Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

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
Full-text PDF

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 [Enhancements On Off] (What's this?)

  • 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

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35Q30, 83C22

Retrieve articles in all journals with MSC (2000): 35Q30, 83C22


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.

American Mathematical Society