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

Solonnikov, Vsevolod A.; Mulone, Giuseppe

Solonnikov, Vsevolod A. ; Mulone, Giuseppe

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Tome 6 (1995), p. 117-132 / Harvested from Biblioteca Digitale Italiana di Matematica

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Abstract
Résumé

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.

Si stabilisce la risolubilità di tre problemi ai valori iniziali e al contorno per il sistema ottenuto linearizzando le equazioni della MHD. Le equazioni contengono termini corrispondenti alle correnti di Hall e di ion-slip. Le soluzioni sono trovate negli spazi di Sobolev $W_{p}^{2, 1} (Q_{T})$ con $p > 5 / 2$ e negli spazi di Holder anisotropi.

Publié le : 1995-06-01

MR 1354225 | Zbl 0834.76094

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     author = {Vsevolod A. Solonnikov and Giuseppe Mulone},
     title = {On the solvability of some initial boundary value problems of magnetofluidmechanics with Hall and ion-slip effects},
     journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
     volume = {6},
     year = {1995},
     pages = {117-132},
     zbl = {0834.76094},
     mrnumber = {1354225},
     language = {en},
     url = {http://dml.mathdoc.fr/item/RLIN_1995_9_6_2_117_0}
}

Solonnikov, Vsevolod A.; 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, Tome 6 (1995) pp. 117-132. http://gdmltest.u-ga.fr/item/RLIN_1995_9_6_2_117_0/

Bibliographie

[1] Pai, S. I., Magnetogasdynamics and Plasma Dynamics. Springer-Verlag, New York1962. | MR 137449 | Zbl 0122.44305

[2] Shercliff, J. A., A Textbook of Magnetohydrodynamics. Pergamon, Oxford1965. | MR 185961

[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. | Zbl 0401.76091

[5] Rionero, S., Sulla stabilità magnetofluidodinamica non lineare asintotica in media in presenza di effetto Hall. Ricerche Mat., vol. 20, 2, 1971, 285-296. | MR 343759 | Zbl 0236.76043

[6] Maiellaro, M. - Palese, L., Electrical anisotropic effects on thermal instability. Int. J. Engng. Sci., vol. 22, 4, 1984, 411-418. | Zbl 0534.76045

[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. | MR 891351 | Zbl 0649.76023

[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. | MR 946029 | Zbl 0666.76153

[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. | MR 1136744 | Zbl 1145.76473

[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. | MR 1060804 | Zbl 0719.76035

[11] Ladyzhenskaya, O. A. - Solonnikov, V. A., Solvability of evolution problems of magnetohydrodynamics. Doklady Acad. Sci. USSR, vol. 124, 1, 1959, 26-28. | Zbl 0091.19804

[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. | MR 170130

[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. | MR 377310 | Zbl 0404.35089

[14] Carbonaro, B., On the linearization principle in magnetohydrodynamics. Ricerche Mat., vol. 28, 1979, 426-437. | MR 578596 | Zbl 0437.76050

[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. | MR 499834 | Zbl 0389.76063

[16] Stedry, M. - Vejvoda, O., Equations of magnetohydrodynamics: periodic solutions. Casopis Pest. Mat., vol. 111, 2, 1986, 177-184. | MR 847316 | Zbl 0608.76108

[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. | MR 415097 | Zbl 0404.35081

[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. | MR 228809 | Zbl 0204.42101

[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. | Zbl 0366.76091

[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. | MR 467012 | Zbl 0388.35035

[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. | MR 918940 | Zbl 0696.35130

[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. | MR 1031983 | Zbl 0717.35069

[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. | MR 1156428 | Zbl 0766.35034

[24] Vishik, M., On strongly elliptic systems of differential equations. Mat. Sbornik, vol. 29, 1951, 615-676 (in Russian). | MR 49440 | Zbl 0044.09503

[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. | MR 211083 | Zbl 0164.12502

[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. | MR 720208 | Zbl 0548.35096

[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. | Zbl 0615.35077