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
2.1.1 Galilean Spacetime
2.1.2 Null Foliation and Acoustical Coordinates
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 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 u.
Completion 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 □trX. Estimates for the Top Order Angular Derivatives of X
8.1 Preliminary
8.1.1 Regularization of The Propagation Equation
8.1.2 Propagation Equations for Higher Order Angular Derivatives
8.1.3 Elliptic Theory on St,u
8.1.4 Preliminary Estimates for the Solutions of the Propagation Equations
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 □μ.
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
10.2.1 L∞ Estimates for Rik ... .Ri1yj
10.2.2 L2 Estimates for Rik... Pi1yj
10.3 Bounds for the quantities Ql and Pl
10.3.1 Estimates for Ql
10.3.2 Estimates for 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 TTi
11.1.1 Basic Lemmas
11.1.2 L∞ Estimates for TTi
11.1.3 L2 Estimates for TTi
11.2 Bounds for Quantities Q''m,l and P''m,l
11.2.1 Bounds for Q''m,l
11.2.2 Bounds for P''m,l
12 Recovery of the Acoustical Assumptions
Estimates for Up to the Next to the Top Order Angular
Derivatives of X and Spatial Derivatives of μ
12.1 Estimates for λi, y'', yi and r. Establishing the Hypothesis H0
12.2 The Coercivity Hypothesis H1, H2 and H2''. Estimates for X''
12.3 Estimates for Higher Order Derivatives of X'' 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
14.4.1 Estimates for the Contribution of 14.56
14.4.2 Estimates for the Contribution of 14.57
14.5 The Borderline Estimates Associated to K1
14.5.1 Estimates for the Contribution of 14.56
14.5.2 Estimates for the Contribution of 14.57
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 J.
Recovery 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
19.1.1 Preliminary
19.1.2 Intrinsic View Point
19.1.3 Invariant Curves
19.1.4 Extrinsic View Point
19.2 The Trichotomy Theorem for Past Null Geodesics Ending at Singular Boundary
19.2.1 Hamiltonian Flow
19.2.2 Asymptotic Behavior
19.3 Transformation of Coordinates
19.4 How H Looks Like in Rectangular Coordinates in Galilean Spacetime
References
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