On the solvability of some initial boundary value problems of magnetofluidmechanics with Hall and ion-slip effects

Vsevolod A. Solonnikov; Giuseppe Mulone

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

  • Volume: 6, Issue: 2, page 117-132
  • ISSN: 1120-6330

Abstract

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The solvability of three linear initial-boundary value problems for the system of equations obtained by linearization of MHD equations is established. The equations contain terms corresponding to Hall and ion-slip currents. The solutions are found in the Sobolev spaces W p 2 , 1 ⁢ Q T with p > 5 / 2 and in anisotropic Holder spaces.

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Solonnikov, Vsevolod A., and Mulone, Giuseppe. "On the solvability of some initial boundary value problems of magnetofluidmechanics with Hall and ion-slip effects." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.2 (1995): 117-132. <http://eudml.org/doc/244197>.

@article{Solonnikov1995,
abstract = {The solvability of three linear initial-boundary value problems for the system of equations obtained by linearization of MHD equations is established. The equations contain terms corresponding to Hall and ion-slip currents. The solutions are found in the Sobolev spaces \( W\_\{p\}^\{2,1\} (Q\_\{T\} ) \) with \( p > 5/2 \) and in anisotropic Holder spaces.},
author = {Solonnikov, Vsevolod A., Mulone, Giuseppe},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Magnetohydrodynamics; Anysotropic currents; Existence; Uniqueness; anisotropic currents; uniqueness; linearized system of MHD equations; Sobolev spaces; anisotropic Hölder spaces},
language = {eng},
month = {6},
number = {2},
pages = {117-132},
publisher = {Accademia Nazionale dei Lincei},
title = {On the solvability of some initial boundary value problems of magnetofluidmechanics with Hall and ion-slip effects},
url = {http://eudml.org/doc/244197},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Solonnikov, Vsevolod A.
AU - Mulone, Giuseppe
TI - On the solvability of some initial boundary value problems of magnetofluidmechanics with Hall and ion-slip effects
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/6//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 2
SP - 117
EP - 132
AB - The solvability of three linear initial-boundary value problems for the system of equations obtained by linearization of MHD equations is established. The equations contain terms corresponding to Hall and ion-slip currents. The solutions are found in the Sobolev spaces \( W_{p}^{2,1} (Q_{T} ) \) with \( p > 5/2 \) and in anisotropic Holder spaces.
LA - eng
KW - Magnetohydrodynamics; Anysotropic currents; Existence; Uniqueness; anisotropic currents; uniqueness; linearized system of MHD equations; Sobolev spaces; anisotropic Hölder spaces
UR - http://eudml.org/doc/244197
ER -

