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Existence, Bifurcation, and Stability of Profiles for Classical and Non-Classical Shock Waves

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Book cover Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems

Abstract

This paper surveys the authors’ recent results on viscous shock waves in PDE systems of conservation laws with non-convexity and non-strict hyperbolicity. Particular attention is paid to the physical model of magnetohydrodynamics. The plan of the paper is as follows. Sections 1 and 2 introduce the classes of systems and the classes of shock waves we consider and recall how profiles for small-amplitude shocks are constructed via center manifold analyses of a corresponding system of ODEs. Section 3 describes the global picture, i. e., large-amplitude shock waves, for the case of magnetohydrodynamics, first the solution set of the Rankine-Hugoniot jump conditions, then a heteroclinic bifurcation occurring in the ODE system for the profiles. Section 4 presents a method for the numerical identification of heteroclinic manifolds, which is applied in Sections 5 and 6 to the case of magnetohydrodynamics. The numerical treatment confirms and details the analytical findings and, more notably, extends them considerably; in particular, it allows to study the existence / non-existence of profiles and the aforementioned heteroclinic bifurcation globally. Section 7 discusses the stability of viscous shock waves; the important nonuniformity of the vanishing viscosity limit for, in particular, non-classical MHD shock waves is not addressed in this paper.

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References

  1. G. Bader, U. Ascher: A new basis implementation for a mixed order boundary value ODE solver, SIAM J. Sci. Stat. Comput. 8, 483-500 (1987).

    Google Scholar 

  2. W.J. Beyn: The numerical computation of connecting orbits in dynamical systems, IMA J. Numer. Anal., 9, 379–405 (1990).

    Article  MathSciNet  Google Scholar 

  3. C. Conley, J. Smoller: On the structure of magnetohydrodynamic shock waves, Commun. Pure Appl. Math. 28 (1974), 367–375.

    Article  MathSciNet  Google Scholar 

  4. H. Freistühler: Small amplitude intermediate magnetohydrodynamic shock waves, Physica Scripta T74 (1998), 26–29.

    Google Scholar 

  5. H. Freistühler: Viscous profiles for Laxian shock waves in Kawashima type systems, Preprint.

    Google Scholar 

  6. H. Freistühler, T. P. Liu: Nonlinear stability of overcompressive shock waves in a rotationally invariant system of viscous conservation laws Commun. Math. Phys. 153 (1993), 147-15.

    Google Scholar 

  7. H. Freistühler, C. Rohde: Numerical Methods for Viscous Profiles of Non-Classical Shock Waves, In: Hyperbolic problems: Theory, Numerics, Applications: Seventh International Conference in Zürich 1998/ ed. by Michael Fey; Rolf Jeltsch (1999).

    Google Scholar 

  8. H. Freistühler, C. Rohde., Numerical Computation of Viscous Profiles for Hyperbolic Conservation Laws, Preprint 1999, submitted to Math. Comput.

    Google Scholar 

  9. H. Freistühler, C. Rohde., A bifurcation analysis of the MHD Rankine-Hugoniot relations for a perfect gas, in preparation.

    Google Scholar 

  10. H. Freistühler, D. Serre: L1 stability of shock waves in scalar viscous conservation laws, Commun. Pure Appl. Math. 51 (1998), 291–301.

    Article  MATH  Google Scholar 

  11. H. Freistühler, P. Szmolyan: Existence and bifurcation of viscous profiles for all intermediate magnetohydrodynamic shock waves, SIAM J. Math. Anal. 26 (1995), 112–128.

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Fries: Nonlinear asymptotic stability of general small-amplitude viscous Laxian shock waves, J. Differ. Equations 146 (1998), 185–202.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Fries: Stability of viscous shock waves associated with non-convex modes, to appear in Arch. Rational Mech. Anal.

    Google Scholar 

  14. R.A. Gardner, K. Zumbrun: The gap lemma and geometric criteria for instability of viscous shock profiles, to appear in Commun. Pure Appl. Math. 51 (1998), 797–855.

    Article  MathSciNet  Google Scholar 

  15. P. Germain: Contribution à la théorie des ondes de choc en magnétodynamique des fluides Off. Nat. Etud. Aéronaut., Publ. 97 (1959).

    Google Scholar 

  16. D. Gilbarg: The existence and limit behavior of the one-dimensional shock layer, Amer. J. Math. 73 (1951), 256–274.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. SFB 256, University of Bonn and Institut für Angewandte Mathematik, University of Freiburg. GRAPE: GRAphics Programming Environment for Mathematical Problems: http://www.mathematik.uni-freiburg.de-/IAM/Research/grape/GENERAL/index.html.

  19. A.M. Il’in, O.A. Oleinik: Behaviour of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of time, Amer. Math. Soc. Translations 42 (1964), 19–23.

    Google Scholar 

  20. C.K.R.T. Jones, R. Gardner, T. Kapitula: Stability of travelling waves for non-convex scalar viscous conservation laws, Commun. Pure Appl. Math. 46 (1993), 505–526.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Kawashima: Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Thesis Kyoto University 1983.

    Google Scholar 

  22. S. Kawashima, A. Matsumura: Stability of shock profiles in viscoelasticity with non-convex constitutive relations, Commun. Pure Appl. Math. 47 (1994), 1547–1569.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Kulikovskii, G. Liubimov: Magnetohydrodynamics, Addison-Wesley: Reading 1965.

    Google Scholar 

  24. T.-P. Liu: The Riemann problem for general systems of conservation laws, J. Differential Equations 18 (1975), 218–234.

    Article  MathSciNet  MATH  Google Scholar 

  25. T.-P. Liu: Nonlinear stability of shock waves for viscous conservation laws, Mem. Am. Math. Soc. 328, Providence: AMS 1985.

    Google Scholar 

  26. T.-P. Liu: Pointwise convergence to shock waves for viscous conservations laws, Commun. Pure Appl. Math. 50 (1997), 1113–1182.

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Matsumura, K. Nishihara: Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Commun. Math. Phys. 165 (1994), 83–96.

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Majda, R. Pego: Stable viscosity matrices for systems of conservation laws, J. Diff. Equations 56 (1985), 229–262.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. A. Szepessy, Z. Xin: Nonlinear stability of viscous shock waves, Arch. Rational Mech. Anal. 122 (1993), 53–103.

    Article  MathSciNet  MATH  Google Scholar 

  32. http://www.mathematik.uni-freiburg.de/IAM/Research/projectskr/mhd/ mhd.html

    Google Scholar 

  33. C.C. Wu: Formation, Structure, and Stability of MHD Intermediate Shocks, J. Geophys. Research 95 (1990), A6 8149–8175.

    Article  Google Scholar 

  34. K. Zumbrun, P. Howard: Pointwise semigroup methods and stability of viscous shock waves, Indiana U. Math. J. 47 (1998), 63–85.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig

    Heinrich Freistühler & Christian Fries

  2. Albert-Ludwigs Universität Freiburg, Freiburg

    Christian Rohde

Authors
  1. Heinrich Freistühler
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  2. Christian Fries
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  3. Christian Rohde
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Editors and Affiliations

  1. Freie Universität Berlin Institut für Mathematik I, Arnimallee 2-6, 14195, Berlin, Germany

    Bernold Fiedler

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Freistühler, H., Fries, C., Rohde, C. (2001). Existence, Bifurcation, and Stability of Profiles for Classical and Non-Classical Shock Waves. In: Fiedler, B. (eds) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56589-2_13

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  • DOI: https://doi.org/10.1007/978-3-642-56589-2_13

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-62524-4

  • Online ISBN: 978-3-642-56589-2

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