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Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations
1. | Laboratoire de Mathématiques, CNRS UMR 6620, Université Blaise Pascal (Clermont-Ferrand 2), 63177 Aubière cedex |
2. | Centre de Mathématiques Appliquées, CNRS, Ecole Polytechnique, 91128 Palaiseau cedex |
References:
[1] |
Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294.
doi: 10.1016/j.jmaa.2008.11.084. |
[2] |
Y. Amirat and K. Hamdache, On a heated incompressible magnetic fluid model, Comm. Pure Appl. Anal., 11 (2012), 675-696.
doi: 10.3934/cpaa.2012.11.675. |
[3] |
Y. Amirat and K. Hamdache, Heat transfer in incompressible magnetic fluid, J. Math. Fluid Mech., 14 (2012), 217-247.
doi: 10.1007/s00021-011-0050-5. |
[4] |
H. I. Andersson and O. A. Valnes, Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole, Acta Mech., 128 (1998), 39-47. |
[5] |
E. Blums, A. Cebers and M. M. Maiorov, "Magnetic Fluids,'' Walter de Gryuter and Co., Berlin-New York, 1997. |
[6] |
G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential and Integral Equations, 14 (2001), 213-229. |
[7] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.
doi: 10.1016/j.matpur.2003.11.004. |
[8] |
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[9] |
B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767. |
[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] |
G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Ii. Nonlinear Steady Problems,'' Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New-York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[12] |
R. Ganguly, S. Sen and I. K. Puri, Heat transfer augmentation using a magnetic fluid under the influence of a line dipole, J. Magn. Magn. Mater., 271 (2004), 63-73. |
[13] |
M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214. |
[14] |
Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, Journal of Functional Analysis, 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[15] |
M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for abstract linear parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 197-202. |
[16] |
P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
[17] |
P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review, E70 (2004), 1-12. |
[18] |
P. N. Kaloni and A. Mahajan, Stability of magnetic fluid motions in a saturated porous medium, Z. Angew. Math. Phys., 62 (2011), 529-538.
doi: 10.1007/s00033-010-0096-x. |
[19] |
H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.
doi: 10.1137/S0036141004442197. |
[20] |
O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. |
[21] |
O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations Math. Monogr., 23, AMS, Providence, R.I., 1968. |
[22] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969. |
[23] |
P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996. |
[24] |
J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937. |
[25] |
Q. Q. A. Pankhurst, J. Connolly, S. K. Jones and J. Dobson, Applications of magnetic nonoparticles in biomedicine, J. Phys. D: Appl. Phys., 36 (2003), R167-R181. |
[26] |
R. E. Rosensweig, "Ferrohydrodynamics,'' Dover Publications, Inc., 1997. |
[27] |
R. E. Rosensweig, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 61-84. |
[28] |
M. I Shliomis, Effective viscosity of magnetic suspension, Sov. Phys. JETP, 44 (1972), 1291-1294. |
[29] |
M. I. Shliomis and B. L. Smorodin, Convective instability of magnetized ferrofluids, J. Magn. and Magn. Mater., 252 (2002), 197-202. |
[30] |
M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111. |
[31] |
M. I. Shliomis, Convective instability of magnetized ferrofluids: Influence of magnetophoresis and soret effect, in "Thermal Nonequilibrium Phenomena in Fluid Mixtures," Lecture Notes in Physics, Springer, Berlin, 584 (2002), 355-371. |
[32] |
L. Sunil, P. Chand, P. Bharti and A. Mahajan, Thermal convection in micropolar ferrofluid in the presence of rotation, J. of Magn. and Magn. Mater., 320 (2008), 316-324. |
[33] |
Z. Tan and Y. Wang, Global analysis for strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 364 (2010), 424-436.
doi: 10.1016/j.jmaa.2009.10.032. |
[34] |
L. Tartar, "Topics in Nonlinear Analysis," Publications Mathématiques d'Orsay 78, 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978. |
[35] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'' Reprint of the 1984 edition, AMS-Chelsea Publishing, Providence, RI, 2001. |
[36] |
E. E Tzirtzilakis and N. G. Kafoussias, Biomagnetic fluid flow over a stretching sheet with nonlinear temperature dependent magnetization, Z. angew. Math. Phys., 54 (2003), 551-565.
doi: 10.1007/s00033-003-1100-5. |
[37] |
M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 3 (2001), 73-78. |
show all references
References:
[1] |
Y. Amirat and K. Hamdache, Strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 353 (2009), 271-294.
doi: 10.1016/j.jmaa.2008.11.084. |
[2] |
Y. Amirat and K. Hamdache, On a heated incompressible magnetic fluid model, Comm. Pure Appl. Anal., 11 (2012), 675-696.
doi: 10.3934/cpaa.2012.11.675. |
[3] |
Y. Amirat and K. Hamdache, Heat transfer in incompressible magnetic fluid, J. Math. Fluid Mech., 14 (2012), 217-247.
doi: 10.1007/s00021-011-0050-5. |
[4] |
H. I. Andersson and O. A. Valnes, Flow of a heated ferrofluid over a stretching sheet in the presence of a magnetic dipole, Acta Mech., 128 (1998), 39-47. |
[5] |
E. Blums, A. Cebers and M. M. Maiorov, "Magnetic Fluids,'' Walter de Gryuter and Co., Berlin-New York, 1997. |
[6] |
G. Carbou and P. Fabrie, Regular solutions for Landau-Lifschitz equation in a bounded domain, Differential and Integral Equations, 14 (2001), 213-229. |
[7] |
Y. Cho, H. J. Choe and H. Kim, Unique solvability of the initial boundary value problems for compressible viscous fluids, J. Math. Pures Appl., 83 (2004), 243-275.
