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