Skip to main content
SpringerLink
Log in

Existence and Stability of Multidimensional Shock Fronts in the Vanishing Viscosity Limit

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract.

In this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation, yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves. More precisely, given a curved Lax shock solution u0 to a hyperbolic system of conservation laws, we construct nearby viscous shock solutions uε to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity ε→0 and satisfy an appropriate (conormal) version of Majda’s stability estimate.

The main new feature of the paper is the derivation of maximal and optimal estimates for the linearization of the parabolic problem about a highly singular approximate solution. These estimates are more robust than the singular estimates obtained in our previous work, and permit us to remove an earlier assumption limiting how much the inviscid shock we start with can deviate from flatness.

The key to the new approach is to work with the full linearization of the parabolic problem, that is, the linearization with respect to both uε and the unknown viscous front, and to allow variation of the front at all stages – not only in the construction of the approximate solution as done in previous work, but also in the final error equation. After reformulating the problem as a transmission problem, we show that the linearized problem can be desingularized and optimal estimates obtained by adding an appropriate extra boundary condition involving the front. The extra condition determines a local evolution rule for the viscous front.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

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

    MATH  Google Scholar 

  2. Alinhac, S.: Existence d’ondes de raréfaction pour des sytèmes quasi-linéaires hyperboliques multidimensionnels. Comm. Partial Diff. Equ. 14, 173–230 (1989)

    MATH  Google Scholar 

  3. Bianchini, S., Bressan, A.: The vanishing viscosity limit for a class of viscous hyperbolic systems. Preprint, 2002

  4. Bony, J.M.: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires. Ann. Sc. E.N.S. Paris 14, 209–246 (1981)

    MATH  Google Scholar 

  5. Chazarain, J., Piriou, A.: 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

  6. Coifmann, R., Meyer, Y.: Au delà des opérateurs pseudo-différentiels. Astérisque 57, (1978)

  7. Coppel, W.A.: Stability and asymptotic behavior of differential equations. D.C. Heath, Boston

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    MATH  Google Scholar 

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

    Google Scholar 

  12. Freistühler, H., Szmolyan, P.: Spectral stability of small shock waves. Arch. Rational Mech. Anal. 164, 287–309 (2002)

    Article  Google Scholar 

  13. Goodman, J.: 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

  14. Goodman, J.: Stability of viscous scalar shock fronts in several dimensions. Trans. Amer. Math. Soc. 311, 683–695 (1989)

    MATH  Google Scholar 

  15. Goodman, J., Miller, J.R.: Long-time behavior of scalar viscous shock fronts in two dimensions. J. Dynam. Diff. Eqns. 11, 255–277 (1999)

    Article  MATH  Google Scholar 

  16. Guès, O., Métivier, G., Williams, M.,Zumbrun, K.: Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability. To appear in J. Amer. Math. Soc.

  17. Guès, O., Métivier, G., Williams, M., Zumbrun, K.: Multidimensional viscous shocks II: The small viscosity limit. Comm. Pure Appl. Math. 57, 141–218 (2004)

    Article  MathSciNet  Google Scholar 

  18. Guès, O., Métivier, G., Williams, M., Zumbrun, K.: Navier–Stokes regularization of multidimensional Euler shocks. In preparation

  19. Guès, O., Williams, M.: Curved shocks as viscous limits: a boundary problem approach. Indiana Univ. Math. J. 51, 421–450 (2002)

    Google Scholar 

  20. Goodman, J., Xin, Z.: Viscous limits for piecewise smooth solutions to systems of conservation laws. Arch. Rational Mech. Anal. 121, 235–265 (1992)

    Article  MATH  Google Scholar 

  21. Gardner, R., Zumbrun, K.: The Gap Lemma and geometric criteria for instability of viscous shock profiles. Comm. Pure. Appl. Math. 51, 797–855 (1998)

    Google Scholar 

  22. Hoff, D., Zumbrun, K.: Multi-dimensional diffusion waves for the Navier-Stokes equations of compressible flow. Indiana Univ. Math. J. 44, 603–676 (1995)

    MATH  Google Scholar 

  23. Hörmander, L.: Lectures on Nonlinear Hyprbolic Differential Equations. Mathématiques et Applications 26, Sringer Verlag, 1997

  24. Kreiss, H.O.: Initial boundary value problems for hyperbolic systems. Comm. Pure Appl. Math. 23, 277–298 (1970)

    MATH  Google Scholar 

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

  26. Métivier, G.: Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d’espace. Trans. Amer. Math. Soc. 296, 431–479 (1986)

    Google Scholar 

  27. Métivier, G., Zumbrun, K.: Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems. To appear in Mem. Am. Math. Soc.

  28. Métivier, G., Zumbrun, K.: Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Disc. Cont. Dyn. Syst. 11, 205–220 (2004)

    MathSciNet  Google Scholar 

  29. Meyer, Y.: Remarques sur un théorème de J.M.Bony. Supplemento al Rendiconti der Circolo Matematico di Palermo, Serie II, No1, 1981

  30. Mokrane, A.: Problèmes mixtes hyperboliques non linéaires. Thesis, Université de Rennes 1, 1987

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

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  33. Rousset, F.: Inviscid boundary conditions and stability of viscous boundary layers. Asymptot. Anal. 26, 285–306 (2001)

    MATH  Google Scholar 

  34. Rousset, F.: Viscous approximation of strong shocks of systems of conservation laws. SIAM J. Math. Anal. 35, 492–519 (2003)

    Article  MATH  Google Scholar 

  35. Taylor, M.: Partial Differential Equations III. Applied Mathematical Sciences 117, Springer, 1996

  36. Yu, S.-H.: Zero-dissipation limit of solutions with shocks for systems of hyperbolic conservation laws. Arch. Rational Mech. Anal. 146, 275–370 (1999)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  38. Zumbrun, K.: 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

  39. Zumbrun, K.: Refined Wave–tracking and Nonlinear Stability of Viscous Lax Shocks. Methods Appl. Anal. 7, 747–768 (2000)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Université de Provence,  

    Olivier Guès

  2. Université de Bordeaux,  

    Guy Métivier

  3. University of North Carolina,  

    Mark Williams

  4. Indiana University, Bloomington

    Kevin Zumbrun

Authors
  1. Olivier Guès
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guy Métivier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mark Williams
    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 Kevin Zumbrun.

Additional information

Communicated by A. Bressan

Acknowledgement The work of O.G. was partially supported by European network HYKE, HPRN-CT-2002-00282. The work of G.M. was partially supported by European network HYKE, HPRN-CT-2002-00282. The work of M.W. was partially supported by NSF grant DMS-0070684. The work of K.Z. was partially supported by NSF grant DMS-0070765.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guès, O., Métivier, G., Williams, M. et al. Existence and Stability of Multidimensional Shock Fronts in the Vanishing Viscosity Limit. Arch. Rational Mech. Anal. 175, 151–244 (2005). https://doi.org/10.1007/s00205-004-0342-5

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-004-0342-5

Keywords

Search

Navigation