doi: 10.1016/j.matpur.2003.11.004. |
[8] |
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98 (1989), 511-547.
doi: 10.1007/BF01393835. |
[9] |
B. A. Finlayson, Convective instability of ferromagnetic fluids, J. Fluid Mech., 40 (1970), 753-767. |
[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] |
G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Ii. Nonlinear Steady Problems,'' Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New-York, 1994.
doi: 10.1007/978-1-4612-5364-8. |
[12] |
R. Ganguly, S. Sen and I. K. Puri, Heat transfer augmentation using a magnetic fluid under the influence of a line dipole, J. Magn. Magn. Mater., 271 (2004), 63-73. |
[13] |
M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214. |
[14] |
Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, Journal of Functional Analysis, 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[15] |
M. Giga, Y. Giga and H. Sohr, $L^p$ estimates for abstract linear parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 197-202. |
[16] |
P. Grisvard, "Elliptic Problems in Nonsmooth Domains,'' Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. |
[17] |
P. N. Kaloni and J. X. Lou, Convective instability of magnetic fluids, Physical Review, E70 (2004), 1-12. |
[18] |
P. N. Kaloni and A. Mahajan, Stability of magnetic fluid motions in a saturated porous medium, Z. Angew. Math. Phys., 62 (2011), 529-538.
doi: 10.1007/s00033-010-0096-x. |
[19] |
H. Kim, A blow-up criterion for the nonhomogeneous incompressible Navier-Stokes equations, SIAM J. Math. Anal., 37 (2006), 1417-1434.
doi: 10.1137/S0036141004442197. |
[20] |
O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. |
[21] |
O. A. Ladyžhenskaya, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,'' Translations Math. Monogr., 23, AMS, Providence, R.I., 1968. |
[22] |
J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires,'' Dunod, Gauthier-Villars, Paris, 1969. |
[23] |
P.-L. Lions, "Mathematical Topics in Fluid Mechanics. Volume 1. Incompressible Models,'' Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996. |
[24] |
J. L. Neuringer and R. E. Rosensweig, Ferrohydrodynamics, Phys. Fluids, 7 (1964), 1927-1937. |
[25] |
Q. Q. A. Pankhurst, J. Connolly, S. K. Jones and J. Dobson, Applications of magnetic nonoparticles in biomedicine, J. Phys. D: Appl. Phys., 36 (2003), R167-R181. |
[26] |
R. E. Rosensweig, "Ferrohydrodynamics,'' Dover Publications, Inc., 1997. |
[27] |
R. E. Rosensweig, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 61-84. |
[28] |
M. I Shliomis, Effective viscosity of magnetic suspension, Sov. Phys. JETP, 44 (1972), 1291-1294. |
[29] |
M. I. Shliomis and B. L. Smorodin, Convective instability of magnetized ferrofluids, J. Magn. and Magn. Mater., 252 (2002), 197-202. |
[30] |
M. I. Shliomis, Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111. |
[31] |
M. I. Shliomis, Convective instability of magnetized ferrofluids: Influence of magnetophoresis and soret effect, in "Thermal Nonequilibrium Phenomena in Fluid Mixtures," Lecture Notes in Physics, Springer, Berlin, 584 (2002), 355-371. |
[32] |
L. Sunil, P. Chand, P. Bharti and A. Mahajan, Thermal convection in micropolar ferrofluid in the presence of rotation, J. of Magn. and Magn. Mater., 320 (2008), 316-324. |
[33] |
Z. Tan and Y. Wang, Global analysis for strong solutions to the equations of a ferrofluid flow model, J. Math. Anal. Appl., 364 (2010), 424-436.
doi: 10.1016/j.jmaa.2009.10.032. |
[34] |
L. Tartar, "Topics in Nonlinear Analysis," Publications Mathématiques d'Orsay 78, 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978. |
[35] |
R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'' Reprint of the 1984 edition, AMS-Chelsea Publishing, Providence, RI, 2001. |
[36] |
E. E Tzirtzilakis and N. G. Kafoussias, Biomagnetic fluid flow over a stretching sheet with nonlinear temperature dependent magnetization, Z. angew. Math. Phys., 54 (2003), 551-565.
doi: 10.1007/s00033-003-1100-5. |
[37] |
M. Zahn, Magnetic fluid and nonoparticle applications to nanotechnology, Journal of Nanoparticle Research, 3 (2001), 73-78. |
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