Planar stability criteria for viscous shock waves of systems with real viscosity

Lecture Notes in Mathematics

Volumn 1911, Issue , 2007, Pages 229-326

  Zumbrun, Kevin ^a  

^a INDIANA UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 34547243052     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-540-72187-1_4     Document Type: Article

References (122)

1. A.A. Barmin and S.A. Egorushkin, Stability of shock waves. Adv. Mech. 15 (1992) No. 1-2, 3-37.
2. S. Benzoni-Gavage, Stability of semi-discrete shock profiles by means of an Evans function in infinite dimensions. J. Dynam. Differential Equations 14 (2002), no. 3, 613-674.
3. S. Benzoni-Gavage, P. Huot, and F. Rousset, Nonlinear stability of semidiscrete. shock waves, preprint (2003).
4. S. Benzoni-Gavage, D. Serre, and K. Zumbrun, Generic types and transitions in hyperbolic initial-boundary-value problems. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 1073-1104.
5. S. Benzoni-Gavage, D. Serre, and K. Zumbrun, Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32 (2001), 929-962.
6. S. Bianchini and A. Bressan, BV solutions for a class of viscous hyperbolic systems. Indiana Univ. Math. J. 49 (2000), no. 4, 1673-1713.
7. S. Bianchini and A. Bressan, The vanishing viscosity limit for a class of viscous hyperbolic systems, preprint (2002).
8. A.M. Blokhin, Strong discontinuities in magnetohydrodynamics. Translated by A. V. Zakharov. Nova Science Publishers, Inc., Commack, NY, 1994. x+150 pp. ISBN: 1-56072-144-8.
9. A. Blokhin and Y. Trakhinin, Stability of strong discontinuities in fluids and MHD. in Handbook of mathematical fluid dynamics, Vol. I, 545-652, North-Holland, Amsterdam, 2002.
10. A.M. Blokhin and Y. Trakhinin, Stability of fast parallel MHD shock waves in polytropic gas. Eur. J. Mech. B Fluids 18 (1999) 197-211.
11. A.M. Blokhin and Y. Trakhinin, Stability of fast parallel and transversal MHD shock waves in plasma with pressure anisotropy. Acta Mech. 135 (1999).
12. A.M. Blokhin and Y. Trakhinin, Hyperbolic initial-boundary value problems on the stability of strong discontinuities in continuum mechanics. Hyperbolic problems: theory, numerics, applications, Vol. I (Zürich, 1998), 77-86, Internat. Ser. Numer. Math., 129, Birkhäuser, Basel, 1999.
13. A.M. Blokhin, Y. Trakhinin, and I.Z. Merazhov, On the stability of shock waves in a continuum with bulk charge. (Russian) Prikl. Mekh. Tekhn. Fiz. 39 (1998) 29-39; translation in J. Appl. Mech. Tech. Phys. 39 (1998) 184-193.
14. A.M. Blokhin, Y. Trakhinin, and I.Z. Merazhov, Investigation on stability of electrohydrodynamic shock waves. Matematiche (Catania) 52 (1997) 87-114 (1998).
15. G. Boillat, On symmetrization of partial differential systems. Appl. Anal. 57 (1995) 17-21.
16. A. Bressan, Center manifold notes, lecture notes (2002).
17. T.J. Bridges, G. Derks, and G. Gottwald, Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys. D 172 (2002), no. 1-4, 190-216.
18. L. Brin, Numerical testing of the stability of viscous shock waves, Doctoral thesis, Indiana University (1998).
19. L. Q. Brin, Numerical testing of the stability of viscous shock waves. Math. Comp. 70 (2001) 235, 1071-1088.
20. L. Brin and K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves. Mat. Contemp. (2003).
21. R. Courant and K.O. Friedrichs, Supersonic flow and shock waves, Springer-Verlag, New York (1976) xvi+464 pp.
22. J. Chazarain-A. Piriou, Introduction to the theory of linear partial differential equations, Translated from the French. Studies in Mathematics and its Applications, 14. North-Holland Publishing Co., Amsterdam-New York, 1982. xiv+559 pp. ISBN: 0-444-86452-0.
23. J.W. Evans, Nerve axon equations: I. Linear approximations. Ind. Univ. Math. J. 21 (1972) 877-885.
24. J.W. Evans, Nerve axon equations: II. Stability at rest. Ind. Univ. Math. J. 22 (1972) 75-90.
25. J.W. Evans, Nerve axon equations: III. Stability of the nerve impulse. Ind. Univ. Math. J. 22 (1972) 577-593.
26. J.W. Evans, Nerve axon equations: IV. The stable and the unstable impulse. Ind. Univ. Math. J. 24 (1975) 1169-1190.
27. J. Francheteau and G. Métivier, Existence de chocs faibles pour des systémes quasi-linéaires hyperboliques multidimensionnels. C.R.AC.Sc. Paris, 327 Série I (1998) 725-728.
