Relativistic hydrodynamics and magnetohydrodynamics: Lectures on the existence of solutions

Cover......Page 1
Series contents......Page 3
Title......Page 4
Dateline......Page 5
Preface......Page 6
Contents......Page 8
1. Einstein Equations......Page 12
2. Elementary Analysis of the Cauchy Problem ......Page 13
3. Harmonic Coordinates ......Page 15
4. Einstein Equations in Harmonic Coordinates ......Page 18
5. Strictly Hyperbolic Matrices ......Page 21
6. Leray Systems ......Page 22
7. The Exterior Case ......Page 25
8. The Case of the Pure Matter ......Page 28
9. Thermodynamical Perfect Fliud ......Page 34
10. The System of the Stream Lines ......Page 36
11. Equation of Continuity and Conservation of the Matter......Page 40
12. Relativistic Helmholtz Equations ......Page 41
13. The System of Relativistic Hydrodynamics and its Characteristics ......Page 42
14. Laplacian of the Current С ......Page 48
15. Existence and Uniqueness Theorem ......Page 53
16. Another Method ......Page 56
17. Incompressible Thermodynamical Fluid ......Page 62
18. Irrotational Motion ......Page 64
19. The Main Differential System for the Fluid ......Page 66
20. The Equation of Continuity and the Differential System of the Stream Lines ......Page 68
21. Relativistic Helmholtz Equations ......Page 71
22. Relations between Hydrodynamic Current and Vorticity Tensor ......Page 73
23. The Operator $\Delta^{(h)}$ ......Page 75
24. The Equations in Harmonic Coordinates with the Lorentz Condition ......Page 76
25. Differential System of the Stream Lines Deduced for the System (II) ......Page 79
26. Characteristics of the System-Analytic Cauchy Problems ......Page 81
27. Helmholtz Equations Deduced from the System (II) ......Page 84
28. Relations between Hydrodynamic Current and Vorticity Tensor Deduced from the System (II) ......Page 85
29. Existence and Uniqueness Theorem ......Page 89
30. Electromagnetic Field with Induction ......Page 94
31. Maxwell Equations and Energy Tensor of Minkowski ......Page 98
32. The Case of the Magnetohydrodynamics ......Page 104
33. The Main Equations of Relativistic Magnetohydrodynamics ......Page 105
34. The Equation of Continuity and the Differential System of the Stream Lines ......Page 107
35. Characteristics of the Relativistic Magnetohydrodynamics ......Page 111
36. Waves and Velocities of Propagation ......Page 118
37. Some Formulas with a Physical Interest ......Page 125
38. The Case of an Incompressible Fluid ......Page 127
39. Equations of the Helmholtz Type ......Page 129
40. Representation of the Wave Cones in $R^3$ ......Page 133
41. Auxiliary Equations, Consequences of the Main System ......Page 138
42. The Equations Corresponding to $f$ and $|\mathbf{h}|^2$ ......Page 143
43. The Equations Corresponding to the $u^\alpha$ ......Page 145
44. A Lemma ......Page 147
45. System (III) of Magnetohydrodynamics ......Page 154
46. The Main System ......Page 161
47. Shock Equations ......Page 162
48. Tangential Shocks ......Page 164
49. Invariants of the Nontangential Shocks ......Page 165
50. Analysis of a Nontangential Shock ......Page 169
51. Analysis of the Alfven Shocks ......Page 173
52. The Vector $U^\beta$ for a Nontangential Shock ......Page 177
53. Relativistic Hugoniot Equation ......Page 180
54. Classical Approximation of the Relativistic Magnetohydrodynamics ......Page 183
55. Thermodynamics of the Shocks ......Page 187
56. Shock Wave and Alfven Waves ......Page 191
57. Compatibility of a Shock Wave with the Alfven Waves ......Page 193
58. The Compressibility Conditions ......Page 197
Conclusion ......Page 202
References ......Page 204
Index ......Page 206