Weak solutions to the equations of stationary magnetohydrodynamic flows in porous media

Youcef Amirat; Laurent Chupin; Rachid Touzani

doi:10.3934/cpaa.2014.13.2445

Article Contents

2014, Volume 13, Issue 6: 2445-2464. Doi: 10.3934/cpaa.2014.13.2445

This issue Previous Article Stability of the linearized MHD-Maxwell free interface problem Next Article Some eigenvalue problems with non-local boundary conditions and applications

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

Received: October 31, 2013

Revised: April 30, 2014

Published: October 2014

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Abstract

We study the differential system which describes the steady flow of an electrically conducting fluid in a saturated porous medium, when the fluid is subjected to the action of a magnetic field. The system consists of the stationary Brinkman-Forchheimer equations and the stationary magnetic induction equation. We prove existence of weak solutions to the system posed in a bounded domain of $\mathbb{R}^3$ and equipped with boundary conditions. We also prove uniqueness in the class of small solutions, and regularity of weak solutions. Then we establish a convergence result, as the Brinkman coefficient (viscosity) tends to 0, of the weak solutions to a solution of the system formed by the Darcy-Forchheimer equations and the magnetic induction equation.

Keywords:

Magnetohydrodynamic flows in porous media,
Brinkman-Forchheimer equations,
Darcy-Forchheimer equations,
Lorentz force,
weak solutions.

Mathematics Subject Classification: Primary: 76S05, 76W05; Secondary: 35Q35.

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