On the stability of shock waves in layered structures at the presence of the electric current

Blokhin, A.; Semenko, R.

doi:10.1090/S0033-569X-09-01155-4

Authors: A. M. Blokhin and R. E. Semenko
Journal: Quart. Appl. Math. 67 (2009), 441-475
MSC (2000): Primary 35Q30, 83C22
DOI: https://doi.org/10.1090/S0033-569X-09-01155-4
Published electronically: May 6, 2009
MathSciNet review: 2547635
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We are concerned with a hydrodynamical model of layered structures at the presence of the electric current. We formulate a linearized stability problem for shock waves and prove its ill-posedness, which means instability of shock waves for the given model of layered structures.

References

Dorovsky V.N., Dorovsky S.V. A hydrodynamic model of water-oil layered systems containing gas. Math. Comput. Mod., 2002, v. 35, pp. 751-757.

V. N. Dorovsky, V. S. Belonosov, and A. S. Belonosov, Numerical investigation of parametric resonance in water-oil structures containing gas, Math. Comput. Modelling 36 (2002), no. 1-2, 203–209. MR 1925071, DOI https://doi.org/10.1016/S0895-7177%2802%2900116-4

Gogosov V.V, Polansky V.A. Electrohydrodynamics: Problem and applications, fundamental equations, discontinuous solutions. Fluid mechanics (Science and technique resume), 1976, 10, pp. 5-85 (Russian).

Dorovsky S.V., Dorovsky V.N. Possibility of electrometry at the investigation on stability of shock waves in layered structures. Geology and Geophysics, 2006, v. 47(7), pp. 892-901 (Russian).

Dorovsky S.V., Dorovsky V.N., Blokhin A.M. Possibility of the methods of electrometry at the investigation on stability of shock waves in layered structures. Geology and Geophysics, v. 47, pp. 1185-1191 (Russian).

Blokhin A.M., Dorovsky V.N. Problems of the mathematical simulation in theory of the multi-velocity continuum. RAS, Sib. Division., United Inst. of Geology, Geophysics and Mineralogy, Inst. of Math., Novosibirsk, 1994 (Russian).

Kats E.N., Lebedev V.V. Dynamics of liquid crystals. Moscow, Nauka, 1988 (Russian).

L. D. Landau and E. M. Lifshits, Teoreticheskaya fizika. Tom VII, 4th ed., “Nauka”, Moscow, 1987 (Russian). Teoriya uprugosti. [Theory of elasticity]; Edited by Lifshits, A. M. Kosevich and L. P. Pitaevskiĭ. MR 912888

L. D. Landau and E. M. Lifshits, Teoreticheskaya fizika. Tom VI, 3rd ed., “Nauka”, Moscow, 1986 (Russian). Gidrodinamika. [Fluid dynamics]. MR 850480

A. M. Blokhin and S. V. Dorovskii, Shock waves stability in layered structures, Comput. Math. Appl. 47 (2004), no. 2-3, 427–440. MR 2048194, DOI https://doi.org/10.1016/S0898-1221%2804%2990035-1

A. M. Blokhin, S. V. Dorovskii, and E. V. Ovechkin, Stability of shock waves in layered structures. II, Comput. Math. Appl. 47 (2004), no. 8-9, 1379–1387. MR 2070991, DOI https://doi.org/10.1016/S0898-1221%2804%2990130-7

Sedov L.N. Mechanics of continua, v.1. Moscow., Nauka, 1970 (Russian).

L. V. Ovsyannikov, Lektsii po osnovam gazovoĭ dinamiki, “Nauka”, Moscow, 1981 (Russian). MR 665918

A. M. Blokhin, I. Z. Merazhov, and Yu. L. Trakhinin, Investigation on stability of electrohydrodynamic shock waves, Matematiche (Catania) 52 (1997), no. 1, 87–114 (1998). MR 1624964

A. M. Blokhin, Integraly ènergii i ikh prilozheniya k zadacham gazovoĭ dinamiki, “Nauka” Sibirsk. Otdel., Novosibirsk, 1986 (Russian). With an appendix by the author and D. L. Tkachëv. MR 888816

Blokhin A.M. The elements of the theory of hyperbolic systems and equations. Novosibirsk. Gos. Univ., Novosibirsk, 1995 (Russian).

S. K. Godunov, Uravneniya matematicheskoĭ fiziki, “Nauka”, Moscow, 1979 (Russian). Second edition, revised and augmented. MR 548574

Richard Bellman, Introduction to matrix analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. MR 0122820