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August  2013, 33(8): 3289-3320. doi: 10.3934/dcds.2013.33.3289

Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations

Youcef Amirat 1, and  Kamel Hamdache 2,

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

Received  May 2012 Revised  September 2012 Published  January 2013

In this paper we study the equations of flow and heat transfer in a magnetic fluid with internal rotations, when the fluid is subjected to the action of an external magnetic field. The system of equations is a combination of the Navier-Stokes equations, the magnetization relaxation equation of Bloch type, the magnetostatic equations and the temperature equation. We prove the local-in-time existence of the unique strong solution to the system equipped with initial and boundary conditions and establish a blow-up criterium for strong solutions. We then prove the global-in-time existence of strong solutions, under smallness assumptions on the initial data and the external magnetic field.
Keywords: Navier-Stokes equations, heat transfer, Magnetic fluid, global weak solutions., incompressible flow.
Mathematics Subject Classification: Primary: 35Q35, 76D05; Secondary: 80A2.
Citation: Youcef Amirat, Kamel Hamdache. Strong solutions to the equations of flow and heat transfer in magnetic fluids with internal rotations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3289-3320. doi: 10.3934/dcds.2013.33.3289
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show all references

References:
[1]

J. Math. Anal. Appl., 353 (2009), 271-294. doi: 10.1016/j.jmaa.2008.11.084.  Google Scholar

[2]

Comm. Pure Appl. Anal., 11 (2012), 675-696. doi: 10.3934/cpaa.2012.11.675.  Google Scholar

[3]

J. Math. Fluid Mech., 14 (2012), 217-247. doi: 10.1007/s00021-011-0050-5.  Google Scholar

[4]

Acta Mech., 128 (1998), 39-47. Google Scholar

[5]

Walter de Gryuter and Co., Berlin-New York, 1997. Google Scholar

[6]

Differential and Integral Equations, 14 (2001), 213-229.  Google Scholar

[7]

J. Math. Pures Appl., 83 (2004), 243-275. doi: 10.1016/j.matpur.2003.11.004.  Google Scholar

[8]

Invent. Math., 98 (1989), 511-547. doi: 10.1007/BF01393835.  Google Scholar

[9]

J. Fluid Mech., 40 (1970), 753-767. Google Scholar

[10]

Springer Tracts in Natural Philosophy, 38, Springer Verlag, New-York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[11]

Springer Tracts in Natural Philosophy, 39, Springer-Verlag, New-York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar

[12]

J. Magn. Magn. Mater., 271 (2004), 63-73. Google Scholar

[13]

J. Reine Angew. Math., 330 (1982), 173-214.  Google Scholar

[14]

Journal of Functional Analysis, 102 (1991), 72-94. doi: 10.1016/0022-1236(91)90136-S.  Google Scholar

[15]

Proc. Japan Acad. Ser. A Math. Sci., 67 (1991), 197-202.  Google Scholar

[16]

Monographs and Studies in Mathematics, 24, Pitman (Advanced Publishing Program), Boston, MA, 1985.  Google Scholar

[17]

Physical Review, E70 (2004), 1-12. Google Scholar

[18]

Z. Angew. Math. Phys., 62 (2011), 529-538. doi: 10.1007/s00033-010-0096-x.  Google Scholar

[19]

SIAM J. Math. Anal., 37 (2006), 1417-1434. doi: 10.1137/S0036141004442197.  Google Scholar

[20]

Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969.  Google Scholar

[21]

Translations Math. Monogr., 23, AMS, Providence, R.I., 1968.  Google Scholar

[22]

Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[23]

Oxford Lecture Series in Mathematics and its Applications, 3, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.  Google Scholar

[24]

Phys. Fluids, 7 (1964), 1927-1937.  Google Scholar

[25]

J. Phys. D: Appl. Phys., 36 (2003), R167-R181. Google Scholar

[26]

Dover Publications, Inc., 1997. Google Scholar

[27]

Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 61-84. Google Scholar

[28]

Sov. Phys. JETP, 44 (1972), 1291-1294. Google Scholar

[29]

J. Magn. and Magn. Mater., 252 (2002), 197-202. Google Scholar

[30]

Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111. Google Scholar

[31]

in "Thermal Nonequilibrium Phenomena in Fluid Mixtures," Lecture Notes in Physics, Springer, Berlin, 584 (2002), 355-371. Google Scholar

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J. of Magn. and Magn. Mater., 320 (2008), 316-324. Google Scholar

[33]

J. Math. Anal. Appl., 364 (2010), 424-436. doi: 10.1016/j.jmaa.2009.10.032.  Google Scholar

[34]

Publications Mathématiques d'Orsay 78, 13, Université de Paris-Sud, Département de Mathématique, Orsay, 1978.  Google Scholar

[35]

Reprint of the 1984 edition, AMS-Chelsea Publishing, Providence, RI, 2001.  Google Scholar

[36]

Z. angew. Math. Phys., 54 (2003), 551-565. doi: 10.1007/s00033-003-1100-5.  Google Scholar

[37]

Journal of Nanoparticle Research, 3 (2001), 73-78. Google Scholar

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