References

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  1. PAI, S. I., Magnetogasdynamics and Plasma Dynamics. Springer-Verlag, New York1962. Zbl0122.44305MR137449
  2. SHERCLIFF, J. A., A Textbook of Magnetohydrodynamics. Pergamon, Oxford1965. MR185961
  3. SUTTON, G. - SHERMAN, A., Engineering Magnetohydrodynamics. McGraw-Hill, 1965. 
  4. MAIELLARO, M., Uniqueness of M. H. D. thermodiffusive mixture flows with Hall and ion-slip effects. Meccanica, vol. 12, 1, 1977, 9-14. Zbl0401.76091
  5. RIONERO, S., Sulla stabilità magnetofluidodinamica non lineare asintotica in media in presenza di effetto Hall. Ricerche Mat., vol. 20, 2, 1971, 285-296. Zbl0236.76043MR343759
  6. MAIELLARO, M. - PALESE, L., Electrical anisotropic effects on thermal instability. Int. J. Engng. Sci., vol. 22, 4, 1984, 411-418. Zbl0534.76045
  7. EBEL, D. - SHEN, M. C., On the linear stability of a toroidal plasma with resistivity, viscosity and Hall current. J. Mat. Anal. Appl., vol. 125, 1987, 81-103. Zbl0649.76023MR891351DOI10.1016/0022-247X(87)90166-1
  8. EBEL, D. - SHEN, M. C., Linearization principle for a toroidal Hall current plasma with viscosity and resistivity. Ann. Mat. Pura Appl., vol. 150, 1988, 39-65. Zbl0666.76153MR946029DOI10.1007/BF01761463
  9. MULONE, G. - SALEMI, F., Some continuous dependence theorems in MHD with Hall and ion-slip currents in unbounded domains. Rend. Accad. Sci. Fis. Mat. Napoli, s. IV, vol. 55, 1988, 139-152. Zbl1145.76473MR1136744
  10. GALDI, G. P. - PADULA, M., A new approach to energy theory in the stability of fluid motion. Arch. Rational Mech. Anal., vol. 110, 1990, 187-286. Zbl0719.76035MR1060804DOI10.1007/BF00375129
  11. LADYZHENSKAYA, O. A. - SOLONNIKOV, V. A., Solvability of evolution problems of magnetohydrodynamics. Doklady Acad. Sci. USSR, vol. 124, 1, 1959, 26-28. Zbl0091.19804
  12. LADYZHENSKAYA, O. A. - SOLONNIKOV, V. A., Resolution of some evolution problems of magnetohydrodynamics for a viscous incompressible fluid. Trudy Math. Inst. Steklov, vol. 59, 1960, 115-173. MR170130
  13. LADYZHENSKAYA, O. A. - SOLONNIKOV, V. A., On linearization principle and on invariant manifolds for problems of magnetohydrodynamics. Zapiski Nauchn. Semin. LOMI, vol. 38, 1973, 46-93 (in Russian); English transl., J. Soviet Math., vol. 8, 1977, 384-422. Zbl0404.35089MR377310
  14. CARBONARO, B., On the linearization principle in magnetohydrodynamics. Ricerche Mat., vol. 28, 1979, 426-437. Zbl0437.76050MR578596
  15. STUPJALIS, L., On solvability of an initial-boundary value problem of magnetohydrodynamics. Zapiski Nauchn. Semin. LOMI, vol. 69, 1977, 219-239 (in Russian); English transl., J. Soviet Math., vol. 10, 1, 1978, 155-170. Zbl0389.76063MR499834
  16. STEDRY, M. - VEJVODA, O., Equations of magnetohydrodynamics: periodic solutions. Casopis Pest. Mat., vol. 111, 2, 1986, 177-184. Zbl0608.76108MR847316
  17. SOLONNIKOV, V. A., Estimates of solutions of nonstationary Navier-Stokes equations. Zapiski Nauchn. Semin. LOMI, vol. 38, 1973, 153-231 (in Russian); English transl., J. Soviet Math., vol. 8, 1, 1977, 467-529. Zbl0404.35081MR415097
  18. SOLONNIKOV, V. A., On L p -estimates of solutions of elliptic and parabolic systems. Trudy Math. Inst. Steklov, vol. 102, 1967 (in Russian); English transl., Proc. Steklov Inst. Math., vol. 102, 1976, 157-185. Zbl0204.42101MR228809
  19. SAKHAEV, SH. - SOLONNIKOV, V. A., Estimates of solutions of a boundary value problem of magnetohydrodynamics. Trudy Math. Inst. Steklov, vol. 127, 1975, 76-92 (in Russian); English transl., Proc. Steklov Inst. Math., vol. 127, 1975, 87-108. Zbl0366.76091
  20. KHACHATRYAN, A. G., Overdetermined parabolic boundary value problems. Zap. Naucn. Sem. Len. LOMI, vol. 69, 1977, 240-277 (in Russian); English transl., J. Soviet Math., vol. 10, n. 1, 1978, 171-193. Zbl0388.35035MR467012
  21. GRUBB, G. - SOLONNIKOV, V. A., Reduction of basic initial-boundary value problems for the Stokes equations to initial-boundary value problems for parabolic systems of pseudodifferential equations. Zapiski Nauchn. Semin. LOMI, vol. 163, 1987, 34-48 (in Russian); English transl., J. Soviet Math., vol. 49, 1990, 1140-1147. Zbl0696.35130MR918940DOI10.1007/BF02208710
  22. GRUBB, G. - SOLONNIKOV, V. A., Reduction of basic initial-boundary value problems for the Navier-Stokes equations to initial-boundary value problems for nonlinear parabolic systems of pseudodifferential equations. Zapiski Nauchn. Semin. LOMI, vol. 171, 1989, 36-52 (in Russian); English transl., Copenh. Univ. Math. Dept. Report., Ser. 1989, n. 5. Zbl0717.35069MR1031983DOI10.1007/BF01671932
  23. GRUBB, G. - SOLONNIKOV, V. A., Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudodifferential methods. Math. Scandinavica, vol. 69, 2, 1991, 217-291. Zbl0766.35034MR1156428
  24. VISHIK, M., On strongly elliptic systems of differential equations. Mat. Sbornik, vol. 29, 1951, 615-676 (in Russian). Zbl0044.09503MR49440
  25. SOLONNIKOV, V. A., On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Math. Inst. Steklov, vol. 83, 1965 (in Russian); English transl., Proc. Steklov Inst. Math., vol. 83, 1965, 1-184. Zbl0164.12502MR211083
  26. MOGILEVSIĬ, I. SH., Estimates of solutions of the general initial-boundary value problem for the linear time-dependent system on Navier-Stokes equations in a bounded domain. Trudy Math. Inst. Steklov, vol. 159, 1983, 61-94 (in Russian); English transl., Proc. Steklov Inst. Math., vol. 159, 1983, 61-95. Zbl0548.35096MR720208
  27. MOGILEVSIĬ, I. SH., On a boundary value problem for the time-dependent Stokes system with general boundary conditions. English transl., Math. USSR Izvestija, vol. 28, 1, 1987, 37-66. Zbl0615.35077

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