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1 Compressible Flow and Non-linear Wave Equations 1

1.1 Euler's Equations 1

1.2 Irrotational Flow and the Nonlinear Wave Equation 2

1.3 The Equation of Variations and the Acoustical Metric 5

1.4 The Fundamental Variations 6

2 The Basic Geometric Construction 11

2.1 Null Foliation Associated with the Acoustical Metric 11

2.1.1 Galilean Spacetime 11

2.1.2 Null Foliation and Acoustical Coordinates 12

2.2 A Geometric Interpretation for Function H 19

3 The Acoustical Structure Equations 21

3.1 The Acoustical Structure Equations 21

3.2 The Derivatives of the Rectangular Components of L and ？ 33

4 The Acoustical Curvature 39

4.1 Expressions for Curvature Tensor 39

4.2 Regularity for the Acoustical Structure Equations asμ→0 42

4.3 A Remark 45

5 The Fundamental Energy Estimate 47

5.1 Bootstrap Assumptions.Statement of the Theorem 47

5.2 The Multiplier Fields K0 and K1.The Associated Energy-Momentum Density Vectorfields 50

5.3 The Error Integrals 60

5.4 The Estimates for the Error Integrals 63

5.5 Treatment of the Integral Inequalities Depending on t and u.Completion of the Proof 76

6 Construction of Commutation Vectorfields 83

6.1 Commutation Vectorfields and Their Deformation Tensors 83

6.2 Preliminary Estimates for the Deformation Tensors 88

7 Outline of the Derived Estimates of Each Order 101

7.1 The Inhomogeneous Wave Equations for the Higher Order Variations.The Recursion Formula for the Source Functions 101

7.2 The First Term in ？n 104

7.3 The Estimates of the Contribution of the First Term in ？n to the Error Integrals 109

8 Regularization of the Propagation Equation for ？trx.Estimates for the Top Order Angular Derivatives of x 129

8.1 Preliminary 129

8.1.1 Regularization of The Propagation Equation 129

8.1.2 Propagation Equations for Higher Order Angular Derivatives 133

8.1.3 Elliptic Theory on St,u 143

8.1.4 Preliminary Estimates for the Solutions of the Propagation Equations 151

8.2 Crucial Lemmas Concerning the Behavior of μ 155

8.3 The Actual Estimates for the Solutions of the Propagation Equations 174

9 Regularization of the Propagation Equation for ？μ.Estimates for the Top Order Spatial Derivatives of μ 185

9.1 Regularization of the Propagation Equation 185

9.2 Propagation Equations for the Higher Order Spatial Derivatives 191

9.3 Elliptic Theory on St,u 202

9.4 The Estimates for the Solutions of the Propagation Equations 214

10 Control of the Angular Derivatives of the First Derivatives of the xi.Assumptions and Estimates in Regard to x 227

10.1 Preliminary 227

10.2 Estimates for yi 238

10.2.1 L∞ Estimates for Rik…Ri1yj 239

10.2.2 L2 Estimates for Rik…Ri1yj 242

10.3 Bounds for the quantities Ql and Pl 251

10.3.1 Estimates for Ql 251

10.3.2 Estimates for Pl 262

11 Control of the Spatial Derivatives of the First Derivatives of the xi.Assumptions and Estimates in Regard to μ 269

11.1 Estimates for T？i 269

11.1.1 Basic Lemmas 269

11.1.2 L∞ Estimates for T？i 287

11.1.3 L2 Estimates for T？i 293

11.2Bounds for Quantities Q′m,l and P′m,l 305

11.2.1 Bounds for Q′m,l 306

11.2.2 Bounds for P′m,l 316

12 Recovery of the Acoustical Assumptions.Estimates for Up to the Next to the Top Order Angular Derivatives of x and Spatial Derivatives ofμ 327

12.1 Estimates for λi,y′i,yi and r.Establishing the Hypothesis HO 327

12.2 The Coercivity Hypothesis H1,H2 and H2′.Estimates for x′ 332

12.3 Estimates for Higher Order Derivatives of x′and μ 351

13 Derivation of the Basic Properties of μ 381

14 The Error Estimates Involving the Top Order Spatial Derivatives of the Acoustical Entities 397

14.1 The Error Terms Involving the Top Order Spatial Derivatives of the Acoustical Entities 397

14.2 The Borderline Error Integrals 404

14.3 Assumption J 405

14.4 The Borderline Estimates Associated to K0 408

14.4.1 Estimates for the Contribution of (14.56) 408

14.4.2 Estimates for the Contribution of (14.57) 417

14.5 The Borderline Estimates Associated to K1 423

14.5.1 Estimates for the Contribution of(14.56) 423

14.5.2 Estimates for the Contribution of(14.57) 446

15 The Top Order Energy Estimates 463

15.1 Estimates Associated to K1 463

15.2 Estimates Associated to K0 477

16 The Descent Scheme 489

17 The Isoperimetric Inequality.Recovery of Assumption J.Recovery of the Bootstrap Assumption Proof of the Main Theorem 503

17.1 Recovery of J—Preliminary 503

17.2 The Isoperimetric Inequality 505

17.3 Recovery of J—Completion 509

17.4 Recovery of the Final Bootstrap Assumption 510

17.5 Completion of the Proof of the Main Theorem 511

18 Sufficient Conditions on the Initial Data for the Formation of a Shock in the Evolution 521

19 The Structure of the Boundary of the Domain of the Maximal Solution 533

19.1 Nature of Singular Hypersurface in Acoustical Differential Structure 533

19.1.1 Preliminary 533

19.1.2 Intrinsic View Point 535

19.1.3 Invariant Curves 537

19.1.4 Extrinsic View Point 539

19.2 The Trichotomy Theorem for Past Null Geodesics Ending at Singular Boundary 543

19.2.1 Hamiltonian Flow 543

19.2.2 Asymptotic Behavior 545

19.3 Transformation of Coordinates 562

19.4 How H Looks Like in Rectangular Coordinates in Galilean Spacetime 575

References 581