Quasilinear Hyperbolic Systems, Compressible Flows, and Waves

Front Page
......Page 1
Contents......Page 5
Preface......Page 9
About the Author......Page 13
1.1 Preliminaries......Page 14
1.2.1 Traffic flow......Page 15
1.2.2 River flow and shallow water equations......Page 16
1.2.3 Gasdynamic equations......Page 17
1.2.4 Relaxing gas flow......Page 18
1.2.5 Magnetogasdynamic equations......Page 20
1.2.6 Hot electron plasma model......Page 23
1.2.8 Relativistic gas model......Page 24
1.2.10 Dusty gases......Page 25
1.2.11 Zero-pressure gasdynamic system......Page 26
2.1 Breakdown of Smooth Solutions......Page 28
2.1.1 Weak solutions and jump condition......Page 30
2.1.2 Entropy condition and shocks......Page 34
2.1.3 Riemann problem......Page 35
2.2.2 Admissibility criterion II (Vanishing viscosity)......Page 38
2.2.3 Admissibility criterion III (Viscous profile)......Page 39
2.2.4 Admissibility criterion IV (Kruzkov)......Page 41
2.2.5 Admissibility criterion V (Oleinik)......Page 42
2.3 Riemann Problem for Nonconvex Flux Function......Page 43
2.4 Irreversibility......Page 45
2.5 Asymptotic Behavior......Page 47
3.1 Genuine Nonlinearity......Page 51
3.2 Weak Solutions and Jump Condition......Page 52
3.3.1 Admissibility criterion I (Entropy pair)......Page 53
3.3.2 Admissibility criterion II (Lax)......Page 54
3.3.4 Contact discontinuity......Page 55
3.4.1 Simple waves......Page 56
3.4.2 Riemann invariants......Page 57
3.4.4 Shock waves......Page 58
3.5 Shallow Water Equations......Page 66
3.5.1 Bores......Page 67
3.5.2 Dilatation waves......Page 69
3.5.3 The Riemann problem......Page 71
3.5.4 Numerical solution......Page 73
3.5.5 Interaction of elementary waves......Page 77
3.5.6 Interaction of elementary waves from different families......Page 78
3.5.7 Interaction of elementary waves from the same family......Page 80
4.1 Waves and Compatibility Conditions......Page 86
4.1.1 Bicharacteristic curves or rays......Page 88
4.1.2 Transport equations for first order discontinuities......Page 89
4.1.3 Transport equations for higher order discontinuities......Page 92
4.1.4 Transport equations for mild discontinuities......Page 93
4.2 Evolutionary Behavior of Acceleration Waves......Page 95
4.2.1 Local behavior......Page 96
4.2.2 Global behavior: The main results......Page 97
4.2.3 Proofs of the main results......Page 100
4.2.4 Some special cases......Page 102
4.3.1 Evolution law for the amplitudes of C[sup(1)] discontinuities......Page 105
4.3.2 Reflected and transmitted amplitudes......Page 107
4.4 Weak Discontinuities in Radiative Gasdynamics......Page 111
4.4.1 Radiation induced waves......Page 112
4.4.2 Modified gasdynamic waves......Page 114
4.4.3 Waves entering in a uniform region......Page 115
4.5.1 Characteristic approach......Page 117
4.5.2 Semi-characteristic approach......Page 120
4.5.3 Singular surface approach......Page 121
4.6 Weak Nonlinear Waves in an Ideal Plasma......Page 123
4.6.1 Centered rarefaction waves......Page 127
4.6.2 Compression waves and shock front......Page 129
4.7 Relatively Undistorted Waves......Page 131
4.7.1 Finite amplitude disturbances......Page 133
4.7.2 Small amplitude waves......Page 134
4.7.3 Waves with amplitude not-so-small......Page 141
5.1 Weakly Nonlinear Geometrical Optics......Page 143
5.1.1 High frequency processes......Page 144
5.1.2 Nonlinear geometrical acoustics solution in a relaxing gas......Page 146
5.2 Far Field Behavior......Page 147
5.3.1 Formula for energy dissipated at shocks......Page 150
5.3.2 Effect of distributional source terms......Page 152
5.3.3 Application to nonlinear geometrical optics......Page 154
5.4 Evolution Equation Describing Mixed Nonlinearity......Page 156
5.4.1 Derivation of the transport equations......Page 157
5.4.2 The ε-approximate equation and transport equation......Page 159
5.4.4 Energy dissipated across shocks......Page 162
5.4.5 Application......Page 165
5.5 Singular Ray Expansions......Page 167
5.6 Resonantly Interacting Waves......Page 170
6. Self-Similar Solutions Involving Discontinuities......Page 174
6.1 Waves in Self-Similar Flows......Page 176
6.1.1 Self-similar solutions and their asymptotic behavior......Page 177
6.1.2 Collision of a C[sup(1)]-wave with a blast wave......Page 182
6.2 Imploding Shocks in a Relaxing Gas......Page 185
6.2.1 Basic equations......Page 186
6.2.2 Similarity analysis by invariance groups......Page 187
6.2.3 Self-similar solutions and constraints......Page 190
6.2.4 Imploding shocks......Page 197
6.2.5 Numerical results and discussion......Page 198
6.3 Exact Solutions of Euler Equations via Lie Group Analysis......Page 205
6.3.1 Symmetry group analysis......Page 206
6.3.2 Euler equations of ideal gas dynamics......Page 207
6.3.3 Solution with shocks......Page 211
7. Kinematics of a Shock of Arbitrary Strength......Page 213
7.1 Shock Wave through an Ideal Gas in 3-Space Dimensions......Page 214
7.1.1 Wave propagation on the shock......Page 218
7.1.2 Shock-shocks......Page 220
7.1.3 Two-dimensional configuration......Page 222
7.1.4 Transport equations for coupling terms......Page 223
7.1.5 The lowest order approximation......Page 226
7.1.6 First order approximation......Page 228
7.2 An Alternative Approach Using the Theory of Distributions......Page 231
7.3 Kinematics of a Bore over a Sloping Beach......Page 238
7.3.1 Basic equations......Page 239
7.3.2 Lowest order approximation......Page 242
7.3.3 Higher order approximations......Page 244
7.3.4 Results and discussion......Page 245
7.3.5 Appendices......Page 251
Biblibography......Page 257