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

Youcef Amirat; Kamel Hamdache

doi:10.3934/dcds.2013.33.3289

Article Contents

2013, Volume 33, Issue 8: 3289-3320. Doi: 10.3934/dcds.2013.33.3289

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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: April 30, 2012

Revised: August 31, 2012

Published: July 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 35Q35, 76D05; Secondary: 80A20.

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