Chapter 6 Stability of strong discontinuities in fluids and MHD

Handbook of Mathematical Fluid Dynamics

Volumn 1, Issue , 2002, Pages 545-652

  Blokhin, Alexander ^a   Trakhinin, Yuri ^a  

^a SOBOLEV INSTITUTE OF MATHEMATICS   (Russian Federation)

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[No Author keywords available]

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EID: 0038077159     PISSN: 18745792     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5792(02)80013-1     Document Type: Chapter

References (150)

1. Russian. Akhiezer A.I., Liubarskii G.Ia., and Polovin R.V. The stability of shock waves in magnetohydrodynamics. Zh. Eksp. Teor. Fiz. 35 (1958) 731-737
2. English translation. Akhiezer A.I., and Liubarskii G.Ia. Soviet Phys. JETP 35 8 (1959) 507-511
3. Alfvén H. On the existence of electromagnetic-hydrodynamic waves. Ark. Mat. Astron. Fys. B 29 2 (1943) 1-7
4. Anile A.M. Relativistic Fluids and Magneto-Fluids. With Applications in Astrophysics and Plasma Physics (1989), Cambridge Univ. Press, Boston
5. Anile A.M., and Russo G. Corrugation stability for plane relativistic shock waves. Phys. Fluids 29 (1986) 2847-2852
6. Anile A.M., and Russo G. Linear stability for plane relativistic shock waves. Phys. Fluids 30 (1987) 1045-1051
7. Anile A.M., and Russo G. Corrugation Stability of Magnetohydrodynamic Shock Waves, Nonlinear Wave Motion. Pitman Monogr. Surv. Pure Appl. Math. Vol. 43 (1989) 11-21
8. Anile A.M., Blokhin A.M., and Trakhinin Yu.L. Investigation of a mathematical model for radiation hydrodynamics. Z. Angew. Math. Phys. 50 (1999) 677-697
9. Bellman R. Introduction to Matrix Analysis. 2nd (1970), McGraw-Hill, Cambridge
10. Benzoni-Gavage S. Stability of multi-dimensional phase transitions in a van der Waals fluid. Nonlinear Anal. 31 (1998) 243-263
11. Benzoni-Gavage S. Stability of subsonic planar phase boundaries in a van der Waals fluid. Arch. Rational Mech. Anal. 150 (1999) 23-55
12. Benzoni-Gavage S., Rousset F., Serre D., and Zumbrun K. Generic types and transitions in hyperbolic initial-boundary vaue problems. Preprint (2000)
13. Report No. 545,1942. Bethe H.A. On the theory of shock waves for an arbitrary equation of state, Office of Scientific Research and Development. Classic Papers in Shock Compression Science (1982), Springer, New York 421-492
14. Russian. Blokhin A.M. A mixed problem for a system of equations of acoustics with boundary conditions on a shock wave. Izv. Sibirsk. Otdel. Akad. Nauk SSSR Ser. Tekhn. Nauk 13 (1979) 25-33
15. Russian. Blokhin A.M. A mixed problem for a three-dimensional system of equations of acoustics with boundary conditions on the shock wave. Dinamika Sploshn. Sredy 46 (1980) 3-13
16. Russian. Blokhin A.M. Estimation of the energy integral of a mixed problem for gas dynamics equations with boundary conditions on the shock wave. Sibirsk. Mat. Zh. 22 4 (1981) 23-51
17. Blokhin A.M. English translation. Siberian Math. J. 22 4 (1981) 501-523
18. Russian. Blokhin A.M. A mixed problem for symmetric t-hyperbolic systems of acoustic type. Dinamika Sploshn. Sredy 52 (1981) 11-29
19. Russian. Blokhin A.M. Uniqueness of the classical solution of a mixed problem for equations of gas dynamics with boundary conditions on a shock wave. Sibirsk. Mat. Zh. 23 5 (1982) 17-30
