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Linear degeneracy and shock waves. (English) Zbl 0756.35048
Jump discontinuities in weak solutions of hyperbolic systems of conservation laws, are investigated, which are associated with linearly degenerate modes. It is assumed that the integral manifolds of the corresponding eigenspace bundle are compact. Multiple eigenvalues of the fields are permitted.
First, the geometry of such linear degenerate fields are established. Then it is shown that there exists also shocks for these modes, which are not of the familiar contact discontinuity type, and that these shocks have sometimes viscous profiles. As an application, this is demonstrated (independent of the dissipation) for certain shocks associated to the rotational Alfvén modes of the system of magnetohydrodynamic equations.

MSC:
35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
76L05 Shock waves and blast waves in fluid mechanics
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[1] Arnold, V.I.: Ordinary differential equations. Cambridge London: The MIT Press 1973 · Zbl 0296.34001
[2] Freistühler, H.: Thesis, Ruhr-Universität Bochum 1987
[3] Freistühler, H.: Instability of vanishing viscosity approximation for hyperbolic systems of conservation laws with rotational invariance. J. Differ. Equations87, 205–226 (1990) · Zbl 0715.35048 · doi:10.1016/0022-0396(90)90001-6
[4] Freistühler, H.: Shocks associated with rotational modes. IMA Vol. Math. Appl.33, 68–69 (1991)
[5] Freistühler, H.: On compact linear degeneracy. IMA preprint #551. University of Minnesota 1989
[6] Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. (Lect. Notes Math., vol. 583) Berlin Heidelberg New York: Springer 1977
[7] Kato, T.: Perturbation theory for linear operators. Berlin Heidelberg New York Tokio: Springer 1984 · Zbl 0531.47014
[8] Lax, P.: Hyperbolic systems of conservations laws II. Commun. Pure Appl. Math.10, 537–566 (1957) · Zbl 0081.08803 · doi:10.1002/cpa.3160100406
[9] Majda, A., Pego, R.: Stable viscosity matrices for systems of conservation laws. J. Differ. Equations56, 229–262 (1985) · Zbl 0543.76100 · doi:10.1016/0022-0396(85)90107-X
[10] Schutz, B.F.: Geometrical methods of mathematical physics. Cambridge: Cambridge University Press 1980 · Zbl 0462.58001
[11] Brio, M.: Propagation of weakly nonlinear magnetoacoustic waves. Wave Motion9, 455–458 (1987) · Zbl 0615.76112 · doi:10.1016/0165-2125(87)90033-3
[12] Cabannes, H.: Theoretical magnetofluiddynamics. New York: Academic Press 1970
[13] Conley, C., Smoller, J.: On the structure of magnetohydrodynamic shock waves II. J. Math. Pures Appl.54, 429–444 (1975) · Zbl 0292.76045
[14] Freistühler, H.: Some remarks on the structure of intermediate magnetohydrodynamic shocks. J. Geophys. Res.95, 3825–3827 (1991) · doi:10.1029/90JA02421
[15] Germain, P.: Contribution à la théorie des ondes de choc en magnétodynamique des fluides. Off. Nat. Etud. Aéronaut., Publ.97 (1959)
[16] Gilbarg, D.: The existence and limit behavior of the one-dimensional shock layer. Am. J. Math.73, 256–274 (1951) · Zbl 0044.21504 · doi:10.2307/2372177
[17] Hesaraaki, M.: The structure of magnetohydrodynamic shockwaves. Mem. Am. Math. Soc.302 (1984)
[18] Kulikovskii, A., Liubimov, G.: Magnetohydrodynamics. Reading Addison-Wesley 1965
[19] Mischaikow, K., Hattori, H.: On the existence of intermediate magnetohydrodynamic shock waves. J. Dyn. Differ. Equations2, 163–175 (1990) · Zbl 0695.76030 · doi:10.1007/BF01057417
[20] Wu, C.C.: On MHD intermediate shocks. Geophys. Res. Lett.14, 668–671 (1987) · doi:10.1029/GL014i006p00668
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