28. H. Freistühler, Some results on the stability of Non-classical shock waves. J. Partial Diff. Eqs. 11 (1998), 23-38.
29. H. Freistühler, Dynamical stability and vanishing viscosity: A case study of a nonstrictly hyperbolic system of conservation laws. Comm. Pure Appl. Math. 45 (1992) 561-582.
30. H. Freistühler, A short note on the persistence of ideal shock waves. Arch. Math. (Basel) 64 (1995), no. 4, 344-352.
31. H. Freistühler and T.-P. Liu, Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws. Commun. Math. Phys. 153 (1993) 147-158.
32. H. Freistühler and P. Szmolyan, Spectral stability of small shock waves. Arch. Ration. Mech. Anal. 164 (2002) 287-309.
33. H. Freistühler and K. Zumbrun, Examples of unstable viscous shock waves. unpublished note, Institut für Mathematik, RWTH Aachen, February 1998.
34. K.O. Friedrichs, Symmetric hyperbolic linear differential equations. Comm. Pure and Appl. Math. 7 (1954) 345-392.
35. K.O. Friedrichs, On the laws of relativistic electro-magneto-fluid dynamics. Comm. Pure and Appl. Math. 27 (1974) 749-808.
36. K.O. Friedrichs and P. Lax, Systems of conservation equations with a convex extension. Proc. nat. Acad. Sci. USA 68 (1971) 1686-1688.
37. R. Gardner and K. Zumbrun, The Gap Lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51 (1998), no. 7, 797-855.
38. I.M. Gelf́and, Some problems in theory of quasilinear equations. Am. Mat. Soc. Trans., Ser. 2, 29 (1963) 295-381.
39. D. Gilbarg, The existence and limit behavior of the one-dimensional shock layer. Amer. J. Math. 73, (1951). 256-274.
40. P. Godillon, Stabilité linéaire des profils pour les systèmes avec relaxation semi-linéaire. Phys. D 148 (2001) 289-316.
41. S.K. Godunov, An interesting class of quasilinear systems. Sov. Math. 2 (1961) 947-948.
42. J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rational Mech. Anal. 95 (1986), no. 4, 325-344.
43. J. Goodman, Remarks on the stability of viscous shock waves, in: Viscous profiles and numerical methods for shock waves. (Raleigh, NC, 1990), 66-72, SIAM, Philadelphia, PA, (1991).
44. J. Goodman, Stability of viscous scalar shock fronts in several dimensions. Trans. Amer. Math. Soc. 311 (1989), no. 2, 683-695.
45. J. Goodman and J.R.. Miller, Long-time behavior of scalar viscous shock fronts in two dimensions. J. Dynam. Differential Equations 11 (1999), no. 2, 255277.
46. J. Goodman and Z. Xin, Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rat. Mech. Anal. 121 (1992), no. 3, 235-265.
47. E. Grenier-F. Rousset, Stability of one-dimensional boundary layers by using Green's functions. Comm. Pure Appl. Math. 54 (2001), 1343-1385.
48. O. Gues, G. Métivier, M. Williams, and K. Zumbrun, Multidimensional viscous shocks I: degenerate symmetrizers and long time stability, preprint (2002).
49. O. Gues, G. Métivier, M. Williams, and K. Zumbrun, Multidimensional viscous shocks II: the small viscosity limit. to appear, Comm. Pure and Appl. Math. (2004).
50. O. Gues, G. Métivier, M. Williams, and K. Zumbrun, A new approach to stability of multidimensional viscous shocks, preprint (2003).
51. O. Gues, G. Métivier, M. Williams, and K. Zumbrun, Navier-Stokes regularization of multidimensional Euler shocks, in preparation.
52. R. Hersh, Mixed problems in several variables. J. Math. Mech. 12 (1963) 317-334.
53. D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin (1981), iv + 348 pp.
54. D. Hoff and K. Zumbrun, Pointwise Green's function 'bounds for multi-dimensional scalar viscous shock fronts. J. Differential Equations 183 (2002) 368-408.
55. D. Hoff and K. Zumbrun, Asymptotic behavior of multi-dimensional scalar viscous shock fronts. Indiana Univ. Math. J. 49 (2000) 427-474.
56. J. Humpherys and K. Zumbrun, Spectral stability of small amplitude shock profiles for dissipative symmetric hyperbolic-parabolic systems. Z. Angew. Math. Phys. 53 (2002) 20-34.
