Skip to main content
SpringerLink
Log in

Stability and Dynamics of Viscous Shock Waves

  • Conference paper
  • First Online:
Nonlinear Conservation Laws and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 153))

Abstract

We examine from a classical dynamical systems point of view stability, dynamics, and bifurcation of viscous shock waves and related solutions of nonlinear pde. The central object of our investigations is the Evans function: its meaning, numerical approximation, and behavior in various asymptotic limits.

Research of K. Zumbrun was partially supported under NSF grants no. DMS-0300487 and DMS-0801745.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. G. Abouseif and T.Y. Toong, Theory of unstable one-dimensional detonations, Combust. Flame 45 (1982) 67–94.

    Google Scholar 

  2. J. Alexander, R. Gardner, and C.K.R.T. Jones, A topological invariant arising in the analysis of traveling waves, J. Reine Angew. Math. 410 (1990) 167–212.

    MATH  MathSciNet  Google Scholar 

  3. A. Azevedo, D. Marchesin, B. Plohr, and K. Zumbrun, Nonuniqueness of solutions of Riemann problems. Z. Angew. Math. Phys. 47 (1996), 977–998.

    MATH  MathSciNet  Google Scholar 

  4. J.C. Alexander and R. Sachs, Linear instability of solitary waves of a Boussinesq-type equation: a computer assisted computation, Nonlinear World 2 (1995) 471–507.

    MATH  MathSciNet  Google Scholar 

  5. L. Allen and T.J. Bridges, Numerical exterior algebra and the compound matrix method, Numer. Math. 92 (2002) 197–232.

    MATH  MathSciNet  Google Scholar 

  6. R.L. Alpert and T.Y. Toong, Periodicity in exothermic hypersonic flows about blunt projectiles, Acta Astron. 17 (1972) 538–560.

    Google Scholar 

  7. J. Anderson, Magnetohydrodynamics Shock Waves, MIT Press, Cambridge, MA (1963).

    Google Scholar 

  8. M. Artola and A. Majda, Nonlinear kink modes for supersonic vortex sheets, Phys. Fluids A 1 (1989), no. 3, 583–596.

    MathSciNet  Google Scholar 

  9. R.L. Alpert and T.Y. Toong, Periodicity in exothermic hypersonic flows about blunt projectiles, Acta Astron. 17 (1972) 538–560.

    Google Scholar 

  10. U.M. Ascher, H. Chin, and S. Reich, Stabilization of DAEs and invariant manifolds, Numer. Math. 67 (1994) 131–149.

    MATH  MathSciNet  Google Scholar 

  11. B. Barker, J. Humpherys, K. Rudd, and K. Zumbrun, Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys. 281 (2008), no. 1, 231–249.

    MATH  MathSciNet  Google Scholar 

  12. B. Barker, J. Humpherys, and K. Zumbrun, One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamics, to appear, J. Diff. Eq.

    Google Scholar 

  13. B. Barker, M. Johnson, P. Noble, M. Rodrigues, and K. Zumbrun, Spectral stability of periodic viscous roll waves, in preparation:

    Google Scholar 

  14. B. Barker, M. Johnson, P. Noble, M. Rodrigues, and K. Zumbrun, Witham averaged equations and modulational stability of periodic solutions of hyperbolic-parabolic balance laws, Proceedings, French GDR meeting on EDP, Port D’Albret, France; to appear.

    Google Scholar 

  15. B. Barker, M. Johnson, M. Rodrigues, and K. Zumbrun, Metastability of solitary roll wave solutions of the St. Venant equations with viscosity, preprint (2010).

    Google Scholar 

  16. B. Barker, O. Lafitte, and K. Zumbrun, Existence and stability of viscous shock profiles for 2-D isentropic MHD with infinite electrical resistivity, to appear, J. Diff. Eq.

    Google Scholar 

  17. B. Barker, M. Lewicka, and K. Zumbrun, Existence and stability of viscoelastic shock profiles, to appear, Arch. Ration. Mech. Anal.

    Google Scholar 

  18. B. Barker and K. Zumbrun, A numerical stability investigation of strong ZND detonations for Majda’s model, preprint (2010).

