Global Propagation of Regular Nonlinear Hyperbolic Waves (eBook)

Preface 6
Contents 8
Introduction 12
1.1 Cauchy Problem 12
1.2 Weak Linear Degeneracy 16
1.3 Some Examples 18
1.4 Main Results for the Cauchy Problem 27
1.5 Normalized Coordinates 29
1.6 Weak Linear Degeneracy and Generalized Null Condition 30
1.7 Nonstrictly Hyperbolic Case 31
1.8 Cauchy Problem on a Semibounded Initial Axis 32
1.9 One-Sided Mixed Initial-Boundary Value Problem 33
1.10 Generalized Riemann Problem 34
1.11 Generalized Nonlinear Initial-Boundary Riemann Problem 35
1.12 Inverse Generalized Riemann Problem 37
1.13 Inverse Piston Problem 37
Preliminaries 39
2.1 Definition of Quasilinear Hyperbolic System 39
2.2 Invariance Under a Smooth Invertible Transformation of Unknown Variables 41
2.3 Genuine Nonlinearity and Linear Degeneracy 43
2.4 Normalized Coordinates 44
2.5 Weak Linear Degeneracy 48
2.6 Decomposition of Waves 51
2.7 Two Lemmas on Ordinary Differential Equations 58
The Cauchy Problem 60
3.1 Necessary Condition to Guarantee the GlobalExistence and Uniqueness of the C1 Solution to theCauchy Problem for the Strictly Hyperbolic System 60
3.2 Some Uniform a Priori Estimates Independent of Normalized Coordinates and Weak Linear Degeneracy for the Strictly Hyperbolic System 63
3.3 Some Uniform a Priori Estimates Depending on Normalized Coordinates and Weak Linear Degeneracy for the Strictly Hyperbolic System 69
3.4 Sufficient Condition to Guarantee the GlobalExistence and Uniqueness of the C1 Solution to theCauchy Problem for the Strictly Hyperbolic System 72
3.5 Global C1 Solution to the Cauchy Problem forthe Hyperbolic System with Characteristics withConstant Multiplicity 76
3.6 Applications 82
The Cauchy Problem (Continued) 87
4.1 Some Uniform a Priori Estimates Independent of Weak Linear Degeneracy 87
4.2 Formation of Singularities of the C1 Solution in theNoncritical Case a < +8
4.3 Blow-Up Mechanism of the C1 Solution in theNoncritical Case a < +8
4.4 Applications 108
4.5 Blow-Up Mechanism of the C1 Solution in theCritical Case a = +8 114
4.6 Remarks 121
Cauchy Problem on a Semibounded Initial Axis 123
5.1 Introduction and Main Results 123
5.2 Proof of Theorem 5.1.1 125
5.3 Application 132
One-Sided Mixed Initial-Boundary Value Problem 134
6.1 Global Existence of the Classical Solution 134
6.2 Formation of Singularities of the C1 Solution 151
6.3 Applications 152
Generalized Riemann Problem 156
7.1 Introduction and Main Results 156
7.2 Preliminaries 161
7.3 Proof of Main Results 165
7.4 Applications 176
Generalized Nonlinear Initial- Boundary Riemann Problem 182
8.1 Introduction and Main Results 182
8.2 Preliminaries 186
8.3 Proof of Theorem 8.1.1 187
8.4 Proof of Theorem 8.1.2 189
Inverse Generalized Riemann Problem 198
9.1 Introduction and Main Results 198
9.2 Generalized Cauchy Problem 202
9.3 Proof of Theorem 9.1.1 209
Inverse Piston Problem 215
10.1 Inverse Piston Problem for the System of One- Dimensional Isentropic Flow 215
10.2 Generalized Cauchy Problem with Cauchy Data Given on a Semibounded Noncharacteristic Curve 236
10.3 Inverse Piston Problem for the System of One- Dimensional Adiabatic Flow 241
References 250
Index 255