20. Blokhin A.M. English translation. Siberian Math. J. 23 5 (1982) 604-615
21. Russian. Blokhin A.M. Symmetrization of Landau equations in the theory of superfluidity of helium II. Dinamika Sploshn. Sredy 68 (1984) 13-34
22. Russian. Blokhin A.M. Energy Integrals and their Applications in Problems of Gas Dynamics (1986), Nauka, Sibirsk. Otdel., New York
23. Blokhin A.M. Strong Discontinuities in Magnetohydrodynamics (1994), Nova Science Publ., Novosibirsk
24. Blokhin A.M. Symmetrization of continuum mechanics equations. Sib. J. Differential Equations 2 (1995) 3-47
25. Blokhin A.M. A new concept of construction of adaptive calculation models for hyperbolic problems. NATO ASI Sen, Ser. C, Math. Phys. Sci. 536 (1999) 23-64
26. Blokhin A.M., and Alaev R.D. Construction of adequate difference models for gas dynamics equations. Siberian J. Comput. Math. 1 2 (1992) 169-189
27. Blokhin A.M., and Dorovsky V.N. Mathematical Modelling in the Theory of Multivelocity Continuum (1995), Nova Science Publ., New York
28. Russian. Blokhin A.M., and Druzhinin I.Yu. Formulation of problems on the stability of discontinuities in magnetohydrodynamics. Boundary Value Problems for Partial Differential Equations (1988), Collect. Sci. Works, New York 16-38
29. Russian. Blokhin A.M., and Druzhinin I.Yu. On the stability of a fast magnetohydrodynamic shock wave for a weak magnetic field. Partial Differential Equations (1989), Collect. Sci. Works, Novosibirsk 15-32
30. Russian. Blokhin A.M., and Druzhinin I.Yu. Stability of shock waves in magnetohydrodynamics. Sibirsk. Mat. Zh. 30 4 (1989) 13-29
31. Blokhin A.M., and Druzhinin I.Yu. English translation. Siberian Math. J. 30 4 (1989) 511-524
32. Russian. Blokhin A.M., and Druzhinin I.Yu. Well-posedness of some linear problems on the stability of strong discontinuities in magnetohydrodynamics. Sibirsk. Mat. Zh. 31 2 (1990) 3-8
33. Blokhin A.M., and Druzhinin I.Yu. English translation. Siberian Math. J. 31 2 (1990) 187-191
34. Russian. Blokhin A.M., and Krymskikh D.A. Symmetrization of equations of magnetohydrodynamics with anisotropic pressure. Boundary Value Problems for Partial Differential Equations (1990), Collect. Sci. Works, Novosibirsk 3-19
35. Novosibirsk. Blokhin A.M., and Krymskikh D.A. Strong discontinuities in superliquid helium. Proc. Inst. Math. 24 (1994) 20-62
36. Russian. Blokhin A.M., and Mishchenko E.V. Symmetrization of relativistic equations of gas dynamics. Dinamika Sploshn. Sredy 88 (1988) 13-22
37. Blokhin A.M., and Mishchenko E.V. Investigation on shock waves stability in relativistic gas dynamics. Matematiche (1993), Catania, Novosibirsk 53-75
38. Russian. Blokhin A.M., and Sokovikov I.G. On a certain approach to constructing difference schemes for quasilinear equations of gas dynamics. Sibirsk. Mat. Zh. 40 6 (1999) 1236-1243
39. Blokhin A.M., and Sokovikov I.G. English translation. Siberian Math. J. 40 6 (1999) 1044-1050
40. Russian. Blokhin A.M., and Trakhinin Yu.L. A rotational discontinuity in magnetohydrodynamics. Sibirsk. Mat. Zh. 34 3 (1993) 3-18
41. Blokhin A.M., and Trakhinin Yu.L. English translation. Siberian Math. J. 34 3 (1993) 395-411
