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Stability of the linearized MHD-Maxwell free interface problem
Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media
1. | Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex |
2. | Université Blaise Pascal & CNRS UMR 6620, Laboratoire de Mathématiques, Campus des Cézeaux, B.P. 80026, F-63177 Aubière cedex |
3. | Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal, 63177 Aubière Cedex, France |
References:
[1] |
Y. Amirat, Écoulements en milieux poreux n'obéissant pas à la loi de Darcy (French) [Flows in porous media not obeying the Darcy law], ESAIM: Math. Mod. Numer. Anal. - Modél. Math. Anal. Numér., 25 (1991), 273-306. |
[2] |
L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes (Italian), Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340. |
[3] |
O. Celebi, V. Kalantarov and D. Ugurlu, On continuous dependence on solutions of the Brinkman-Forchheimer equations, Appl. Math. Lett., 19 (2006), 801-807.
doi: 10.1016/j.aml.2005.11.002. |
[4] |
P. E. Druet, Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects, Czechoslovak Math. J., 59 (2009), 791-825.
doi: 10.1007/s10587-009-0048-9. |
[5] |
G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, Dunod, 1972. |
[6] |
G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[7] |
P. Fabrie, Régularité de la solution de l'équation de Darcy-Forchheimer, Nonlinear Anal., 13 (1989), 1025-1051.
doi: 10.1016/0362-546X(89)90093-X. |
[8] |
M. Firdaouss, J. L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers, J. Fluid Mech., 343 (1997), 331-350.
doi: 10.1017/S0022112097005843. |
[9] |
F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 459 (2003), 3195-3202.
doi: 10.1098/rspa.2003.1169. |
[10] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems, Springer tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[11] |
C. Geindreau and J. L. Auriault, Magnetohydrodynamic flows in porous media, J. Fluid Mech., 466 (2002), 343-63.
doi: 10.1017/S0022112002001404. |
[12] |
V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds, Ann. Mat. Pura Appl., 4 (1979), 159-198.
doi: 10.1007/BF02411693. |
[13] |
J. F. Gerbeau and C. Le Bris, A coupled system arising in magnetohydrodynamics, Appl. Math. Lett., 12 (1999), 53-57.
doi: 10.1016/S0893-9659(98)00172-4. |
[14] |
T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206. |
[15] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[16] |
J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Son, New York, 1975, (Third Edition, 1999). |
[17] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Rev. second edition, 1969. |
[18] |
P. Lehmann, R. Moreau, D. Camel and R. Bolcato, Modification of interdendritic convection in directional solidifcation by a uniform magnetic field, Acta Materialia, 46 (1998), 4067-4079. |
[19] |
P. Lehmann, R. Moreau, D. Camel and R. Bolcato, A simple analysis of the effect of convection on the structure of the mushy zone in the case of horizontal Bridgman solidification. Comparison with experimental results,, \emph{J. Cryst. Growth, 183 (): 690.
|
[20] |
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969. |
[21] |
P. L. Lions, Mathematical topics in fluid mechanics. Volume 1. Incompressible models, Oxford Science Publications, 1996. |
[22] |
A. J. Meir and P. G. Schmidt, On electromagnetically and thermally driven liquid-metal flows, Nonlinear anal., 47 (2001), 3281-3294.
doi: 10.1016/S0362-546X(01)00445-X. |
[23] |
R. J. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, 1990.
doi: 10.1007/978-94-015-7883-7. |
[24] |
L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 455 (1999), 2173-2190.
doi: 10.1098/rspa.1999.0398. |
[25] |
L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Stud. Appl. Math., 102 (1999), 419-439.
doi: 10.1111/1467-9590.00116. |
[26] |
V. R. Prasad, A. Beg and B. Vasu, Thermo-diffusion and diffusion-thermo effects on MHD free convection flow past a vertical porous plate in a non-Darcy porous medium, Chemical Engineering Journal, 173 (2011) 598-606. |
[27] |
B. Saramito, Stabilité d'un Plasma: Modélisation mathématique et simulation numérique (French) [Stability of a plasma: mathematical modelling and numerical simulation], Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, 1994. |
[28] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[29] |
J. Simon, Régularité de la solution d'un problème aux limites non linéaires (French) [Regularity of the solution of a nonlinear boundary problem], Ann. Fac. Sci. Toulouse Math., 3 (1981), 247-274. |
[30] |
L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay, 78.13, 1978. |
[31] |
R. Temam, Navier-Stokes equations, 3rd Edition, North-Holland, Amsterdam, 1984. Reedited in the AMS-Chelsea Series, Amer. Math. Soc., Providence, RI, 2001. |
[32] |
S. Whitaker, The Forchheimer equation: a theoretical development, Transport in Porous Media, 25 (1996), 27-62. |
[33] |
K. Zaïdat, Influence d'un champ magnétique glissant sur la solidification dirigée des alliages métalliques binaires, PhD Thesis, Institut National Polytechnique de Grenoble, 2005. Available from: http://tel.archives-ouvertes.fr/tel-00011040 |
show all references
References:
[1] |
Y. Amirat, Écoulements en milieux poreux n'obéissant pas à la loi de Darcy (French) [Flows in porous media not obeying the Darcy law], ESAIM: Math. Mod. Numer. Anal. - Modél. Math. Anal. Numér., 25 (1991), 273-306. |
[2] |
L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes (Italian), Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340. |
[3] |
O. Celebi, V. Kalantarov and D. Ugurlu, On continuous dependence on solutions of the Brinkman-Forchheimer equations, Appl. Math. Lett., 19 (2006), 801-807.
