The structure of MFD shock waves for rectilinear motion in some models of plasma

Hesaaraki, Mahmoud

doi:10.1090/S0002-9947-1995-1297528-4

The structure of MFD shock waves for rectilinear motion in some models of plasma
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by Mahmoud Hesaaraki

Trans. Amer. Math. Soc. 347 (1995), 3423-3452

DOI: https://doi.org/10.1090/S0002-9947-1995-1297528-4

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Abstract:

The mathematical question of the existence of structure for "fast", "slow" and "intermediate" MFD shock waves in the case of rectilinear motion in some model of plasma is stated in terms of a six-dimensional system of ordinary differential equations, which depends on five viscosity parameters. In this article we shall show that this system is gradient-like. Then by using the Conley theory we prove that the fast and the slow shocks always possess structure. Moreover, the intermediate shocks do not admit structure. Some limiting cases for singular viscosities are investigated. In particular, we show how the general results in the classical one fluid MHD theory are obtained when "the plasma viscosities" $\beta$ and $\chi$ tend to zero.

References

J. Edward Anderson, Magnetohydrodynamic shock waves, The M.I.T. Press, Cambridge, Mass., 1963. MR 0149795
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
C. Conley and R. Easton, Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc. 158 (1971), 35–61. MR 279830, DOI 10.1090/S0002-9947-1971-0279830-1
Charles C. Conley and Joel A. Smoller, On the structure of magnetohydrodynamic shock waves, Comm. Pure Appl. Math. 27 (1974), 367–375. MR 368586, DOI 10.1002/cpa.3160270306
Charles C. Conley and Joel A. Smoller, On the structure of magnetohydrodynamic shock waves. II, J. Math. Pures Appl. (9) 54 (1975), no. 4, 429–443. MR 454376
Charles C. Conley and Joel A. Smoller, Viscosity matrices for two-dimensional nonlinear hyperbolic systems, Comm. Pure Appl. Math. 23 (1970), 867–884. MR 274956, DOI 10.1002/cpa.3160230603
Charles C. Conley and Joel A. Smoller, Shock waves as limits of progressive wave solutions of higher order equations, Comm. Pure Appl. Math. 24 (1971), 459–472. MR 283414, DOI 10.1002/cpa.3160240402

Topological methods in the theory of shock waves

A model of some plasma shock structures

Shock waves, jump relations, and structure

12

P. Germain, Shock waves and shock-wave structure in magneto-fluid dynamics, Rev. Mod. Phys. 32 (1960), 951–958. MR 0129723, DOI 10.1103/revmodphys.32.951
Mahmoud Hesaaraki, The structure of MFD shock waves in a model of two fluids, Nonlinearity 6 (1993), no. 1, 1–24. MR 1203049
Mahmud Hesaaraki, The structure of shock waves in magnetohydrodynamics, Mem. Amer. Math. Soc. 49 (1984), no. 302, v+96. MR 743547, DOI 10.1090/memo/0302
Mahmoud Hesaaraki, The structure of shock waves in magnetohydrodynamics for purely transverse magnetic fields, SIAM J. Appl. Math. 51 (1991), no. 2, 412–428. MR 1095026, DOI 10.1137/0151020
M. Hesaaraki, The structure of MFD shock waves in a model of plasma, J. Math. Pures Appl. (9) 72 (1993), no. 4, 377–404. MR 1228998

Magnetohydrodynamics

Roger Peyret, Sur la structure du choc lent dans un schéma à deux fluides non dissipatif, C. R. Acad. Sci. Paris 258 (1964), 2973–2976 (French). MR 164557
Roger Peyret, Sur la structure du choc lent dans un schéma à deux fluides avec dissipation, C. R. Acad. Sci. Paris 258 (1964), 3178–3181 (French). MR 161544
Dietmar Salamon, Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291 (1985), no. 1, 1–41. MR 797044, DOI 10.1090/S0002-9947-1985-0797044-3
Joel Smoller, Shock waves and reaction-diffusion equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York-Berlin, 1983. MR 688146
Joel A. Smoller and Charles C. Conley, Viscosity matrices for two-dimensional non-linear hyperbolic systems. II, Amer. J. Math. 94 (1972), 631–650. MR 320537, DOI 10.2307/2373748
Joel A. Smoller and Charles C. Conley, Shock waves as limits of progressive wave solutions of higher order equations. II, Comm. Pure Appl. Math. 25 (1972), 133–146. MR 306721, DOI 10.1002/cpa.3160250203

Compressible-fluid dynamics

Gordon Thomas Whyburn, Analytic Topology, American Mathematical Society Colloquium Publications, Vol. 28, American Mathematical Society, New York, 1942. MR 0007095