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 Introduction
 1 Compressible Flow and Non-linear Wave Equations
  1.1 Euler’s Equations
  1.2 Irrotational Flow and the Nonlinear Wave Equation
  1.3 The Equation of Variations and the Acoustical Metric
  1.4 The Fundamental Variations
 2 The Basic Geometric Construction
  2.1 Null Foliation Associated with the Acoustical Metric
  2.2 A Geometric Interpretation for Function H
 3 The Acoustical Structure Equations
  3.1 The Acoustical Structure Equations
  3.2 The Derivatives of the Rectangular Components of L and ˆ T
 4 The Acoustical Curvature
  4.1 Expressions for Curvature Tensor
  4.2 Regularity for the Acoustical Structure Equations as μ → 0
  4.3 A Remark
 5 The Fundamental Energy Estimate
  5.1 Bootstrap Assumptions. Statement of the Theorem
  5.2 The Multiplier Fields K0 and K1. The Associated Energy-Momentum Density Vectorfields
  5.3 The Error Integrals
  5.4 The Estimates for the Error Integrals
  5.5 Treatment of the Integral Inequalities Depending on t and pletion of the Proof
 6 Construction of Commutation Vectorfields
  6.1 Commutation Vectorfields and Their Deformation Tensors
  6.2 Preliminary Estimates for the Deformation Tensors
 7 Outline of the Derived Estimates of Each Order
  7.1 The Inhomogeneous Wave Equations for the Higher Order Variations. The Recursion Formula for the Source Functions
  7.2 The First Term in ˜ρn
  7.3 The Estimates of the Contribution of the First Term in ˜ρn to the Error Integrals
 8 Regularization of the Propagation Equation for /timates for the Top Order Angular Derivatives of x
  8.1 Preliminary
  8.2 Crucial Lemmas Concerning the Behavior of μ
  8.3 The Actual Estimates for the Solutions of the Propagation Equations
 9 Regularization of the Propagation Equation for /xμ.Estimates for the Top Order Spatial Derivatives of μ
  9.1 Regularization of the Propagation Equation
  9.2 Propagation Equations for the Higher Order Spatial Derivatives
  9.3 Elliptic Theory on St,u
  9.4 The Estimates for the Solutions of the Propagation Equations
 10 Control of the Angular Derivatives of the First Derivatives of the xi. Assumptions and Estimates in Regard to x
  10.1 Preliminary
  10.2 Estimates for yi
  10.3 Bounds for the quantities Ql and Pl
 11 Control of the Spatial Derivatives of the First Derivatives of the xi. Assumptions and Estimates in Regard to μ
  11.1 Estimates for T ˆ T i
  11.2 Bounds for Quantities Q′m,l and P′m,l
 12 Recovery of the Acoustical timates for Up to the Next to the Top Order Angular Derivatives of x and Spatial Derivatives of μ
  12.1 Estimates for λi, y′i, yi and r. Establishing the Hypothesis H0
  12.2 The Coercivity Hypothesis H1, H2 and H2′. Estimates for χ′
  12.3 Estimates for Higher Order Derivatives of χ′ and μ
 13 Derivation of the Basic Properties of μ
 14 The Error Estimates Involving the Top Order Spatial Derivatives of the Acoustical Entities
  14.1 The Error Terms Involving the Top Order Spatial Derivatives of the Acoustical Entities
  14.2 The Borderline Error Integrals
  14.3 Assumption J
  14.4 The Borderline Estimates Associated to K0
  14.5 The Borderline Estimates Associated to K1
 15 The Top Order Energy Estimates
  15.1 Estimates Associated to K1
  15.2 Estimates Associated to K0
 16 The Descent Scheme
 17 The Isoperimetric Inequality. Recovery of Assumption covery of the Bootstrap Assumption. Proof of the Main Theorem
  17.1 Recovery of J—Preliminary
  17.2 The Isoperimetric Inequality
  17.3 Recovery of J—Completion
  17.4 Recovery of the Final Bootstrap Assumption
  17.5 Completion of the Proof of the Main Theorem
 18 Sufficient Conditions on the Initial Data for the Formation of a Shock in the Evolution
 19 The Structure of the Boundary of the Domain of the Maximal Solution
  19.1 Nature of Singular Hypersurface in Acoustical Differential Structure
  19.2 The Trichotomy Theorem for Past Null Geodesics Ending at Singular Boundary
  19.3 Transformation of Coordinates
  19.4 How H Looks Like in Rectangular Coordinates in Galilean Spacetime
 References