    Google Scholar 

  19. G.K. Batchelor, An introduction to fluid dynamics, Cambridge Mathematical Library. Cambridge University Press, Cambridge, paperback edition, 1999.

    MATH  Google Scholar 

  20. A.A. Barmin and S.A. Egorushkin, Stability of shock waves, Adv. Mech. 15 (1992) no. 1–2, 3–37.

    MathSciNet  Google Scholar 

  21. M. Beck, B. Sandstede, and K. Zumbrun, Nonlinear stability of timeperiodic shock waves, Arch. Rat. Mechanics and Anal. 196 (2010) no. 3, 1011–1076.

    MATH  MathSciNet  Google Scholar 

  22. M. Beck, H.J. Hupkes, B. Sandstede, and K. Zumbrun, Nonlinear stability of semi-discrete shocks for general numerical schemes, SIAM J. Math. Anal. 42 (2010), no. 2, 857–903.

    MATH  MathSciNet  Google Scholar 

  23. S. Benzoni-Gavage, Semi-discrete shock profiles for hyperbolic systems of conservation laws, Phys. D 115 (1998) 109–123.

    MATH  MathSciNet  Google Scholar 

  24. S. Benzoni-Gavage and P. Huot, Existence of semi-discrete shocks, Discrete Contin. Dyn. Syst. 8 (2002) 163–190.

    MATH  MathSciNet  Google Scholar 

  25. S. Benzoni-Gavage, P. Huot, and F. Rousset, Nonlinear stability of semidiscrete shock waves, SIAM J. Math. Anal. 35 (2003) 639–707.

    MATH  MathSciNet  Google Scholar 

  26. S. Benzoni-Gavage, F. Rousset, 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.

    MATH  MathSciNet  Google Scholar 

  27. S. Benzoni-Gavage, D. Serre, and K. Zumbrun, Transition to instability of planar viscous shock fronts, to appear, Z.A.A.

    Google Scholar 

  28. W.-J. Beyn, The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Analysis 9 (1990) 379–405.

    MathSciNet  Google Scholar 

  29. A. Bourlioux, A. Majda, and V. Roytburd, Theoretical and numerical structure for unstable one-dimensional detonations. SIAM J. Appl. Math. 51 (1991) 303–343.

    MATH  MathSciNet  Google Scholar 

  30. 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.

    MATH  MathSciNet  Google Scholar 

  31. L. Brin, Numerical testing of the stability of viscous shock waves. Doctoral thesis, Indiana University (1998).

    Google Scholar 

  32. L.Q. Brin, Numerical testing of the stability of viscous shock waves. Math. Comp. 70 (2001) 235, 1071–1088.

    MathSciNet  Google Scholar 

  33. L. Brin and K. Zumbrun, Analytically varying eigenvectors and the stability of viscous shock waves. Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001). Mat. Contemp. 22 (2002), 19–32.

    MATH  MathSciNet  Google Scholar 

  34. H. Cabannes, Theoretical Magnetofluiddynamics, Academic PRess, New York (1970).

    Google Scholar 

  35. N. Costanzino, J. Humpherys, T. Nguyen, and K. Zumbrun, Spectral stability of noncharacteristic boundary layers of isentropic Navier-Stokes equations, Arch. Rat. Mechanics and Anal. 192 (2009), no. 1–3, 119–136.

    MathSciNet  Google Scholar 

  36. A. Davey, An automatic orthonormalization method for solving stiff boundary value problems, J. Comput. Phys. 51 (1983) 343–356.

    MATH  MathSciNet  Google Scholar 

  37. J.W. Demmel, L. Dieci, and M.J. Friedman, Computing connecting orbits via an improved algorithm for continuing invariant subspaces. SIAM J. Sci. Comput. 22 (2000) 81–94.

    MATH  MathSciNet  Google Scholar 

  38. L. Dieci and T. Eirola, Applications of smooth orthogonal factorizations of matrices. In Numerical methods for bifurcation problems and largescale dynamical systems (Minneapolis, MN, 1997). Springer, IMA Vol. Math. Appl. 119 (2000) 141–162.