42. Russian. Blokhin A.M., and Trakhinin Yu.L. A rotational discontinuity in magnetohydrodynamics with anisotropic pressure. I. Sibirsk. Mat. Zh. 35 1 (1994) 12-23
43. Blokhin A.M., and Trakhinin Yu.L. English translation. Siberian Math. J. 35 1 (1994) 9-20
44. Russian. Blokhin A.M., and Trakhinin Yu.L. A rotational discontinuity in magnetohydrodynamics with anisotropic pressure. II. Sibirsk. Mat. Zh. 35 2 (1994) 278-287
45. Blokhin A.M., and Trakhinin Yu.L. English translation. Siberian Math. J. 35 2 (1994) 250-259
46. Blokhin A.M., and Trakhinin Yu.L. Investigation of the well-posedness of the mixed problem on the stability of fast Shockwaves in magnetohydrodynamics. Matematiche (Catania) 49 1 (1994) 123-141
47. Russian. Blokhin A.M., and Trakhinin Yu.L. Stability of shock waves for one model of radiation hydrodynamics. Prikl. Mekh. Tekhn. Phiz. 37 6 (1996) 3-14
48. Blokhin A.M., and Trakhinin Yu.L. English translation. J. Appl. Mech. Tech. Phys. 37 6 (1996) 775-784
49. Blokhin A.M., and Trakhinin Yu.L. Stability of fast parallel and transversal MHD shock waves in plasma with pressure anisotropy. Acta Mechanica 135 (1999) 57-71
50. Blokhin A.M., and Trakhinin Yu.L. Stability of fast parallel MHD shock waves in polytropic gas. Eur. J. Mech. B/Fluids 18 2 (1999) 197-211
51. Catania. Blokhin A.M., Merazhov I.Z., and Trakhinin Yu.L. Investigation of stability of electrodynamic shock waves. Matematiche 52 1 (1997) 87-114
52. Russian. Blokhin A.M., Merazhov I.Z., and Trakhinin Yu.L. On the stability of shock waves in the continuum medium with a volume charge. Prikl. Mekh. Tekhn. Fiz. 39 2 (1998) 29-39
53. Blokhin A.M., Merazhov I.Z., and Trakhinin Yu.L. English translation. J. Appl. Mech. Tech. Phys. 39 2 (1998) 184-193
54. Blokhin A.M., Romano V., and Trakhinin Yu.L. Some mathematical properties of radiating gas model obtained with a variable Eddington factor. Z. Angew. Math. Phys. 47 (1996) 639-658
55. Blokhin A.M., Romano V., and Trakhinin Yu.L. Stability of shock waves in relativistic radiation hydrodynamics. Ann. Inst. H. Poincaré Phys. Théor. 67 2 (1997) 145-180
56. Boillat G. Sur Vexistence recherche d'équations de conservation supplémentals pour les systémes hyperboliques. C. R. Acad. Sci. Paris Sér. A 278 (1974) 909-912
57. Chazarain J., and Piriou A. Introduction to the theory of linear partial differential equations (1982), North- Holland
58. Chew G.F., Goldberger M.L., and Low F.E. The Boltzmann equation and the one-fluid hydromagnetic equations in the absence of particle collisions. Proc. Roy. Soc. London Ser. A. 236 (1956) 112-118
59. Courant R., and Hilbert D. Methods of Mathematical Physics (1962), Interscience Publishers, Amsterdam
60. Russian. D'yakov S.P. On stability of shock waves. Zh. Eksp. Teor. Fiz. 27 3 (1954) 288-296
61. D'yakovti S.P. English translation. Atomic Energy Research Establishment AERE Lib./trans. (1956) 648
62. Russian. Egorushkin A. Nonlinear instability of a spontaneously radiating shock wave. Izv. Akad. Nauk SSSR,Mekh. Zhidk. Gaza No. 3 (1984) 110-118
63. Fluid Dyn. 19 3 (1984) 436-443
64. Russian. Egorushkin A., and Kulikovsky A.G. On the stability of solutions of some boundary value problems for hyperbolic equations. Prikl. Mat. Mekh. 56 1 (1992) 40-51
