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Qualitative Behavior of Solutions for Thermodynamically Consistent Stefan Problems with Surface Tension

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Abstract

We study the qualitative behavior of a thermodynamically consistent two-phase Stefan problem with surface tension and with or without kinetic undercooling. It is shown that these problems generate local semiflows in well-defined state manifolds. If a solution does not exhibit singularities in a sense made precise herein, it is proved that it exists globally in time and its orbit is relatively compact. In addition, stability and instability of equilibria are studied. In particular, it is shown that multiple spheres of the same radius are unstable, reminiscent of the onset of Ostwald ripening.

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Authors and Affiliations

  1. Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, Theodor-Lieser-Strasse 5, 06120, Halle, Germany

    Jan Prüss & Rico Zacher

  2. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Gieri Simonett

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  1. Jan Prüss
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  2. Gieri Simonett
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Correspondence to Gieri Simonett.

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Communicated by F. Otto

The research of G.S. was partially supported by the NSF Grant DMS-0600870. The research of R.Z. was partially supported by the Deutsche Forschungsgemeinschaft (DFG)

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Prüss, J., Simonett, G. & Zacher, R. Qualitative Behavior of Solutions for Thermodynamically Consistent Stefan Problems with Surface Tension. Arch Rational Mech Anal 207, 611–667 (2013). https://doi.org/10.1007/s00205-012-0571-y

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