On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities

Jan Prüss; Yoshihiro Shibata; Senjo Shimizu; Gieri Simonett

doi:10.3934/eect.2012.1.171

Article Contents

2012, Volume 1, Issue 1: 171-194. Doi: 10.3934/eect.2012.1.171

This issue Previous Article Modeling of a nonlinear plate Next Article Hyperbolic Navier-Stokes equations I: Local well-posedness

On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities

1.
Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, D-60120 Halle
2.
Department of Mathematics and Research Institute of Science and Engineering, JST CREST, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555
3.
Department of Mathematics, Shizuoka University, Shizuoka 422-8529
4.
Department of Mathematics, Vanderbilt University, Nashville, TN 37240

Received: September 2011

Revised: November 2011

Published: June 2012

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Abstract

The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of maximal $L_p$-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

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Mathematics Subject Classification: Primary: 35R35, Secondary: 35Q30, 76D45, 76T05, 80A2.

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