Small time-periodic solutions of equations of magnetohydrodynamics as a singularly perturbed problem

Milan Štědrý; Otto Vejvoda

Aplikace matematiky (1983)

  • Volume: 28, Issue: 5, page 344-356
  • ISSN: 0862-7940

Abstract

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This paper deals with a system of equations describing the motion of viscous electrically conducting incompressible fluid in a bounded three dimensional domain whose boundary is perfectly conducting. The displacement current appearing in Maxwell’s equations, ϵ E t is not neglected. It is proved that for a small periodic force and small positive there exists a locally unique periodic solution of the investigated system. For ϵ 0 , these solutions are shown to convergeto a solution of the simplified (and usually considered) system of equations of magnetohydrodynamics.

How to cite

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Štědrý, Milan, and Vejvoda, Otto. "Small time-periodic solutions of equations of magnetohydrodynamics as a singularly perturbed problem." Aplikace matematiky 28.5 (1983): 344-356. <http://eudml.org/doc/15314>.

@article{Štědrý1983,
abstract = {This paper deals with a system of equations describing the motion of viscous electrically conducting incompressible fluid in a bounded three dimensional domain whose boundary is perfectly conducting. The displacement current appearing in Maxwell’s equations, $\epsilon E_t$ is not neglected. It is proved that for a small periodic force and small positive there exists a locally unique periodic solution of the investigated system. For $\epsilon \rightarrow 0$, these solutions are shown to convergeto a solution of the simplified (and usually considered) system of equations of magnetohydrodynamics.},
author = {Štědrý, Milan, Vejvoda, Otto},
journal = {Aplikace matematiky},
keywords = {electrically conducting; bounded three dimensional domain; boundary perfectly conducting; displacement current; Maxwell’s equations; small periodic force; small positive epsilon; locally unique periodic solution; electrically conducting; bounded three dimensional domain; boundary perfectly conducting; displacement current; Maxwell's equations; small periodic force; small positive epsilon; locally unique periodic solution},
language = {eng},
number = {5},
pages = {344-356},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Small time-periodic solutions of equations of magnetohydrodynamics as a singularly perturbed problem},
url = {http://eudml.org/doc/15314},
volume = {28},
year = {1983},
}

TY - JOUR
AU - Štědrý, Milan
AU - Vejvoda, Otto
TI - Small time-periodic solutions of equations of magnetohydrodynamics as a singularly perturbed problem
JO - Aplikace matematiky
PY - 1983
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 28
IS - 5
SP - 344
EP - 356
AB - This paper deals with a system of equations describing the motion of viscous electrically conducting incompressible fluid in a bounded three dimensional domain whose boundary is perfectly conducting. The displacement current appearing in Maxwell’s equations, $\epsilon E_t$ is not neglected. It is proved that for a small periodic force and small positive there exists a locally unique periodic solution of the investigated system. For $\epsilon \rightarrow 0$, these solutions are shown to convergeto a solution of the simplified (and usually considered) system of equations of magnetohydrodynamics.
LA - eng
KW - electrically conducting; bounded three dimensional domain; boundary perfectly conducting; displacement current; Maxwell’s equations; small periodic force; small positive epsilon; locally unique periodic solution; electrically conducting; bounded three dimensional domain; boundary perfectly conducting; displacement current; Maxwell's equations; small periodic force; small positive epsilon; locally unique periodic solution
UR - http://eudml.org/doc/15314
ER -

References

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  1. N. G. Van Kampen B. U. Felderhof, Theoretical Methods in Plasma Physics, North-Holland Publishing Company - Amsterdam, 1967. (1967) Zbl0159.29601
  2. O. A. Ladyženskaja V. A. Solonnikov, Solutions of some non-stationary problems of.magnetohydrodynamics for incompressible fluid, (Russian.) Trudy Mat. Inst. V. A. Steklova, 59 (1960), 115-173. (1960) MR0170130
  3. O. A. Ladyženskaja V. A. Solonnikov, On the principle of linearization and invariant manifolds in problems of magnetohydrodynamics, (Russian.) Zapiski naučnych seminarov LOMI, 38 (1973), 46-93. (1973) MR0377310
  4. O. A. Ladyženskaja, Mathematical Problems of the Dynamics of Viscous Incompressible Liquid, (Russian.) Nauka, Moskva, 1970. (1970) MR0271559
  5. A. Milani, On a singular perturbation problem for the linear Maxwell equations, Quaderni di Matematica, Università di Torino, n° 20, 1980, 11-16. (1980) Zbl0478.35010
  6. A. Milani, On a singular perturbation problem for the Maxwell equations in a multiply connected domain, Rend. Sem. Mat. Univers. Politecn. Torino, 38, 1 (1980), 123-132. (1980) Zbl0464.35006MR0608934
  7. J. A. Shercliff, A Textbook of Magnetohydrodynamics, Pergamon, Oxford 1965. (1965) MR0185961
  8. L. Stupjalis, A nonstationary problem of magnetohydrodynamics, (Russian.) Zapiski naučnych seminarov LOMI, 52 (1975), 175-217. (1975) Zbl0399.76096MR0464896
  9. L. Stupjalis, On solvability of an initial-boundary value problem of magnetohydrodynamics, (Russian.) Zapiski naučnych seminarov LOMÍ, 69 (1977), 219-239. (1977) Zbl0363.76085MR0499834
  10. L. Stupjalis, A nonstationary problem of magnetohydrodynamics in the case of two spatial variables, (Russian.) Trudy Mat. Inst. V. A. Steklova, 147 (1980), 156-168. (1980) MR0573906

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