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Evolution Equations and Control Theory (EECT)
 

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

Pages: 171 - 194, Volume 1, Issue 1, June 2012      doi:10.3934/eect.2012.1.171

 
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Jan Prüss - Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, D-60120 Halle, Germany (email)
Yoshihiro Shibata - Department of Mathematics and Research Institute of Science and Engineering, JST CREST, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan (email)
Senjo Shimizu - Department of Mathematics, Shizuoka University, Shizuoka 422-8529, Japan (email)
Gieri Simonett - Department of Mathematics, Vanderbilt University, Nashville, TN 37240, United States (email)

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.

Keywords:  Two-phase Navier-Stokes equations, surface tension, phase transitions, entropy, well-posedness, time weights.
Mathematics Subject Classification:  Primary: 35R35, Secondary: 35Q30, 76D45, 76T05, 80A2.

Received: September 2011;      Revised: November 2011;      Available Online: March 2012.

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