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Stability of Equilibrium Shapes in Some Free Boundary Problems Involving Fluids

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Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

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Abstract

In this chapter the motion of two-phase, incompressible, viscous fluids with surface tension is investigated. Three cases are considered: (1) the case of heat-conducting fluids, (2) the case of isothermal fluids, and (3) the case of Stokes flows. In all three situations, the equilibrium states in the absence of outer forces are characterized and their stability properties are analyzed. It is shown that the equilibrium states correspond to the critical points of a natural physical or geometric functional (entropy, available energy, surface area) constrained by the pertinent conserved quantities (total energy, phase volumes). Moreover, it is shown that solutions which do not develop singularities exist globally and converge to an equilibrium state.

The research of G.S. was partially supported by the NSF Grant DMS-1265579.

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, 37240, Nashville, TN, USA

    Gieri Simonett

  2. Fakultät für Mathematik, Universität Regensburg, D-93040, Regensburg, Germany

    Mathias Wilke

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  1. Gieri Simonett
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  2. Mathias Wilke
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Correspondence to Gieri Simonett .

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Editors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku, Tokyo, Japan

    Yoshikazu Giga

  2. Université de Toulon IMATH, Toulon, France

    Antonín Novotný

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Simonett, G., Wilke, M. (2018). Stability of Equilibrium Shapes in Some Free Boundary Problems Involving Fluids. In: Giga, Y., Novotný, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-13344-7_28

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