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

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

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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.

**1.**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.**2.**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**, https://doi.org/10.1016/S0895-7177(02)00116-4**3.**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).**4.**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).**5.**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).**6.**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).**7.**Kats E.N., Lebedev V.V. Dynamics of liquid crystals. Moscow, Nauka, 1988 (Russian).**8.**L. D. Landau and E. M. Lifshits,*\cyr Teoreticheskaya fizika. Tom VII*, 4th ed., “Nauka”, Moscow, 1987 (Russian). \cyr Teoriya uprugosti. [Theory of elasticity]; Edited by Lifshits, A. M. Kosevich and L. P. Pitaevskiĭ. MR**912888****9.**L. D. Landau and E. M. Lifshits,*\cyr Teoreticheskaya fizika. Tom VI*, 3rd ed., “Nauka”, Moscow, 1986 (Russian). \cyr Gidrodinamika. [Fluid dynamics]. MR**850480****10.**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**, https://doi.org/10.1016/S0898-1221(04)90035-1**11.**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**, https://doi.org/10.1016/S0898-1221(04)90130-7**12.**Sedov L.N. Mechanics of continua, v.1. Moscow., Nauka, 1970 (Russian).**13.**L. V. Ovsyannikov,*\cyr Lektsii po osnovam gazovoĭ dinamiki*, “Nauka”, Moscow, 1981 (Russian). MR**665918****14.**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****15.**A. M. Blokhin,*\cyr 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****16.**Blokhin A.M. The elements of the theory of hyperbolic systems and equations. Novosibirsk. Gos. Univ., Novosibirsk, 1995 (Russian).**17.**S. K. Godunov,*\cyr Uravneniya matematicheskoĭ fiziki*, “Nauka”, Moscow, 1979 (Russian). Second edition, revised and augmented. MR**548574****18.**Richard Bellman,*Introduction to matrix analysis*, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. MR**0122820**

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Additional Information

**A. M. Blokhin**

Affiliation:
Institute of Mathematics, Novosibirsk State University, Novosibirsk, 630090, Russia

Email:
blokhin@math.nsc.ru

**R. E. Semenko**

Affiliation:
Novosibisk State University, Novosibirsk, 630090, Russia

Email:
rsem86@mail.ru

DOI:
https://doi.org/10.1090/S0033-569X-09-01155-4

Keywords:
Layered structures,
anisotropic dielectrics,
electrohydrodynamical approximaton,
linear problem,
shock waves.

Received by editor(s):
February 12, 2008

Published electronically:
May 6, 2009

Article copyright:
© Copyright 2009
Brown University

The copyright for this article reverts to public domain 28 years after publication.