    MathSciNet  Google Scholar 

  39. L. Dieci and T. Eirola, On smooth decompositions of matrices. SIAM J. Matrix Anal. Appl. 20 (1999) 800–819.

    MATH  MathSciNet  Google Scholar 

  40. L. Dieci and M.J. Friedman, Continuation of invariant subspaces. Numer. Lin. Alg. Appl. 8 (2001) 317–327.

    MATH  MathSciNet  Google Scholar 

  41. L.O. Drury, Numerical solution of Orr-Sommerfeld-type equations, J. Comput. Phys. 37 (1980) 133–139.

    MATH  MathSciNet  Google Scholar 

  42. J.J. Erpenbeck, Stability of steady-state equilibrium detonations, Phys. Fluids 5 (1962), 604–614.

    MathSciNet  Google Scholar 

  43. J.J. Erpenbeck, Stability of idealized one-reaction detonations, Phys. Fluids 7 (1964).

    Google Scholar 

  44. J.J. Erpenbeck, Stability of step shocks. Phys. Fluids 5 (1962) no. 10, 1181–1187.

    MATH  MathSciNet  Google Scholar 

  45. J.W. Evans, Nerve axon equations: I. Linear approximations. Ind. Univ. Math. J. 21 (1972) 877–885.

    MATH  Google Scholar 

  46. J.W. Evans, Nerve axon equations: II. Stability at rest. Ind. Univ. Math. J. 22 (1972) 75–90.

    MATH  Google Scholar 

  47. J.W. Evans, Nerve axon equations: III. Stability of the nerve impulse. Ind. Univ. Math. J. 22 (1972) 577–593.

    MATH  Google Scholar 

  48. J.W. Evans, Nerve axon equations: IV. The stable and the unstable impulse. Ind. Univ. Math. J. 24 (1975) 1169–1190.

    MATH  Google Scholar 

  49. J.W. Evans and J.A. Feroe, Traveling waves of infinitely many pulses in nerve equations, Math. Biosci., 37 (1977) 23–50.

    MATH  Google Scholar 

  50. W. Fickett, Stability of the square wave detonation in a model system. Physica 16D (1985) 358–370.

    MathSciNet  Google Scholar 

  51. W. Fickett, Detonation in miniature, 133–182, in The mathematics of combustion, Frontiers in App. Math. (1985) SIAM, Philadelphia ISBN: 0-89871-053-7.

    Google Scholar 

  52. W. Fickett and W.C. Davis, Detonation, University of California Press, Berkeley, CA (1979): reissued as Detonation: Theory and experiment, Dover Press, Mineola, New York (2000), ISBN 0-486-41456-6.

    Google Scholar 

  53. Fickett and Wood, Flow calculations for pulsating one-dimensional detonations. Phys. Fluids 9 (1966) 903–916.

    Google Scholar 

  54. H. Freistühler and P. Szmolyan, Spectral stability of small shock waves, Arch. Ration. Mech. Anal. 164 (2002) 287–309.

    MATH  MathSciNet  Google Scholar 

  55. R. Gardner, On the structure of the spectra of periodic traveling waves, J. Math. Pures Appl. 72 (1993), 415–439.

    MATH  MathSciNet  Google Scholar 

  56. R. Gardner, On the detonation of a combustible gas, Trans. Amer. Math. Soc. 277 (1983), no. 2, 431–468.

    MATH  MathSciNet  Google Scholar 

  57. R. Gardner and C.K.R.T. Jones, A stability index for steady state solutions of boundary value problems for parabolic systems, J. Diff. Eqs. 91 (1991), no. 2, 181–203.

    MATH  MathSciNet  Google Scholar 

  58. R. Gardner and C.K.R.T. Jones, Traveling waves of a perturbed diffusion equation arising in a phase field model, Ind. Univ. Math. J. 38 (1989), no. 4, 1197–1222.

