| Preface | 6 |
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| Contents | 9 |
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| 1 The Physical Experiments and Their Mathematical Modelling | 13 |
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| 1.1 A hint on the experiments | 13 |
| 1.2 The mathematical framework | 17 |
| 2 The Mathematical Setting: A Survey of the Main Theorems | 30 |
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| 2.1 Small e problem | 30 |
| 2.2 Vortex lattice | 35 |
| 2.3 Flow around an obstacle | 38 |
| 3 Two-Dimensional Model for a Rotating Condensate | 40 |
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| 3.1 Main results | 41 |
| 3.2 Preliminaries | 46 |
| 3.3 Bounded number of vortices | 49 |
| 3.4 Refined structure of vortices | 57 |
| 3.5 Lower bound | 66 |
| 3.6 Upper bound | 72 |
| 3.7 Final expansion and properties of vortices | 77 |
| 3.8 Open Questions | 86 |
| 4 Other Trapping Potentials | 89 |
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| 4.1 Non radial harmonic potential | 90 |
| 4.2 Quartic potential | 92 |
| 4.3 Open questions | 108 |
| 5 High-Velocity and Quantum Hall Regime | 109 |
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| 5.1 Introduction | 110 |
| 5.2 Regular lattice | 117 |
| 5.3 Distorted lattice | 121 |
| 5.4 Infinite number of zeros | 127 |
| 5.5 Other trapping potentials | 129 |
| 5.6 Open questions | 129 |
| 6 Three-Dimensional Rotating Condensate | 132 |
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| 6.1 Numerical simulations | 133 |
| 6.2 Formal derivation of the reduced energy E[. ] | 135 |
| 6.3 G convergence results | 141 |
| 6.4 Single Vortex line, study of E[. ] | 151 |
| 6.5 A few open questions | 163 |
| 7 Superfluid Flow Around an Obstacle | 166 |
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| 7.1 Mathematical setting | 167 |
| 7.2 Proof of Theorem 7.1 | 176 |
| 7.3 Proof of Theorem 7.2 | 189 |
| 8 Further Open Problems | 203 |
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| 8.1 Setting in the whole space for the Thomas Fermi regime | 203 |
| 8.2 Other scalings | 204 |
| 8.3 Other models | 205 |
| References | 207 |
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| Index | 212 |
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