65. English translation. J. Appl. Math. Mech. 56 1 (1992) 36-45
66. Erpenbeck J.J. Stability of step shocks. Phys. Fluids 5 (1962) 1181-1187
67. Russian. Filippova O.L. Stability of plane MHD shock waves in an ideal gas. Izv. Akad. Nauk SSSR,Mekh. Zhidk. Gaza No. 6 (1991) 128-136
68. Filippova O.L. English translation. Fluid Dyn. 26 6 (1991) 897-904
69. Fowles G.R. Stimulated and spontaneous emission of acoustic waves from shock fronts. Phys. Fluids 24 (1981) 220-227
70. Freeman N.C. A theory of the stability of plane shock waves. Proc. Roy. Soc. London Ser. A 228 (1955) 341-362
71. Freistühler H., and Liu T.-P. Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws. Comm. Math. Phys. 153 (1993) 147-158
72. Freistühler H. The persistence of ideal shock waves. Appl. Math. Lett. 7 6 (1994) 7-11
73. Freistühler H. A short note on the persistence of ideal shock waves. Arch. Math. 64 (1995) 344-352
74. Freistühler H. Some results on the stability of non-classical shock waves. J. Partial Differential Equations 11 (1998) 25-38
75. Friedrichs K.O. Symmetric hyperbolic linear differential equations. Comm. Pure and Appl. Math. 27 (1974) 123-131
76. Friedrichs K.O., and Lax P.D. System of conservation equations with a convex extension. Proc. Nat. Acad. Sci. U.S.A. (1971) 1686-1688
77. Gardner C.S., and Kruskal M.D. Stability of plane magnetohydrodynamic shocks. Phys. Fluids 7 (1964) 700-706
78. Gilbarg D. The existence and limit behavior of the one-dimensional shock layer. Amer. J. Math. 73 (1951) 256-274
79. Russian. Godunov S.K. An interesting class of quasi-linear systems. Dokl. Akad. Nauk SSSR 139 (1961) 521-523
80. Russian. Godunov S.K. Symmetrization of magnetohydrodynamics equations, Chislennye Metody Mekhaniki Sploshnoi Sredy. Novosibirsk 3 (1972) 26-34
81. Russian. Godunov S.K. Equations of mathematical physics (1979), Nauka, New York
82. French translation: Equations de la Physique Mathematique. traduit du russe par Edouard Gloukhian (1973), Mir Moscou
83. Russian. Godunov S.K., and Gordienko V.M. A mixed problem for the wave equation. Trudy Sem. S.L. Soboleva 2 (1977) 5-31
84. Russian. Godunov S.K., Zabrodin A.V., Ivanov M.Ya., Krajko A., and Prokopov G.P. Numerical Solving of Multidimensional Gas Dynamics Problems (1976), Nauka, Moscow
85. Moscou M. French translation: Resolution Numerique des Problemes Multidimensionnels de la Dynamique des Gaz, sous la redaction de S. Godounov. traduit du russe par Valeri Platonov (1979)
86. Gordienko V.M. Symmetrization of a mixed problem for a hyperbolic equation of second order with two spatial variables. Siberian Math. J. 22 (1981) 231-248
87. Harten A. On the symmetric form of systems of conservation laws with entropy. J. Comp. Phys. 49 1 (1983) 151-164
88. Hoffman F., and Teller E. Magnetohydrodynamic shocks. Phys. Rev. 80 4 (1950) 696-703
89. Russian. Iordanskii S.V. On the stability of a planar steady shock wave. Prikl. Mat. Mekh. 21 (1957) 465-472
90. Russian. Iordanskii S.V. On compression waves in magnetohydrodynamics. Dokl. Akad. Nauk SSSR (1958) 610-612
91. Iordanskii S.V. English translation. Soviet Phys. Dokl. 3 (1959) 736-738
92. Isaacson E.L., Marchesin D., and Plohr J. Transitional waves for conservation laws. SIAM J. Math. Anal. 21 (1990) 837-866