    MathSciNet  Google Scholar 

  59. R. A. 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.

    MathSciNet  Google Scholar 

  60. F. Gesztesy, Y. Latushkin, and K. Zumbrun, Derivatives of (Modified) Fredholm Determinants and Stability of Standing and Traveling Waves, J. Math. Pures Appl. (9) 90 (2008), no. 2, 160–200.

    MATH  MathSciNet  Google Scholar 

  61. F. Gilbert and G. Backus, Propagator matrices in elastic wave and vibration problems, Geophysics, 31 (1966) 326–332.

    Google Scholar 

  62. P. Godillon, Green’s function pointwise estimates for the modified Lax-Friedrichs scheme, M2ANMath. Model. Numer. Anal.37 (2003) 1–39.

    MATH  MathSciNet  Google Scholar 

  63. G.H. Golub and C.F. Van Loan, Matrix computations, Johns Hopkins University Press, Baltimore (1996).

    MATH  Google Scholar 

  64. 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).

    Google Scholar 

  65. O. Gues, G. Metivier, M. Williams, and K. Zumbrun, Multidimensional viscous shocks I: degenerate symmetrizers and long time stability, Journal of the Amer. Math. Soc. 18 (2005), 61–120.

    MATH  MathSciNet  Google Scholar 

  66. O. Guès, G. Métivier, M. Williams, and K. Zumbrun, Existence and stability of noncharacteristic hyperbolic-parabolic boundary-layers, Arch. for Ration. Mech. Anal. 197 (2010), no. 1, 1–87.

    MATH  Google Scholar 

  67. D. Henry, Geometric theory of semilinear parabolic equations. Lecture Notes in Mathematics, Springer-Verlag, Berlin (1981), iv + 348 pp.

    MATH  Google Scholar 

  68. Olga Holtz, Hermite-Biehler, Routh-Hurwitz, and total positivity, Linear algebra and its applications 372 (2003) 105–110.

    MATH  MathSciNet  Google Scholar 

  69. E. Hopf, Abszweigung einer periodischen Lösung von einer stationaeren Lösung eines Differentialsystems, Akad. Wiss. (Leipzig) 94 (1942), 2–11.

    Google Scholar 

  70. E. Hopf, Selected works of Eberhard Hopf with commentaries, Edited by Cathleen S. Morawetz, James B. Serrin and Yakov G. Sinai. American Mathematical Society, Providence, RI, 2002. xxiv+386 pp. ISBN: 0-8218-2077-X. (Commentary by M. Golubitsky and P.H. Rabinowitz.)

    Google Scholar 

  71. P. Howard and M. Raoofi, Pointwise asymptotic behavior of perturbed viscous shock profiles, Adv. Differential Equations (2006) 1031–1080.

    Google Scholar 

  72. P. Howard, M. Raoofi, and K. Zumbrun, Sharp pointwise bounds for perturbed viscous shock waves, J. Hyperbolic Differ. Equ. (2006) 297–373.

    Google Scholar 

  73. P. Howard and K. Zumbrun, Stability of undercompressive viscous shock waves, in press, J. Differential Equations 225 (2006), no. 1, 308–360.

    MATH  MathSciNet  Google Scholar 

  74. J. Humpherys, O. Lafitte, and K. Zumbrun, Stability of viscous shock profiles in the high Mach number limit, CMP 293 (2010), no. 1, 1–36.

    MATH  MathSciNet  Google Scholar 

  75. J. Humpherys, G. Lyng, and K. Zumbrun, Spectral stability of ideal gas shock layers, Arch. for Rat. Mech. Anal. 194 (2009), no. 3, 1029–1079.

    MATH  MathSciNet  Google Scholar 

  76. J. Humpherys, G. Lyng, and K. Zumbrun, Multidimensional spectral stability of large-amplitude Navier-Stokes shocks, in preparation.

    Google Scholar 

  77. J. Humpherys, G. Lyng, and K. Zumbrun, Spectral stability of combustion waves: the Majda model, in preparation.

    Google Scholar 

  78. 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.

    MATH  MathSciNet  Google Scholar 

  79. J. Humpherys and K. Zumbrun, An efficient shooting algorithm for Evans function calculations in large systems, Phys. D 220 (2006), no. 2, 116–126.

