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Weak Solutions to Unsteady and Steady Models of Conductive Magnetic Fluids

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Abstract

We study a nonlinear coupling system of partial differential equations describing the dynamic of a magnetic fluid with internal rotations. The present mathematical model generalizes those discussed previously in the literature since actually the fluid is electrically conducting inducing additional nonlinearities in the problem and the dynamics of the magnetic field is described by the quasi-static Maxwell equations instead of the usual magnetostatic ones. We prove existence of weak solutions with finite energy first for the unsteady problem then for the steady one.

Keywords

Navier–Stokes equations Bloch–Torrey equation Quasi-static Maxwell equations Magnetization Internal rotations 

Mathematics Subject Classification

35Q30 35Q35 35Q79 76D05 76W05 76U05 

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.DVRCLéonard de Vinci Pôle UniversitaireParis la Défense CedexFrance
  2. 2.Laboratoire AMNEDP, Faculté de MathématiquesUniversité USTHBAlgiersAlgeria

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