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 ？

4 The Acoustical Curvature

4.1 Expressions for Curvature Tensor

4.2 Regularity for the Acoustical Structure Equations asμ→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 ？n

7.3 The Estimates of the Contribution of the First Term in ？n 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…Ri1yj

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 T？i

11.1.1 Basic Lemmas

11.1.2 L∞ Estimates for T？i

11.1.3 L2 Estimates for T？i

11.2Bounds 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 ofμ

12.1 Estimates for λi,y′i,yi and r.Establishing the Hypothesis HO

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