    MATH  MathSciNet  Google Scholar 

  80. J. Humpherys and K. Zumbrun, Efficient numerical stability analysis of detonation waves in ZND, in preparation.

    Google Scholar 

  81. J. Humpherys, K. Zumbrun, and Björn Sandstede, Efficient computation of analytic bases in Evans function analysis of large systems, Numer. Math. 103 (2006), no. 4, 631–642.

    MATH  MathSciNet  Google Scholar 

  82. A. Jeffrey, Magnetohydrodynamics, University Mathematical Texts, no. 33 Oliver & Boyd, Edinburgh-London; Interscience Publishers Inc. John Wiley & Sons, Inc., New York 1966 viii+252 pp.

    MATH  Google Scholar 

  83. M. Johnson and K. Zumbrun, Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg-de Vries Equation, to appear, Studies in Appl. Math.

    Google Scholar 

  84. M. Johnson and K. Zumbrun, Nonlinear stability of periodic traveling wave solutions of systems of viscous conservation laws in the generic case, J. Diff. Eq. 249 (2010), no. 5, 1213–1240.

    MATH  MathSciNet  Google Scholar 

  85. M. Johnson and K. Zumbrun, Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction diffusion equations, preprint (2010).

    Google Scholar 

  86. M. Johnson and K. Zumbrun, Nonlinear stability and asymptotic behavior of periodic traveling waves of multidimensional viscous conservation laws in dimensions one and two, preprint (2009).

    Google Scholar 

  87. M. Johnson and K. Zumbrun, Stability of KP waves, to appear, SIAM J. for Math. Anal.

    Google Scholar 

  88. M. Johnson, K. Zumbrun, and J.C. Bronski, On the modulation equations and stability of periodic GKdV waves via bloch decompositions, to appear Physica D.

    Google Scholar 

  89. M. Johnson, K. Zumbrun, and P. Noble, Nonlinear stability of viscous roll waves, preprint (2010).

    Google Scholar 

  90. T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin Heidelberg (1985).

    Google Scholar 

  91. A.R. Kasimov and D.S. Stewart, Spinning instability of gaseous detonations. J. Fluid Mech. 466 (2002), 179–203.

    MATH  MathSciNet  Google Scholar 

  92. S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics. thesis, Kyoto University (1983).

    Google Scholar 

  93. 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.

    MATH  MathSciNet  Google Scholar 

  94. 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.

    Google Scholar 

  95. H.I. Lee and D.S. Stewart, Calculation of linear detonation instability: one-dimensional instability of plane detonation, J. Fluid Mech., 216 (1990) 103–132.

    MATH  Google Scholar 

  96. G. Lyng and K. Zumbrun, One-dimensional stability of viscous strong detonation waves, Arch. Ration. Mech. Anal. 173 (2004), no. 2, 213–277.

    MATH  MathSciNet  Google Scholar 

  97. G. Lyng and K. Zumbrun, A stability index for detonation waves in Majda’s model for reacting flow, Physica D, 194 (2004), 1–29.

    MATH  MathSciNet  Google Scholar 

  98. G. Lyng, M. Raoofi, B. Texier, and K. Zumbrun, Pointwise Green Function Bounds and stability of combustion waves, J. Differential Equations 233 (2007) 654–698.

    MATH  MathSciNet  Google Scholar 

  99. A. Majda, The stability of multi-dimensional shock fronts – a new problem for linear hyperbolic equations, Mem. Amer. Math. Soc. 275 (1983).

    Google Scholar 

  100. A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983).

    Google Scholar 

  101. A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, New York (1984), viii+ 159 pp.

    MATH  Google Scholar 

  102. A. Majda, A qualitative model for dynamic combustion, SIAM J. Appl. Math., 41 (1981), 70–91.

    MATH  MathSciNet  Google Scholar 

  103. A. Majda and R. Pego, Stable viscosity matrices for systems of conservation laws. J. Diff. Eqs. 56 (1985) 229–262.

    MATH  MathSciNet  Google Scholar 

  104. A. Majda and R. Rosales, A theory for the spontaneous formation of Mach stems in reacting shock fronts, I. The basic perturbation analysis, SIAM J. Appl. Math. 43 (1983), no. 6, 1310–1334.

