Skip to main content
Springer Nature Link
Log in

Stability of Viscous Shocks in Isentropic Gas Dynamics

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier–Stokes equations, or p-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength.

By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there are no unstable spectra in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number M ≈ 3000 for 1 ≤ γ ≤ 3. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant γ. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alexander J., Gardner R., Jonesm C.: A topological invariant arising in the stability analysis of travelling waves. J. Reine Angew. Math. 410, 167–212 (1990)

    MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

  4. Benzoni-Gavage, S., Serre, D., Zumbrun, K.: Alternate Evans functions and viscous shock waves. SIAM J. Math. Anal. 32(5), 929–962 (electronic) (2001)

    Google Scholar 

  5. Bridges T.J., Derks G., Gottwald G.: Stability and instability of solitary waves of the fifth-order KdV equation: a numerical framework. Phys. D 172(1–4), 190–216 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brin, L.Q.: Numerical testing of the stability of viscous shock waves. PhD thesis, Indiana University, Bloomington, 1998

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

  8. Brin, L.Q., Zumbrun, K.: Analytically varying eigenvectors and the stability of viscous shock waves. In: Seventh Workshop on Partial Differential Equations, Part I (Rio de Janeiro, 2001), P.L. Dias, D. Marchesin, A. Nelson, C. Tomei, eds. Mat. Contemp. 22, 19–32, (2002)

  9. Evans J.W.: Nerve axon equations. I. Linear approximations. Indiana Univ. Math. J. 21, 877–885 (1971)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  12. Evans J.W.: Nerve axon equations. IV. The stable and the unstable impulse. Indiana Univ. Math. J. 24(12), 1169–1190 (1974)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  14. Gardner R., Jones C.K.R.T.: A stability index for steady state solutions of boundary value problems for parabolic systems. J. Differ. Eqs. 91(2), 181–203 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gardner R.A., Zumbrun K.: The gap lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure Appl. Math. 51(7), 797–855 (1998)

    Article  MathSciNet  Google Scholar 

  16. Goodman J.: Nonlinear asymptotic stability of viscous shock profiles for conservation laws. Arch. Rat. Mech. Anal. 95(4), 325–344 (1986)

    Article  MATH  Google Scholar 

  17. Howard P., Zumbrun K.: Pointwise estimates and stability for dispersive-diffusive shock waves. Arch. Rat. Mech. Anal. 155(2), 85–169 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Humpherys J., Sandstede B., Zumbrun K.: Efficient computation of analytic bases in Evans function analysis of large systems. Numer. Math. 103(4), 631–642 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Humpherys J., Zumbrun K.: An efficient shooting algorithm for evans function calculations in large systems. Physica D 220(2), 116–126 (2006)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  20. Kato, T.: Perturbation theory for linear operators. Classics in Mathematics. Berlin: Springer-Verlag, 1995 (Reprint of the 1980 edition)

  21. Mascia C., Zumbrun K.: Pointwise Green function bounds for shock profiles of systems with real viscosity. Arch. Rat. Mech. Anal. 169(3), 177–263 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Mascia C., Zumbrun K.: Stability of large-amplitude viscous shock profiles of hyperbolic-parabolic systems. Arch. Rat. Mech. Anal. 172(1), 93–131 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mascia C., Zumbrun K.: Stability of small-amplitude shock profiles of symmetric hyperbolic-parabolic systems. Commun. Pure Appl. Math. 57(7), 841–876 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Matsumura A., Nishihara K.: On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas. Japan J. Appl. Math. 2(1), 17–25 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  25. Pego R.L., Smereka P., Weinstein M.I.: Oscillatory instability of traveling waves for a KdV-Burgers equation. Phys. D 67(1–3), 45–65 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  26. Pego R.L., Weinstein M.I.: Eigenvalues, and instabilities of solitary waves. Philos. Trans. Roy. Soc. London Ser. A 340(1656), 47–94 (1992)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  27. Smoller, J.: Shock waves and reaction-diffusion equations. New York: Springer-Verlag, Second edition, 1994

  28. Texier B., Zumbrun K.: Relative Poincaré-Hopf bifurcation and galloping instability of traveling waves. Methods Appl. Anal. 12(4), 349–380 (2005)

    MATH  MathSciNet  Google Scholar 

  29. Texier, B., Zumbrun, K.: Galloping instability of viscous shock waves. http://arxiv.org/list/math.Ap/0609331, 2006

  30. Texier, B., Zumbrun, K.: Hopf bifurcation of viscous shock waves in compressible gas- and magnetohydrodynamics. Toappear Arch. Rat. Mech. Anal., doi:10.1007/s00205-008-0012-x

  31. Zumbrun, K.: Multidimensional stability of planar viscous shock waves. Lecture notes, TMR summer school, 1999

  32. Zumbrun K., Howard P.: Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ. Math. J. 47(3), 741–871 (1998)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Blake Barker & Jeffrey Humpherys

  2. Department of Eng. Sci. and App. Math., Northwestern University, Evanston, IL, 60208-3125, USA

    Keith Rudd

  3. Department of Mathematics, Indiana University, Bloomington, IN, 47402, USA

    Kevin Zumbrun

Authors
  1. Blake Barker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeffrey Humpherys
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Keith Rudd
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Kevin Zumbrun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeffrey Humpherys.

Additional information

Communicated by P. Constantin

This work was supported in part by the National Science Foundation award numbers DMS-0607721 and DMS-0300487.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barker, B., Humpherys, J., Rudd, K. et al. Stability of Viscous Shocks in Isentropic Gas Dynamics. Commun. Math. Phys. 281, 231–249 (2008). https://doi.org/10.1007/s00220-008-0487-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0487-4

Keywords

Search

Navigation