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A free boundary approach to the Rosensweig instability of ferrofluids

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Abstract

We establish the existence of saddle points for a free boundary problem describing the two-dimensional free surface of a ferrofluid undergoing normal field instability. The starting point is the ferrohydrostatic equations for the magnetic potentials in the ferrofluid and air, and the function describing their interface. These constitute the strong form for the Euler–Lagrange equations of a convex–concave functional, which we extend to include interfaces that are not necessarily graphs of functions. Saddle points are then found by iterating the direct method of the calculus of variations and applying classical results of convex analysis. For the existence part, we assume a general nonlinear magnetization law; for a linear law, we also show, via convex duality, that the saddle point is a constrained minimizer of the relevant energy functional.

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Authors and Affiliations

  1. CNRS, Centrale Marseille, I2M, Aix Marseille Univ, 39 Rue Frederic Joliot Curie, Marseille Cedex 13, France

    Enea Parini

  2. Institut für Mathematik, Universität Kassel, 34132, Kassel, Germany

    Athanasios Stylianou

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  1. Enea Parini
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  2. Athanasios Stylianou
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Correspondence to Athanasios Stylianou.

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Parini, E., Stylianou, A. A free boundary approach to the Rosensweig instability of ferrofluids. Z. Angew. Math. Phys. 69, 32 (2018). https://doi.org/10.1007/s00033-018-0924-y

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  • DOI: https://doi.org/10.1007/s00033-018-0924-y

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