# The Formation of Shocks in 3-Dimensional Fluids

### Demetrios Christodoulou

ETH Zürich, Switzerland

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FrontmatterDownload pp. i–v | |

ContentsDownload pp. vii–viii | |

Prologue and SummaryDownload pp. 1–21 | |

1 | Relativistic Fluids and Nonlinear Wave Equations. The Equations of Variationpp. 23–37 |

2 | The Basic Geometric Constructionpp. 39–52 |

3 | The Acoustical Structure Equationspp. 53–83 |

4 | The Acoustical Curvaturepp. 85–98 |

5 | The Fundamental Energy Estimatepp. 99–137 |

6 | Construction of the Commutation Vectorfieldspp. 139–168 |

7 | Outline of the Derived Estimates of Each Orderpp. 169–201 |

8 | Regularization of the Propagation Equation for $d $tr$χ$. Estimates for the Top Order Angular Derivatives of $χ$pp. 203–273 |

9 | Regularization of the Propagation Equation for $Δ μ$. Estimates for the Top Order Spatial Derivatives of $μ$pp. 275–328 |

10 | Control of the Angular Derivatives of the First Derivatives of the $x_{i}$. Assumptions and Estimates in Regard to $χ$pp. 329–472 |

11 | Control of the Spatial Derivatives of the First Derivatives of the $x_{i}$. Assumptions and Estimates in Regard to $μ$pp. 473–664 |

12 | Recovery of the Acoustical Assumptions. Estimates for Up to the Next to the Top Order Angular Derivatives of $χ$ and Spatial Derivatives of $μ$pp. 665–740 |

13 | The Error Estimates Involving the Top Order Spatial Derivatives of the Acoustical Entities. The Energy Estimates. Recovery of the Bootstrap Assumptions. Statement and Proof of the Main Theorem: Existence up to Shock Formationpp. 741–892 |

14 | Sufficient Conditions on the Initial Data for the Formation of a Shock in the Evolutionpp. 893–926 |

15 | The Nature of the Singular Hypersurface. The Invariant Curves. The Trichotomy Theorem. The Structure of the Boundary of the Domain of the Maximal Solutionpp. 927–975 |

Epiloguepp. 977–986 | |

Bibliographypp. 987–988 | |

Indexpp. 989–992 |