Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Computing the refined stability condition

Authors: Nicholas B. Anderson, Allison M. Lindgren and Gregory D. Lyng
Journal: Quart. Appl. Math. 73 (2015), 1-21
MSC (2010): Primary 35Pxx, 35B40, 47Fxx
DOI: https://doi.org/10.1090/S0033-569X-2015-01362-2
Published electronically: January 21, 2015
MathSciNet review: 3322724
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The classical (inviscid) stability analysis of shock waves is based on the Lopatinskiĭ determinant, $ \Delta $--a function of frequencies whose zeros determine the stability of the underlying shock. A careful analysis of $ \Delta $ shows that in some cases the stable and unstable regions of parameter space are separated by an open set of parameters. Zumbrun and Serre [Indiana Univ. Math. J. 48 (1999), 937-992] have shown that by taking account of viscous effects not present in the definition of $ \Delta $, it is possible to determine the precise location in the open, neutral set of parameter space at which stability is lost. In particular, they show that the transition to instability under suitably localized perturbations is determined by an ``effective viscosity'' coefficient given in terms of the second derivative of the associated Evans function, the viscous analogue of $ \Delta $. Here, in the simplest possible setting, we propose and implement two approaches toward the practical computation of this coefficient. Moreover, in a special case, we derive an exact solution of the relevant differential equations.

References [Enhancements On Off] (What's this?)

  • [1] N. Anderson, S. Bagley, A Lindgren, G. Lyng, S. Mukherjee, D. Swedberg, and M. Xu, The refined stability condition for gas dynamics, 2012, in preparation.
  • [2] Miguel Artola and Andrew J. Majda, Nonlinear development of instabilities in supersonic vortex sheets. I. The basic kink modes, Phys. D 28 (1987), no. 3, 253-281. MR 914450 (88i:76025), https://doi.org/10.1016/0167-2789(87)90019-4
  • [3] A. A. Barmin and S. A. Egorushkin, Stability of shock waves, Adv. Mech. 15 (1992), no. 1-2, 3-37 (English, with English and Russian summaries). MR 1220256 (94j:76042)
  • [4] Sylvie Benzoni-Gavage and Denis Serre, Multidimensional hyperbolic partial differential equations: First order systems and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Oxford, 2007. MR 2284507 (2008k:35002)
  • [5] Sylvie Benzoni-Gavage, Denis Serre, and Kevin Zumbrun, Alternate Evans functions and viscous shock waves, SIAM J. Math. Anal. 32 (2001), no. 5, 929-962. MR 1828312 (2002b:35144), https://doi.org/10.1137/S0036141099361834
  • [6] Sylvie Benzoni-Gavage, Denis Serre, and Kevin Zumbrun, Transition to instability of planar viscous shock fronts: the refined stability condition, Z. Anal. Anwend. 27 (2008), no. 4, 381-406. MR 2448741 (2009g:35179), https://doi.org/10.4171/ZAA/1361
  • [7] S. P. Dyakov, On the stability of shock waves, Ž. Eksper. Teoret. Fiz. 27 (1954), 288-295 (Russian).
  • [8] Jerome J. Erpenbeck, Stability of step shocks, Phys. Fluids 5 (1962), 1181-1187. MR 0155515 (27 #5449)
  • [9] Jonathan Goodman and Judith R. Miller, Long-time behavior of scalar viscous shock fronts in two dimensions, J. Dynam. Differential Equations 11 (1999), no. 2, 255-277. MR 1695245 (2000e:35147), https://doi.org/10.1023/A:1021977329306
  • [10] David Hoff and Kevin Zumbrun, Asymptotic behavior of multidimensional scalar viscous shock fronts, Indiana Univ. Math. J. 49 (2000), no. 2, 427-474. MR 1793680 (2001j:35195), https://doi.org/10.1512/iumj.2000.49.1942
  • [11] Jeffrey Humpherys, Gregory Lyng, and Kevin Zumbrun, Spectral stability of ideal-gas shock layers, Arch. Ration. Mech. Anal. 194 (2009), no. 3, 1029-1079. MR 2563632 (2011b:35329), https://doi.org/10.1007/s00205-008-0195-4
  • [12] A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Applied Mathematical Sciences, vol. 53, Springer-Verlag, New York, 1984. MR 748308 (85e:35077)
  • [13] Guy Métivier and Kevin Zumbrun, Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems, Mem. Amer. Math. Soc. 175 (2005), no. 826, vi+107. MR 2130346 (2006f:35168)
  • [14] Denis Serre, Systems of conservation laws. 2: Geometric structures, oscillations, and initial boundary value problems., Cambridge University Press, Cambridge, 2000. Translated from the 1996 French original by I. N. Sneddon. MR 1775057 (2001c:35146)
  • [15] Kevin Zumbrun, Multidimensional stability of planar viscous shock waves, Advances in the theory of shock waves, Progr. Nonlinear Differential Equations Appl., vol. 47, Birkhäuser Boston, Boston, MA, 2001, pp. 307-516. MR 1842778 (2002k:35200)
  • [16] Kevin Zumbrun, Stability of large-amplitude shock waves of compressible Navier-Stokes equations: With an appendix by Helge Kristian Jenssen and Gregory Lyng, Handbook of mathematical fluid dynamics. Vol. III, North-Holland, Amsterdam, 2004, pp. 311-533. MR 2099037 (2006f:35229)
  • [17] Kevin Zumbrun, Planar stability criteria for viscous shock waves of systems with real viscosity, Hyperbolic systems of balance laws, Lecture Notes in Math., vol. 1911, Springer, Berlin, 2007, pp. 229-326. MR 2348937 (2008k:35315), https://doi.org/10.1007/978-3-540-72187-1_4
  • [18] Kevin Zumbrun, The refined inviscid stability condition and cellular instability of viscous shock waves, Phys. D 239 (2010), no. 13, 1180-1187. MR 2644683 (2012e:35157), https://doi.org/10.1016/j.physd.2010.03.006
  • [19] K. Zumbrun and D. Serre, Viscous and inviscid stability of multidimensional planar shock fronts, Indiana Univ. Math. J. 48 (1999), no. 3, 937-992. MR 1736972 (2001h:35122), https://doi.org/10.1512/iumj.1999.48.1765

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35Pxx, 35B40, 47Fxx

Retrieve articles in all journals with MSC (2010): 35Pxx, 35B40, 47Fxx

Additional Information

Nicholas B. Anderson
Affiliation: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071-3036
Email: nanderson7@gmail.com

Allison M. Lindgren
Affiliation: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071-3036
Email: lindgren.allison.m@gmail.com

Gregory D. Lyng
Affiliation: Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071-3036
Email: glyng@uwyo.edu

DOI: https://doi.org/10.1090/S0033-569X-2015-01362-2
Received by editor(s): December 14, 2012
Published electronically: January 21, 2015
Article copyright: © Copyright 2015 Brown University

American Mathematical Society