93. Jeffrey A. Quasilinear Hyperbolic Systems and Waves (1976), Pitman, Moscow
94. Kato T. The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Rational Mech. Anal. 58 (1975) 181-205
95. Russian. Kontorovich V.M. On the shock waves stability. Zh. Eksp. Teor. Fiz. 33 (1957) 1525-1526
96. Kontorovich V.M. English translation. Soviet Phys. JETP 33 6 (1959) 1179-1180
97. Russian. Kontorovich V.M. Stability of shock waves in relativistic hydrodynamics. Zh. Eksp. Teor. Fiz. 34 (1958) 186-194
98. Kontorovich V.M. English translation. Soviet Phys. JETP 34 7 (1960) 127-132
99. Kreiss H.-O. Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math. 23 (1970) 277-296
100. Russian. Kulikovskii A.G., and Lyubimov G.A. Gas-ionizing magnetohydrodynamic shock waves. Dokl. Akad. Nauk SSSR 129 (1959) 52-55
101. English translation. Soviet Phys. Dokl. 4 (1960) 1185-1188
102. Kulikovskii A.G., and Lyubimov G.A. Magnetohydrodynamics (1965), Addison-Wesley, MA, New York
103. Russian. Ladyzhenskaja O.A. Boundary-Value Problems of Mathematical Physics (1973), Nauka
104. Russian. Landau L.D. On the stability of tangential discontinuities in a compressible fluid. Dokl. Akad. Nauk SSSR 44 4 (1944) 151-153
105. Landau L.D., and Lifshiz E.M. Electrodynamics of Continuous Media, Course of Theoretical Physics (1960), Pergamon Press, Moscow
106. Landau L.D., and Lifshiz E.M. Fluid Mechanics, Course of Theoretical Physics (1997), Pergamon Press, Oxford
107. Lankaster P. Theory of Matrices (1969), Academic Press, New York
108. Lax P.D. Hyperbolic systems of conservation laws (II). Comm. Pure Appl. Math. 10 (1957) 537-566
109. No. 11. Lax P.D. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Philadelphia. SIAM Reg. Conf. (1973)
110. Lax P.D., and Philips PS. Local boundary conditions for dissipative symmetric linear operators. Comm. Pure Appl. Math. 13 (1960) 427-455
111. Lessen M., and Deshpande M.V. Stability of magnetohydrodynamic shocks waves. J. Plasma Physics 1 4 (1967) 463-472
112. Liu T.R. Nonlinear stability and instability of overcompressive shock waves. IMA Vol. Math. Appl. 52 (1993) 159-167
113. Majda A. The stability of multi-dimensional shock fronts - a new problem for linear hyperbolic equations. Mem. Amer. Math. Soc. 41 275 (1983)
114. Majda A. The existence of multi-dimensional shock fronts. Mem. Amer. Math. Soc. 43 281 (1983)
115. Majda A. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (1984), Springer, New York
116. Majda A., and Osher S. Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary. Comm. Pure Appl. Math. 28 (1975) 607-675
117. Majda A., and Rosales R. A theory for spontaneous Mach stem formation in reacting shock fronts. I, The basic perturbation analysis. SIAM J. Appl. Math. 43 (1983) 1310-1334
118. Majda A., and Rosales R. A theory for spontaneous Mach-stem formation in reacting shock fronts. II.Steady-wave bifurcations and the evidence for breakdown. Stud. Appl. Math. 71 (1984) 117-148
119. Metivier G. The block structure condition for symmetric hyperbolic systems. Bull. London Math. Soc. 32 (2000) 689-702
120. G. Métivier, Stability of Multidimensional Shocks, Recent Advances in the Theory of Shock Waves, Progress in Nonlinear Differential Equations, Birkhäuser, to appear.