57. K. Jenssen and G. Lyng, Evaluation of the Lopatinski determinant for multi-dimensional Euler equations. Appendix to [Z.3] (see below).
58. K. Jenssen and G. Lyng, Low frequency multi-dimensional stability of viscous detonation waves, in preparation.
59. F. John, Formation of singularities in one-dimensional nonlinear wave propagation. Comm. Pure Appl. Math. 27 (1974), 377-405.
60. C.K.R.T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system. Trans. Amer. Math. Soc. 286 (1984), 431-469.
61. T. Kapitula and B. Sandstede, Stability of bright solitary-wave solutions to perturbed nonlinear Schrdinger equations. Phys. D 124 (1998), no. 1-3, 58-103.
62. T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin Heidelberg (1985).
63. S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. thesis, Kyoto University (1983).
64. S. Kawashima and Y. Shizuta, On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws. Tohoku Math. J. 40 (1988) 449-464.
65. S. Klainerman and A. Majda, Formation of singularities for wave equations including the nonlinear vibrating string. Comm. Pure Appl. Math. 33 (1980), no. 3, 241-263.
66. H.O. Kreiss, Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math. 23 (1970) 277-298.
67. G. Kreiss and H.O. Kreiss, Stability of systems of viscous conservation laws. Comm. Pure Appl. Math. 51 (1998), no. 11-12, 1397-1424.
68. G. Kreiss and M. Liefvendahl, Numerical investigation of examples of unstable viscous shock waves. Hyperbolic problems: theory, numerics, applications, Vol. I, II (Magdeburg, 2000), 613-621, Internat. Ser. Numer. Math., 140, 141, Birkhäuser, Basel, 2001.
69. P.D. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 11. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. v+48 pp.
70. P.D. Lax, Hyperbolic systems of conservation laws. II. Comm. Pure Appl. Math. 10 1957 537-566.
71. T.-P. Liu, Pointwise convergence to shock waves for viscous conservation laws. Comm. Pure Appl. Math. 50 (1997), no. 11, 1113-1182.
72. T.-P. Liu, The entropy condition and the admissibility of shocks. J. Math. Anal. Appl. 53 (1976) 78-88.
73. T.-P. Liu, Nonlinear stability and instability of overcompressive shock waves, in: Shock induced transitions and phase structures in general media. 159-167, IMA Vol. Math. Appl., 52, Springer, New York, 1993.
74. T.-P. Liu, Interactions of Hyperbolic waves, in: Viscous profiles and numerical methods for shock waves. ed: M. Shearer, Philadelphia, SIAM (1991) 66-72.
75. T.P. Liu and K. Zumbrun, Nonlinear stability of an undercompressive shock for complex Burgers equation. Comm. Math. Phys. 168 (1995), no. 1, 163-186.
76. T.P. Liu and K. Zumbrun, On nonlinear stability of general undercompressive viscous shock waves. Comm. Math. Phys. 174 (1995), no. 2, 319-345.
77. T.-P. Liu and Y. Zeng, Large time behavior of solutions for general quasilinear hyperbolic-parabolic systems of conservation laws. AMS memoirs 599 (1997).
78. G. Lyng, One-dimensional stability of combustion waves. Thesis, Indiana University (2003).
79. G. Lyng and K. Zumbrun, A stability index for detonation waves in Majda's model for reacting flow, preprint (2003).
80. G. Lyng and K. Zumbrun, On the one-dimensional stability of viscous strong detonation waves. preprint (2003).
81. A. Majda, The stability of multi-dimensional shock fronts - a new problem for linear hyperbolic equations. Mem. Amer. Math. Soc. 275 (1983).
82. A. Majda, The existence of multi-dimensional shock fronts. Mem. Amer. Math. Soc. 281 (1983).
83. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables. Springer-Verlag, New York (1984), viii+ 159 pp.
84. A. Majda and R. Pego, Stable viscosity matrices for systems of conservation laws. J. Diff. Eqs. 56 (1985) 229-262.
85. C. Mascia and K. Zumbrun, Pointwise Green's function bounds and stability of relaxation shocks. Indiana Univ. Math. J. 51 (2002), no. 4, 773-904.
86. C. Mascia and K. Zumbrun, Stability of shock profiles of dissipative symmetric hyperbolic-parabolic systems. preprint (2001).
87. C. Mascia and K. Zumbrun, Pointwise Green's function bounds for shock profiles with degenerate viscosity. Arch. Rational Mech. Anal., to appear.
88. C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles of hyperbolic-parabolic systems. Arch. Rational Mech. Anal., to appear.
