# American Institute of Mathematical Sciences

March 2019, 18(2): 709-734. doi: 10.3934/cpaa.2019035

## Weak solutions to stationary equations of heat transfer in a magnetic fluid

 1 Laboratoire de Mathématiques Blaise Pascal, Université Clermont Auvergne, CNRS UMR 6620, Campus universitaire des Cézeaux, 3, place Vasarely, 63178, Aubière, France 2 Pôle universitaire Léonard de Vinci. DVRC. 92916 Paris la Défense Cedex

* Corresponding author

Received  January 2018 Revised  July 2018 Published  October 2018

We consider the differential system describing the stationary heat transfer in a magnetic fluid in the presence of a heat source and an external magnetic field. The system consists of the stationary incompressible Navier-Stokes equations, the magnetostatic equations and the stationary heat equation. We prove, for the differential system posed in a bounded domain of $\mathbb{R}^3$ and equipped with Fourier boundary conditions, the existence of weak solutions by using a regularization of the Kelvin force and the thermal power.

Citation: Youcef Amirat, Kamel Hamdache. Weak solutions to stationary equations of heat transfer in a magnetic fluid. Communications on Pure & Applied Analysis, 2019, 18 (2) : 709-734. doi: 10.3934/cpaa.2019035
##### References:
 [1] R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012. [2] 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. [3] Y. Amirat and K. Hamdache, Global weak solutions to the equations of thermal convection in micropolar fluids subjected to Hall current, Nonlinear Analysis, Series A: Theory, Methods & Applications, 102 (2014), 186-207. doi: 10.1016/j.na.2014.02.001. [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] B. Ducomet and E. Feireisl, On the dynamics of gaseous stars, Arch. Rational Mech. Anal., 174 (2004), 221-266. doi: 10.1007/s00205-004-0326-5. [6] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004. [7] E. Feireisl and D. Prazak, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, Springfield, MO, 2010. [8] E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics, (2006), 1-14. [9] 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. [10] 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, 1994. doi: 10.1007/978-1-4612-5364-8. [11] P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, 1985. [12] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [13] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969. [14] P. B. Mucha and M. Pokorny, Weak solutions to equations of steady compressible heat conducting fluids, Mathematical Models and Methods in Applied Sciences, 20 (2010), 785-813. doi: 10.1142/S0218202510004441. [15] P. B. Mucha and M. Pokorny, On the steady compressible Navier-Stokes-Fourier system, Commun. Math. Phys, 288 (2009), 349-377. doi: 10.1007/s00220-009-0772-x. [16] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004. [17] 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), 167-181. [18] A. Prignet, Conditions aux limites non homogènes pour des problènmes elliptiques avec second membre mesure, Ann. Fac. Sciences Toulouse, 6 (1997), 297-318. [19] R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc. 1997. [20] R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (SpringerVerlag, Heidelberg), 594, S. Odenbache Ed., (2002), 61-84. [21] M. I. Shliomis, in Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111. [22] R. Temam, Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984. [23] 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.

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##### References:
 [1] R. Alexandre and C. Villani, On the Boltzmann equation for long-range interactions, Comm. Pure Appl. Math., 55 (2002), 30-70. doi: 10.1002/cpa.10012. [2] 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. [3] Y. Amirat and K. Hamdache, Global weak solutions to the equations of thermal convection in micropolar fluids subjected to Hall current, Nonlinear Analysis, Series A: Theory, Methods & Applications, 102 (2014), 186-207. doi: 10.1016/j.na.2014.02.001. [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] B. Ducomet and E. Feireisl, On the dynamics of gaseous stars, Arch. Rational Mech. Anal., 174 (2004), 221-266. doi: 10.1007/s00205-004-0326-5. [6] E. Feireisl, Dynamics of Viscous Compressible Fluids, Oxford University Press, 2004. [7] E. Feireisl and D. Prazak, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS Series on Applied Mathematics, 4, Springfield, MO, 2010. [8] E. Feireisl and J. Málek, On the Navier-Stokes equations with temperature-dependent transport coefficients, Differential Equations and Nonlinear Mechanics, (2006), 1-14. [9] 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. [10] 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, 1994. doi: 10.1007/978-1-4612-5364-8. [11] P. Grisvard, Elliptic Problems in Non Smooth Domains, Pitman, 1985. [12] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. [13] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, 1969. [14] P. B. Mucha and M. Pokorny, Weak solutions to equations of steady compressible heat conducting fluids, Mathematical Models and Methods in Applied Sciences, 20 (2010), 785-813. doi: 10.1142/S0218202510004441. [15] P. B. Mucha and M. Pokorny, On the steady compressible Navier-Stokes-Fourier system, Commun. Math. Phys, 288 (2009), 349-377. doi: 10.1007/s00220-009-0772-x. [16] A. Novotný and I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford University Press, 2004. [17] 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), 167-181. [18] A. Prignet, Conditions aux limites non homogènes pour des problènmes elliptiques avec second membre mesure, Ann. Fac. Sciences Toulouse, 6 (1997), 297-318. [19] R. E. Rosensweig, Ferrohydrodynamics, Dover Publications, Inc. 1997. [20] R. E. Rosensweig, Basic equations for magnetic fluids with internal rotations, in Ferrofluids: Magnetically Controllable Fluids and Their Applications, Lecture Notes in Physics (SpringerVerlag, Heidelberg), 594, S. Odenbache Ed., (2002), 61-84. [21] M. I. Shliomis, in Ferrofluids: Magnetically controllable fluids and their applications, Lecture Notes in Physics (Springer-Verlag, Heidelberg), S. Odenbach Ed., 594 (2002), 85-111. [22] R. Temam, Navier-Stokes Equations, 3rd (revised) edition, Elsevier Science Publishers B.V., Amsterdam, 1984. [23] 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.
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