wave equations with source |
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28 | (7) |
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3.4 Sup-norm estimate for Klein-Gordon equations |
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35 | (1) |
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3.5 Weighted Hardy inequality along the hyperboloidal foliation |
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36 | (4) |
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3.6 Sobolev inequality on hyperboloids |
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40 | (1) |
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3.7 Hardy inequality for hyperboloids |
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41 | (2) |
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3.8 Commutator estimates for admissible vector fields |
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43 | (2) |
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4 Quasi-Null Structure of the Einstein-Massive Field System on Hyperboloids |
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45 | (24) |
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4.1 Einstein equations in wave coordinates |
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45 | (6) |
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4.2 Analysis of the support |
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51 | (2) |
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4.3 A classification of nonlinearities in the Einstein-massive field system |
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53 | (5) |
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4.4 Estimates based on commutators and homogeneity |
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58 | (1) |
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4.5 Basic structure of the quasi-null terms |
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59 | (1) |
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4.6 Metric components in the semi-hyperboloidal frame |
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60 | (2) |
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4.7 Wave gauge condition in the semi-hyperboloidal frame |
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62 | (3) |
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4.8 Revisiting the structure of the quasi-null terms |
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65 | (4) |
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5 Initialization of the Bootstrap Argument |
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69 | (4) |
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5.1 The bootstrap assumption and the basic estimates |
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69 | (3) |
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5.2 Estimates based on integration along radial rays |
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72 | (1) |
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6 Direct Control of Nonlinearities in the Einstein Equations |
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73 | (4) |
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73 | (1) |
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74 | (3) |
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7 Direct Consequences of the Wave Gauge Condition |
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77 | (12) |
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77 | (4) |
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81 | (3) |
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84 | (5) |
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8 Second-Order Derivatives of the Spacetime Metric |
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89 | (10) |
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89 | (2) |
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91 | (1) |
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92 | (3) |
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8.4 Conclusion for general second-order derivatives |
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95 | (1) |
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95 | (4) |
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9 Sup-Norm Estimate Based on Characteristics |
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99 | (6) |
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9.1 Main statement in this section |
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99 | (4) |
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9.2 Application to quasi-null terms |
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103 | (2) |
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10 Low-Order Refined Energy Estimate for the Spacetime Metric |
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105 | (8) |
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105 | (2) |
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10.2 Main estimate established in this section |
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107 | (3) |
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10.3 Application of the refined energy estimate |
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110 | (3) |
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11 Low-Order Refined Sup-Norm Estimate for the Metric and Scalar Field |
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113 | (12) |
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11.1 Main estimates established in this section |
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113 | (1) |
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11.2 First refinement on the metric components |
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114 | (2) |
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11.3 First refinement for the scalar field |
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116 | (2) |
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11.4 Second refinement for the scalar field and the metric |
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118 | (2) |
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11.5 A secondary bootstrap argument |
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120 | (5) |
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12 High-Order Refined L2 Estimates |
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125 | (14) |
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12.1 Objective of this section and preliminary |
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125 | (6) |
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12.2 Main estimates in this section |
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131 | (5) |
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12.3 Applications to the derivation of refined decay estimates |
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136 | (3) |
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13 High-Order Refined Sup-Norm Estimates |
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139 | (8) |
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139 | (2) |
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13.2 Main estimate in this section |
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141 | (6) |
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14 Low-Order Refined Energy Estimate for the Scalar Field |
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147 | (4) |
| Appendix A Revisiting the wave-Klein-Gordon model |
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151 | (2) |
| Appendix B Sup-norm estimate for the wave equations |
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153 | (6) |
| Appendix C Sup-norm estimate for the Klein-Gordon equation |
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159 | (6) |
| Appendix D Commutator estimates for the hyperboloidal frame |
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165 | (6) |
| Bibliography |
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171 | |
Lisainfo e-raamatute kohta