89. C. Mascia and K. Zumbrun, Stability of large-amplitude shock profiles for general relaxation systems. preprint (2003).
90. R. Menikoff and B. Plohr, The Riemann problem for fluid flow of real materials. Rev. Modern Phys. 61 (1989), no. 1, 75-130.
91. G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d'espace. Trans. Amer. Math. Soc. 296 (1986) 431-479.
92. G. Métivier, Stability of multidimensional shocks. Advances in the theory of shock waves, 25-103, Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser Boston, Boston, MA, 2001.
93. G.Métivier. The Block Structure Condition for Symmetric Hyperbolic Problems. Bull. London Math.Soc., 32 (2000), 689-702
94. G.Métivier. Stability of multidimensional weak shocks, Comm. Partial Diff. Equ. 15 (1990) 983-1028.
95. G.Métivier-K.Zumbrun, Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems. preprint (2002).
96. G.Métivier-K.Zumbrun, Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. to appear, J. Discrete. Cont. Dyn. Systems (2004).
97. G.Métivier-K.Zumbrun, Hyperbolic boundary value problems for symmetric systems with variable multiplicities, in preparation.
98. A. Pazy, Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44, Springer-Verlag, New York-Berlin, (1983) viii+279 pp. ISBN: 0-387-90845-5.
99. R.L. Pego, Stable viscosities and shock profiles for systems of conservation laws. Trans. Amer. Math. Soc. 282 (1984) 749-763.
100. R. L. Pego-M.I. Weinstein, Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340 (1992), 47-94.
101. R. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude viscous shock profiles. to appear, J. Discrete and Continuous Dynamical Systems.
102. K. Promislow and N. Kutz, Bifurcation and asymptotic stability in the large detuning limit of the optical parametric oscillator. Nonlinearity 13 (2000) 675-698.
103. 0-semigroups. Trans. Amer. Math. Soc. 284 (1984), no. 2, 847-857.
104. Lord Rayleigh (J.W. Strutt), On the stability, or instability, of certain fluid motions, II, Scientific Papers, 3 (1887) 2-23, Cambridge University Press.
105. D. Sattinger, On the stability of waves of nonlinear parabolic systems. Adv. Math. 22 (1976) 312-355.
106. T. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations. Arch. Rational Mech. Anal. 86 (1984), no. 4, 369-381.
107. J. Smoller, Shock waves and reaction-diffusion equations. Second edition, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, 258. Springer-Verlag, New York, 1994. xxiv+632 pp. ISBN: 0-387-94259-9.
108. A. Szepessy and Z. Xin, Nonlinear stability of viscous shock waves. Arch. Rat. Mech. Anal. 122 (1993) 53-103.
109. A. Szepessy and K. Zumbrun, Stability of rarefaction waves in viscous media. Arch. Rat. Mech. Anal. 133 (1996) 249-298.
110. M.Taylor. Partial Differential EquationsIII.. Applied Mathematical Sciences 117, Springer, 1996.
111. J. von Neumann, Collected works. Vol. VI: Theory of games, astrophysics, hydrodynamics and meteorology. General editor: A. H. Taub. A Pergamon Press Book The Macmilkn Co., New York 1963 x+538 pp. (1 plate).
112. M. Williams, Stability of multidimensional viscous shock waves. C.I.M.E. summer school notes (2003).
113. W.-A. Yong, Singular perturbations of first-order hyperbolic systems. PhD Thesis, Universität Heidelberg (1992).
114. K. Yosida,, Functional analysis. Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995, xii+501 pp. ISBN: 3-540-58654-7.
115. S.-H. Yu. Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Ration. Mech. Anal. 146 (1999), no. 4, 275-370.
116. K. Zumbrun, Stability of viscous shock waves. Lecture Notes, Indiana University (1998).
117. K. Zumbrun, Refined Wave-tracking and Nonlinear Stability of Viscous Lax Shocks. Methods Appl. Anal. 7 (2000) 747-768.
118. K. Zumbrun, Multidimensional stability of planar viscous shock waves. Advances in the theory of shock waves, 307-516, Progr. Nonlinear Differential Equations Appl., 47, Birkhuser Boston, Boston, MA, 2001.
119. K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, for Handbook of Fluid Dynamics, preprint (2003).
120. K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves. Indiana Mathematics Journal V47 (1998), 741-871.
121. K. Zumbrun-P. Howard, Erruta to: "Pointwise semigroup methods, and stability of viscous shock waves" [Indiana Univ. Math. J. 47 (1998), no. 3, 741-871. Indiana Univ. Math. J. 51 (2002), no. 4, 1017-1021.
122. K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts. Indiana Univ. Math. J. 48 (1999) 937-992.