# 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 $\cancel{d}$tr$\chi$. Estimates for the Top Order Angular Derivatives of $\chi$pp. 203–273 9 Regularization of the Propagation Equation for $\cancel{\Delta}\mu$. Estimates for the Top Order Spatial Derivatives of $\mu$pp. 275–328 10 Control of the Angular Derivatives of the First Derivatives of the $x^i$. Assumptions and Estimates in Regard to $\chi$pp. 329–472 11 Control of the Spatial Derivatives of the First Derivatives of the $x^i$. Assumptions and Estimates in Regard to $\mu$pp. 473–664 12 Recovery of the Acoustical Assumptions. Estimates for Up to the Next to the Top Order Angular Derivatives of $\chi$ and Spatial Derivatives of $\mu$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