doi: 10.1016/j.aml.2005.11.002. |
[4] |
P. E. Druet, Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects, Czechoslovak Math. J., 59 (2009), 791-825.
doi: 10.1007/s10587-009-0048-9. |
[5] |
G. Duvaut and J. L. Lions, Les inéquations en mécanique et en physique, Dunod, 1972. |
[6] |
G. Duvaut and J. L. Lions, Inéquations en thermoélasticité et magnétohydrodynamique, Arch. Rational Mech. Anal., 46 (1972), 241-279. |
[7] |
P. Fabrie, Régularité de la solution de l'équation de Darcy-Forchheimer, Nonlinear Anal., 13 (1989), 1025-1051.
doi: 10.1016/0362-546X(89)90093-X. |
[8] |
M. Firdaouss, J. L. Guermond and P. Le Quéré, Nonlinear corrections to Darcy's law at low Reynolds numbers, J. Fluid Mech., 343 (1997), 331-350.
doi: 10.1017/S0022112097005843. |
[9] |
F. Franchi and B. Straughan, Continuous dependence and decay for the Forchheimer equations, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 459 (2003), 3195-3202.
doi: 10.1098/rspa.2003.1169. |
[10] |
G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. I. Linearized Steady Problems, Springer tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[11] |
C. Geindreau and J. L. Auriault, Magnetohydrodynamic flows in porous media, J. Fluid Mech., 466 (2002), 343-63.
doi: 10.1017/S0022112002001404. |
[12] |
V. Georgescu, Some boundary value problems for differential forms on compact Riemannian manifolds, Ann. Mat. Pura Appl., 4 (1979), 159-198.
doi: 10.1007/BF02411693. |
[13] |
J. F. Gerbeau and C. Le Bris, A coupled system arising in magnetohydrodynamics, Appl. Math. Lett., 12 (1999), 53-57.
doi: 10.1016/S0893-9659(98)00172-4. |
[14] |
T. Giorgi, Derivation of the Forchheimer law via matched asymptotic expansions, Transport in Porous Media, 29 (1997), 191-206. |
[15] |
C. He and Z. Xin, On the regularity of weak solutions to the magnetohydrodynamic equations, J. Differential Equations, 213 (2005), 235-254.
doi: 10.1016/j.jde.2004.07.002. |
[16] |
J. D. Jackson, Classical Electrodynamics, Second Edition, John Wiley & Son, New York, 1975, (Third Edition, 1999). |
[17] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, Rev. second edition, 1969. |
[18] |
P. Lehmann, R. Moreau, D. Camel and R. Bolcato, Modification of interdendritic convection in directional solidifcation by a uniform magnetic field, Acta Materialia, 46 (1998), 4067-4079. |
[19] |
P. Lehmann, R. Moreau, D. Camel and R. Bolcato, A simple analysis of the effect of convection on the structure of the mushy zone in the case of horizontal Bridgman solidification. Comparison with experimental results,, \emph{J. Cryst. Growth, 183 (): 690.
|
[20] |
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969. |
[21] |
P. L. Lions, Mathematical topics in fluid mechanics. Volume 1. Incompressible models, Oxford Science Publications, 1996. |
[22] |
A. J. Meir and P. G. Schmidt, On electromagnetically and thermally driven liquid-metal flows, Nonlinear anal., 47 (2001), 3281-3294.
doi: 10.1016/S0362-546X(01)00445-X. |
[23] |
R. J. Moreau, Magnetohydrodynamics, Kluwer Academic Publishers, 1990.
doi: 10.1007/978-94-015-7883-7. |
[24] |
L. E. Payne, J. C. Song and B. Straugham, Continuous dependence and convergence results for Brinkman and Forchheimer models with variable viscosity, R. Soc. Lond. Proc. Ser. A, Math. Phys. Eng. Sci., 455 (1999), 2173-2190.
doi: 10.1098/rspa.1999.0398. |
[25] |
L. E. Payne and B. Straugham, Convergence and continuous dependence for the Brinkman-Forchheimer equations, Stud. Appl. Math., 102 (1999), 419-439.
doi: 10.1111/1467-9590.00116. |
[26] |
V. R. Prasad, A. Beg and B. Vasu, Thermo-diffusion and diffusion-thermo effects on MHD free convection flow past a vertical porous plate in a non-Darcy porous medium, Chemical Engineering Journal, 173 (2011) 598-606. |
[27] |
B. Saramito, Stabilité d'un Plasma: Modélisation mathématique et simulation numérique (French) [Stability of a plasma: mathematical modelling and numerical simulation], Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, 1994. |
[28] |
M. Sermange and R. Temam, Some mathematical questions related to the MHD equations, Comm. Pure Appl. Math., 36 (1983), 635-664.
doi: 10.1002/cpa.3160360506. |
[29] |
J. Simon, Régularité de la solution d'un problème aux limites non linéaires (French) [Regularity of the solution of a nonlinear boundary problem], Ann. Fac. Sci. Toulouse Math., 3 (1981), 247-274. |
[30] |
L. Tartar, Topics in Nonlinear Analysis, Publications Mathématiques d'Orsay, 78.13, 1978. |
[31] |
R. Temam, Navier-Stokes equations, 3rd Edition, North-Holland, Amsterdam, 1984. Reedited in the AMS-Chelsea Series, Amer. Math. Soc., Providence, RI, 2001. |
[32] |
S. Whitaker, The Forchheimer equation: a theoretical development, Transport in Porous Media, 25 (1996), 27-62. |
[33] |
K. Zaïdat, Influence d'un champ magnétique glissant sur la solidification dirigée des alliages métalliques binaires, PhD Thesis, Institut National Polytechnique de Grenoble, 2005. Available from: http://tel.archives-ouvertes.fr/tel-00011040 |
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