    MATH  MathSciNet  Google Scholar 

  105. A. Majda and R. Rosales, 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), no. 2, 117–148.

    MATH  MathSciNet  Google Scholar 

  106. C. Lattanzio, C. Mascia, T. Nguyen, R. Plaza, and K. Zumbrun, Stability of scalar radiative shock profiles, SIAM J. for Math. Anal. 41 (2009/2010), no. 6, 2165–2206.

    MathSciNet  Google Scholar 

  107. 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.

    MATH  MathSciNet  Google Scholar 

  108. C. Mascia and K. Zumbrun, Stability of shock profiles of dissipative symmetric hyperbolic-parabolic systems, preprint (2001): Comm. Pure Appl. Math. 57 (2004), no. 7, 841–876.

    MATH  MathSciNet  Google Scholar 

  109. C. Mascia and K. Zumbrun, Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Ration. Mech. Anal. 169 (2003), no. 3, 177–263.

    MATH  MathSciNet  Google Scholar 

  110. C. Mascia and K. Zumbrun, Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems, Arch. Ration. Mech. Anal. 172 (2004), no. 1, 93–131.

    MATH  MathSciNet  Google Scholar 

  111. A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math. 2 (1985) 17–25.

    MATH  MathSciNet  Google Scholar 

  112. A. Matsumura and Y. Wang, Asymptotic stability of viscous shock wave for a one-dimensional isentropic model of viscous gas with density dependent viscosity, preprint (2010).

    Google Scholar 

  113. T. Nguyen, R. Plaza, and K. Zumbrun, Stability of radiative shock profiles for hyperbolic-elliptic coupled systems, Physica D. 239 (2010), no. 8, 428–453.

    MATH  MathSciNet  Google Scholar 

  114. U.B. McVey and T.Y. Toong, Mechanism of instabilities in exothermic blunt-body flows, Combus. Sci. Tech. 3 (1971) 63–76.

    Google Scholar 

  115. 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.

    Google Scholar 

  116. G. Métivier and K. Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (2005), no. 826, vi+107 pp.

    MathSciNet  Google Scholar 

  117. G. Métivier and K. Zumbrun, Existence of semilinear relaxation shocks, J. Math. Pures. Appl. (9) 92 (2009), no. 3, 209–231.

    MATH  MathSciNet  Google Scholar 

  118. G. Métivier and K. Zumbrun, Existence and sharp localization in velocity of small-amplitude Boltzmann shocks, Kinet. Relat. Models 2 (2009), no. 4, 667–705.

    MATH  MathSciNet  Google Scholar 

  119. B.S. Ng and W.H. Reid, An initial value method for eigenvalue problems using compound matrices, J. Comput. Phys. 30 (1979) 125–136.

    MATH  MathSciNet  Google Scholar 

  120. B.S. Ng and W.H. Reid, A numerical method for linear two-point boundary value problems using compound matrices, J. Comput. Phys. 33 (1979) 70–85.

    MATH  MathSciNet  Google Scholar 

  121. B.S. Ng and W.H. Reid, On the numerical solution of the Orr-Sommerfeld problem: asymptotic initial conditions for shooting methods, J. Comput. Phys., 38 (1980) 275–293.

    MATH  MathSciNet  Google Scholar 

  122. B.S. Ng and W.H. Reid, The compound matrix method for ordinary differential systems, J. Comput. Phys. 58 (1985) 209–228.

    MATH  MathSciNet  Google Scholar 

  123. M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation laws with viscosity: Analysis of the Evans function, Arch. Ration. Mech. Anal. 166 (2003), no. 2, 99–166.

    MATH  MathSciNet  Google Scholar 

  124. M. Oh and K. Zumbrun, Stability of periodic solutions of viscous conservation laws: pointwise bounds on the Green function, Arch. Ration. Mech. Anal. (2002), no. 2, 167–196.