121. Mizohata S. The Theory of Partial Differential Equations (1973), Cambridge Univ. Press, New York
122. Thése. Mokrane A. Problem es mixtes hyperboliques non lineaires (1987), Université de Rennes, New York
123. Russian. Ovsyannikov L.V. Lectures on the Fundamentals of Gas Dynamics (1981), Nauka
124. Russian. Polovin R.V., and Demutskij V.P. Fundamentals of Magnetohydrodynamics (1987), Atomizdat, Moscow
125. Ralston F.V. Note on a paper of Kreiss. Comm. Pure Appl. Math. 24 (1971) 759-762
126. 2 is a continuable initial condition for Kreiss mixed problems. Comm. Pure Appl. Math. 25 (1971) 265-285
127. 0Rosales R., and Majda A. Weakly nonlinear detonation waves. SIAM J. Appl. Math. 43 (1983) 1086-1118
128. Providence. Rozhdestvenskii B.L., and Janenko N.N. Systems of Quasilinear Equations and Their Applications to Gas Dynamics. Transl. Math. Monogr. Amer. Math. Soc (1983)
129. Ruggeri T., and Strumia A. Main field and convex covariant density for quasilinear hyperbolic systems. Relativistic fluid dynamics. Ann. Inst. H. Poincaré Sect. A (N.S.) 34 1 (1981) 65-84
130. Ruggeri T., and Strumia A. Convex covariant entropy density, symmetric conservative form, and shock waves in relativistic magnetohydrodynamics. J. Math. Phys. 22 (1981) 1824-1827
131. Russo G., and Anile A.M. Stability properties of relativistic shock waves: basic results. Phys. Fluids 30 (1987) 2406-2413
132. Sablé-Tougeron M. Existence pour un probleme de I'elastodynamique Neumann non lineaire en dimension 2. Arch. Rational Mech. Anal. 101 (1988) 261-292
133. Sedov L.I. Continuum Mechanics. World Scientific, River Edge, NJ (1997)
134. Shearer M., Schaeffer D.G., Marchesin D., and Paes-Leme P. Solution of the Riemann problem for a prototype 2 x 2 system of non-strictly hyperbolic consevation laws. Arch. Rational Mech. Anal. 97 (1987) 299-320
135. Providence. Sobolev S.L. Applications of Functional Analysis in Mathematical Physics. Transl. Math. Monogr. Amer. Math. Soc (1963)
136. Russian. Syrovatskij S.I. The instability of tangential discontinuities in a compressible fluid. Zh. Eksp. Teor. Fiz. 27 (1954) 121-123
137. Russian. Syrovatskij S.L. The stability of shock waves in magnetohydrodynamics. Zh. Eksp. Teor. Fiz. 35 (1958) 1466-1470
138. Syrovatskij S.L. English translation. SovietPhys. JETP 35 8 (1959) 1024-1027
139. Taub A.H. Relativistic fluid mechanics. Ann. Rev. Fluid Mech. 10 (1978) 301-332
140. Thoene K.S. Astrophys. J. 179 (1973) 897
141. Trakhinin Yu.L. On stability of shock waves in relativistic magnetohydrodynamics. Quart. Appl. Math. 59 (2001) 25-45
142. Preprint 14. Trakhinin Yu.L. A complete 2D stability analysis of fast MHD shocks in an ideal gas. English translation submitted for publication. (2002), Novosibirsk State University, Moscow
143. Vinokur M. Conservation equations of gasdynamics in curvilinear coordinate systems. J. Comput. Phys. 14 (1974) 105-125
144. Russian. Vol'pert A.I., and Hudjaev S.I. On the Cauchy problem for composite systems of nonlinear differential equations. Mat. Sb. 87 4 (1973) 504-528
145. Vinokur M. English translation: Math. USSR Sb. 16 (1973) 517-544
146. Whitham G.B. Linear and Nonlinear Waves (1974), Wiley, Russian
147. Wu C.C. New theory of MHD shock waves. Viscous Profiles and Numerical Methods for Shock Waves (Proc. Workshop, Raleigh/NC (USA) 1990) (1991) 209-236
148. Russian. Zaidel R.M. On the stability of planar shock waves. Prikl. Mekh. Tekhn. Fiz. 8 4 (1967) 30-39
149. Zumbrun K., and Howard P. Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. (1998) 741-748
150. Zumbrun K., and Serre D. Viscous and inviscid stability of multidimensional planar viscous shock waves. Indiana Univ. Math. J. 48 (1999) 937-992