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Some Qualitative Properties of the Solution to the Magnetohydrodynamic Equations for Nonlinear Bipolar Fluids

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Abstract

In this article we study the long-time behaviour of a system of nonlinear Partial Differential Equations (PDEs) modelling the motion of incompressible, isothermal and conducting bipolar fluids in presence of magnetic field. We mainly prove the existence of a global attractor denoted by \(\mathcal{A}\) for the semigroup associated to the aforementioned system of nonlinear PDEs. We also show that this semigroup is uniformly differentiable on \(\mathcal{A}\). This fact enables us to go further and prove that the attractor \(\mathcal{A}\) is of finite-dimensional and we give an explicit bounds for its Hausdorff and fractal dimensions.

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Acknowledgement

The author’s research is supported by the Austrian Science Foundation through the Lise-Meitner-Programm M1487.

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  1. Department of Mathematics and Information Technology, Montanuniversität Leoben, Franz Josef Strasse 18, 8700, Leoben, Austria

    Paul André Razafimandimby

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  1. Paul André Razafimandimby
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Correspondence to Paul André Razafimandimby.

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Razafimandimby, P.A. Some Qualitative Properties of the Solution to the Magnetohydrodynamic Equations for Nonlinear Bipolar Fluids. Acta Appl Math 138, 213–240 (2015). https://doi.org/10.1007/s10440-014-9964-2

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  • DOI: https://doi.org/10.1007/s10440-014-9964-2

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