Skip to main content
Log in

Nonlinear stability of viscous shock waves

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

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Chern, I. L., Convergence to diffusion waves for Lax-Friedrichs method,Math. Comp. 59 (1991) 107–119.

    Google Scholar 

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

    Google Scholar 

  3. Goodman, J., Remarks on the stability of viscous shock waves,Viscous Profiles and Numerical Methods for Shock Waves, Ed.M. Shearer, SIAM, Philadelphia, 1991, 66–72.

    Google Scholar 

  4. Hardy, G. H., Littlewood, J. E. &Polya, G.,Inequalities, Cambridge Univ. Press, 1934.

  5. Il'in, A. M. &Oleinik, O. A., Behavior of the solution of the Cauchy problem for certain quasi linear equations for unbounded increase of time,Amer. Math. Soc. Translations, Ser. 242 (1964) 19–23.

    Google Scholar 

  6. Jones, C. K. R. T., Gardner, R. &Kapitula, T., Stability of travelling waves for nonconvex scalar viscous conservation laws, preprint.

  7. Kawashima, S. &Matsumura, A., Asymptotic stability of travelling wave solutions of systems for one-dimensional gas motion,Comm. Math. Phys. 101 (1985) 97–127.

    Google Scholar 

  8. Lax, P., Hyperbolic systems of conservation laws, II,Comm. Pure Appl. Math. 10 (1957) 537–566.

    Google Scholar 

  9. Liu, T.-P., Nonlinear stability of shock waves for viscous conservation laws,Memoirs of Amer. Math. Soc. 328 (1986).

  10. Liu, T.-P., Linear and nonlinear large time behavior of solutions of general systems of conservation laws,Comm. Pure Appl. Math. 30 (1977) 767–796.

    Google Scholar 

  11. Liu, T.-P., Shock waves for compressible Navier-Stokes are stable,Comm. Pure Appl. Math. 39 (1986) 565–594.

    Google Scholar 

  12. Liu, T.-P., Interactions of nonlinear hyperbolic waves, inNonlinear Analysis, Eds.F.-C. Liu &T.-P. Liu, World Scientific, 1991, 171–184.

  13. Liu, T.-P. &Xin, Z., Stability of viscous shock waves associated with a system of nonstrictly hyperbolic conservation laws,Comm. Pure Appl. Math. 45 (1992) 361–388.

    Google Scholar 

  14. Matsumura, A. &Nishihara, K., On a stability of travelling wave solution of a one-dimensional model system for compressible viscous gas,Japan J. Appl. Math. 3 (1986) 1–13.

    Google Scholar 

  15. Osher, S. &Ralston, J.,L 1 stability of travelling waves with applications to convective porous media flow,Comm. Pure Appl. Math. 35 (1982) 737–751.

    Google Scholar 

  16. Pego, R., Linearized stability of extreme shock profiles for systems of conservation laws with viscosity, Trans. Amer. Math. Soc.280 (1983) 431–461.

    Google Scholar 

  17. Pego, R., Remarks on the stability of shock profiles for conservation laws with dissipation, Trans. Amer. Math. Soc.291 (1985) 353–361.

    Google Scholar 

  18. Sattinger, D. H., On the stability of waves of nonlinear parabolic systems,Advances in Math. 22 (1976) 312–355.

    Google Scholar 

  19. Smoller, J.,Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.

    Google Scholar 

  20. Szepessy, A., On the stability of finite element methods for shock waves,Comm. Pure Appl. Math. 45 (1992) 923–946.

    Google Scholar 

  21. Szepessy, A., Stability of numerical shock waves, in preparation.

  22. Weinberger, H. F., preprint.

  23. Xin, Z. P., On the linearized stability of viscous shock profiles for systems of conservation laws,J. Diff. Eqs. 100 (1992) 119–136.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by L.Arkeryd

Rights and permissions

Reprints and permissions

About this article

Cite this article

Szepessy, A., Xin, Z. Nonlinear stability of viscous shock waves. Arch. Rational Mech. Anal. 122, 53–103 (1993). https://doi.org/10.1007/BF01816555

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01816555

Keywords

Navigation