    MathSciNet  Google Scholar 

  125. M. Oh and K. Zumbrun, Low-frequency stability analysis of periodic traveling-wave solutions of viscous conservation laws in several dimensions, Z. Anal. Anwend. 25 (2006), no. 1, 1–21.

    MATH  MathSciNet  Google Scholar 

  126. M. Oh and K. Zumbrun, Stability and asymptotic behavior of periodic traveling wave solutions of viscous conservation laws in several dimensions, Arch. for Rat. Mech. Anal. 196 (2010), no. 1, 1–20; Errata, Arch. for Rat. Mech. Anal. 196 (2010), no. 1, 21–23.

    MATH  MathSciNet  Google Scholar 

  127. T. Nguyen and K. Zumbrun, Long-time stability of large-amplitude noncharacteristic boundary layers for hyperbolic parabolic systems, J. Math. Pures Appl. (9) 92 (2009), no. 6, 547–598.

    MATH  MathSciNet  Google Scholar 

  128. T. Nguyen and K. Zumbrun, Long-time stability of multi-dimensional noncharacteristic viscous boundary layers, to appear, Comm. Math. Phys.

    Google Scholar 

  129. R. Plaza and K. Zumbrun, An Evans function approach to spectral stability of small-amplitude shock profiles, J. Disc. and Cont. Dyn. Sys. 10 (2004), 885–924.

    MATH  MathSciNet  Google Scholar 

  130. R.L. Pego, Stable viscosities and shock profiles for systems of conservation laws. Trans. Amer. Math. Soc. 282 (1984) 749–763.

    MATH  MathSciNet  Google Scholar 

  131. R.L. Pego and M.I. Weinstein, Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340 (1992), 47–94.

    MATH  MathSciNet  Google Scholar 

  132. D. Sattinger, On the stability of waves of nonlinear parabolic systems, Adv. Math. 22 (1976) 312–355.

    MATH  MathSciNet  Google Scholar 

  133. M. Raoofi, Lp asymptotic behavior of perturbed viscous shock profiles, J. Hyperbolic Differ. Equ. 2 (2005), no. 3, 595–644.

    MATH  MathSciNet  Google Scholar 

  134. M. Raoofi and K. Zumbrun, Stability of undercompressive viscous shock profiles of hyperbolic-parabolic systems, J. Differential Equations (2009) 1539–1567.

    Google Scholar 

  135. D. Serre, La transition vers l’instabilité pour les ondes de chocs multidimensionnelles, Trans. Amer. Math. Soc. 353 (2001) 5071–5093.

    MATH  MathSciNet  Google Scholar 

  136. D. Serre, Systems of conservation laws. 1. Hyperbolicity, entropies, shock waves, Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 1999. xxii+263 pp. ISBN: 0-521-58233-4.

    Google Scholar 

  137. D. Serre, Systems of conservation laws. 2. Geometric structures, oscillations, and initial-boundary value problems, Translated from the 1996 French original by I. N. Sneddon. Cambridge University Press, Cambridge, 2000. xii+269 pp. ISBN: 0-521-63330-3.

    Google Scholar 

  138. 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.

    MATH  Google Scholar 

  139. J. Stoer and R. Bulirsch, Introduction to numerical analysis, Springer-Verlag, New York (2002).

    MATH  Google Scholar 

  140. G.I. Taylor, Proceedings of the Royal Society, 1910 Volume.

    Google Scholar 

  141. B. Texier and K. Zumbrun, Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves, Methods Anal. and Appl. 12 (2005), no. 4, 349–380.

    MATH  MathSciNet  Google Scholar 

  142. B. Texier and K. Zumbrun, Galloping instability of viscous shock waves, Physica D. 237 (2008) 1553-1601.

    MATH  MathSciNet  Google Scholar 

  143. B. Texier and K. Zumbrun, Hopf bifurcation of viscous shock waves in gas dynamics and MHD, Arch. Ration. Mech. Anal. 190 (2008) 107–140.

    MATH  MathSciNet  Google Scholar 

  144. B. Texier and K. Zumbrun, Transition to longitudinal instability of detonation waves is generically associated with Hopf bifurcation to timeperiodic galloping solutions, preprint (2008).

