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Nonlinear stability of viscous shock waves

  • Anders Szepessy
  • Zhouping Xin
Article

Keywords

Neural Network Shock Wave Complex System Nonlinear Dynamics Electromagnetism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Chern, I. L., Convergence to diffusion waves for Lax-Friedrichs method,Math. Comp. 59 (1991) 107–119.Google Scholar
  2. 2.
    Goodman, J., Nonlinear asymptotic stability of viscous shock profiles for conservation laws,Arch. Rational Mech. Anal. 95 (1986) 325–344.Google Scholar
  3. 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. 4.
    Hardy, G. H., Littlewood, J. E. &Polya, G.,Inequalities, Cambridge Univ. Press, 1934.Google Scholar
  5. 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. 6.
    Jones, C. K. R. T., Gardner, R. &Kapitula, T., Stability of travelling waves for nonconvex scalar viscous conservation laws, preprint.Google Scholar
  7. 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. 8.
    Lax, P., Hyperbolic systems of conservation laws, II,Comm. Pure Appl. Math. 10 (1957) 537–566.Google Scholar
  9. 9.
    Liu, T.-P., Nonlinear stability of shock waves for viscous conservation laws,Memoirs of Amer. Math. Soc. 328 (1986).Google Scholar
  10. 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. 11.
    Liu, T.-P., Shock waves for compressible Navier-Stokes are stable,Comm. Pure Appl. Math. 39 (1986) 565–594.Google Scholar
  12. 12.
    Liu, T.-P., Interactions of nonlinear hyperbolic waves, inNonlinear Analysis, Eds.F.-C. Liu &T.-P. Liu, World Scientific, 1991, 171–184.Google Scholar
  13. 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. 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. 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. 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. 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. 18.
    Sattinger, D. H., On the stability of waves of nonlinear parabolic systems,Advances in Math. 22 (1976) 312–355.Google Scholar
  19. 19.
    Smoller, J.,Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.Google Scholar
  20. 20.
    Szepessy, A., On the stability of finite element methods for shock waves,Comm. Pure Appl. Math. 45 (1992) 923–946.Google Scholar
  21. 21.
    Szepessy, A., Stability of numerical shock waves, in preparation.Google Scholar
  22. 22.
    Weinberger, H. F., preprint.Google Scholar
  23. 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

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Anders Szepessy
    • 1
    • 2
  • Zhouping Xin
    • 1
    • 2
  1. 1.NADARoyal Institute of TechnologyStockholm
  2. 2.Courant InstituteNew York

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