Geodesic
Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects
Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 791-825.

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We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded in a larger domain, relevant for the global temperature- and magnetic field- distributions. Material inhomogeneities in this larger region lead to transmission relations for the electromagnetic fields and the heat flux on inner boundaries. In the presence of transparent materials, the radiative heat transfer is important and leads to a nonlocal and nonlinear jump relation for the heat flux. We prove the existence of weak solutions, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small.
Classification : 35A01, 35D30, 35J55, 35Q30, 35Q35, 35Q60, 76W05
Keywords: nonlinear elliptic system; magnetohydrodynamics; natural interface conditions; nonlinear heat equation; nonlocal radiation boundary conditions 
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     author = {Druet, Pierre-\'Etienne},
     title = {Existence for the stationary {MHD-equations} coupled to heat transfer with nonlocal radiation effects},
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     pages = {791--825},
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     zbl = {1224.35337},
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     url = {http://geodesic.mathdoc.fr/item/CMJ_2009__59_3_a17/}
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Druet, Pierre-Étienne. Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 791-825. http://geodesic.mathdoc.fr/item/CMJ_2009__59_3_a17/