    Google Scholar 

  145. M.Williams, Heteroclinic orbits with fast transitions: a new construction of detonation profiles, to appear, Indiana Mathematics Journal.

    Google Scholar 

  146. K. Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves, 307–516, Progr. Nonlinear Differential Equations Appl., 47, Birkhäuser Boston, Boston, MA, 2001.

    Google Scholar 

  147. K. Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations, with an appendix by Helge Kristian Jenssen and Gregory Lyng, in Handbook of mathematical fluid dynamics. Vol. III, 311–533, North-Holland, Amsterdam (2004).

    Google Scholar 

  148. K. Zumbrun, Stability of noncharacteristic boundary layers in the standing shock limit, to appear, Trans. Amer. Math. Soc.

    Google Scholar 

  149. K. Zumbrun, Conditional stability of unstable viscous shock waves in compressible gas dynamics and MHD, to appear, Arch. Ration. Mech. Anal.

    Google Scholar 

  150. K. Zumbrun, Planar stability criteria for viscous shock waves of systems with real viscosity, in Hyperbolic Systems of Balance Laws, CIME School lectures notes, P. Marcati ed., Lecture Note in Mathematics 1911, Springer (2004).

    Google Scholar 

  151. K. Zumbrun, Dynamical stability of phase transitions in the p-system with viscosity-capillarity, SIAM J. Appl. Math. 60 (2000), no. 6, 1913–1924 (electronic).

    MATH  MathSciNet  Google Scholar 

  152. K. Zumbrun, Conditional stability of unstable viscous shocks, J. Differential Equations 247 (2009), no. 2, 648–671.

    MATH  MathSciNet  Google Scholar 

  153. K. Zumbrun, Center stable manifolds for quasilinear parabolic pde and conditional stability of nonclassical viscous shock waves, preprint (2008).

    Google Scholar 

  154. K. Zumbrun, The refined inviscid stability condition and cellular instability of viscous shock waves, to appear, Physica D.

    Google Scholar 

  155. K. Zumbrun, Numerical error analysis for Evans function computations: a numerical gap lemma, centered-coordinate methods, and the unreasonable effectiveness of continuous orthogonalization, preprint (2009).

    Google Scholar 

  156. K. Zumbrun, A local greedy algorithm and higher-order extensions for global continuation of analytically varying subspaces, To appear, Quarterly Appl. Math.

    Google Scholar 

  157. K. Zumbrun, Stability of detonation waves in the ZND limit, to appear, Arch. Ration. Mech. Anal.

    Google Scholar 

  158. K. Zumbrun, High-frequency asymptotics and stability of ZND detonations in the high-overdrive and small-heat release limits, preprint (2010).

    Google Scholar 

  159. K. Zumbrun and P. Howard, Pointwise semigroup methods and stability of viscous shock waves. Indiana Mathematics Journal V47 (1998), 741–871; Errata, Indiana Univ. Math. J. 51 (2002), no. 4, 1017–1021.

    MATH  MathSciNet  Google Scholar 

  160. K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J. 48 (1999) 937–992.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Indiana University, Bloomington, IN, 47405, USA

    Kevin Zumbrun

Authors
  1. Kevin Zumbrun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kevin Zumbrun .

Editor information

Editors and Affiliations

  1. McAllister Bldg 201, University Park, 16802-6401, Pennsylvania, USA

    Alberto Bressan

  2. , Oxford Centre for Nonlinear PDE, Mathema, University of Oxford, St Giles 24–29, Oxford, OX1 3LB, Montserrat

    Gui-Qiang G. Chen

  3. , School of Mathematics, University of Minnesota, Church Street S.E, 206, Minneapolis, 55455, Minnesota, USA

    Marta Lewicka

  4. , Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260, Pennsylvania, USA

    Dehua Wang

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer Science+Business Media, LLC

About this paper

Cite this paper

Zumbrun, K. (2011). Stability and Dynamics of Viscous Shock Waves. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_5

Download citation

Publish with us

Policies and ethics

Search

Navigation