# American Institute of Mathematical Sciences

June  2012, 1(1): 171-194. doi: 10.3934/eect.2012.1.171

## 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, Germany 2 Department of Mathematics and Research Institute of Science and Engineering, JST CREST, Waseda University, Ohkubo 3-4-1, Shinjuku-ku, Tokyo 169-8555, Japan 3 Department of Mathematics, Shizuoka University, Shizuoka 422-8529, Japan 4 Department of Mathematics, Vanderbilt University, Nashville, TN 37240

Received  September 2011 Revised  November 2011 Published  March 2012

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.
Citation: Jan Prüss, Yoshihiro Shibata, Senjo Shimizu, Gieri Simonett. On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities. Evolution Equations & Control Theory, 2012, 1 (1) : 171-194. doi: 10.3934/eect.2012.1.171
##### References:
 [1] D. Bothe, J. Prüss, $L_p$-theory for a class of non-Newtonian fluids,, SIAM J. Math. Anal., 39 (2007), 379.  doi: 10.1137/060663635.  Google Scholar [2] D. M. Anderson, P. Cermelli, E. Fried, M. E. Gurtin and G. B. McFadden, General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids,, J. Fluid Mech., 581 (2007), 323.  doi: 10.1017/S0022112007005587.  Google Scholar [3] E. DiBenedetto and A. Friedman, Conduction-convection problems with change of phase,, J. Differential Equations, 62 (1986), 129.   Google Scholar [4] E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems with change of phase,, Arch. Rational Mech. Anal., 123 (1993), 99.  doi: 10.1007/BF00695273.  Google Scholar [5] R. Denk, M. Hieber and J. Prüss, "$\mathcal R$-boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type,", AMS Memoirs, 788 (2003).   Google Scholar [6] R. Denk, M. Hieber and J. Prüss, Optimal $L^ p$-$L^ q$-estimates for parabolic boundary value problems with inhomogeneous data,, Math. Z., 257 (2007), 193.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [7] K.-H. Hoffmann and V. N. Starovoitov, The Stefan problem with surface tension and convection in Stokes fluid,, Adv. Math. Sci. Appl., 8 (1998), 173.   Google Scholar [8] K.-H. Hoffmann and V. N. Starovoitov, Phase transitions of liquid-liquid type with convection,, Adv. Math. Sci. Appl., 8 (1998), 185.   Google Scholar [9] M. Ishii, "Thermo-Fluid Dynamics of Two-Phase Flow,", Collection de la Direction des Etudes et Recherches d'Electricte de France, (1975).   Google Scholar [10] M. Ishii and H. Takashi, "Thermo-Fluid Dynamics of Two-Phase Flow,", Springer, (2006).   Google Scholar [11] M. Köhne, J. Prüss and M. Wilke, Qualitative behaviour of solutions for the two-phase Navier-Stokes equations with surface tension,, Math. Annalen, ().   Google Scholar [12] Y. Kusaka, On a limit problem of the Stefan problem with surface tension in a viscous incompressible fluid flow,, Adv. Math. Sci. Appl., 12 (2002), 665.   Google Scholar [13] Y. Kusaka and A. Tani, On the classical solvability of the Stefan problem in a viscous incompressible fluid flow,, SIAM J. Math. Anal., 30 (1999), 584.  doi: 10.1137/S0036141098334936.  Google Scholar [14] Y. Kusaka and A. Tani, Classical solvability of the two-phase Stefan problem in a viscous incompressible fluid flow,, Math. Models Methods Appl. Sci., 12 (2002), 365.  doi: 10.1142/S0218202502001696.  Google Scholar [15] M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions,, Math. Nachr., ().   Google Scholar [16] J. Prüss, Maximal regularity for evolution equations in $L_p$-spacess,, Conf. Sem. Mat. Univ. Bari, 285 (2003), 1.   Google Scholar [17] J. Prüss and S. Shimizu, Incompressible two-phase flows with phase transition: Non-equal densities,, submitted., ().   Google Scholar [18] J. Prüss and G. Simonett, Maximal regularity for evolution equations in weighted $L_p$-spaces,, Archiv Math., 82 (2004), 415.   Google Scholar [19] J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension,, SIAM J. Math. Anal., 40 (2008), 675.  doi: 10.1137/070700632.  Google Scholar [20] J. Prüss and G. Simonett, On the two-phase Navier-Stokes equations with surface tension,, Interfaces & Free Bound, 12 (2010), 311.   Google Scholar [21] J. Prüss and G. Simonett, Analytic solutions for the two-phase Navier-Stokes equations with surface tension,, Progr. Nonlin. Diff. Eqns. Appl., 80 (2011), 507.   Google Scholar [22] J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension,, \arXiv{1101.3763}, ().   Google Scholar [23] Y. Shibata and S. Shimizu, Resolvent estimates and maximal regularity of the interface problem for the Stokes system in a bounded domain,, preprint, (2009).   Google Scholar [24] N. Tanaka, Two-phase free boundary problem for viscous incompressible thermo-capillary convection,, Japan J. Mech., 21 (1995), 1.   Google Scholar [25] H. Triebel, "Theory of Function Spaces II,", Birkhäuser Verlag, (1992).   Google Scholar

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##### References:
 [1] D. Bothe, J. Prüss, $L_p$-theory for a class of non-Newtonian fluids,, SIAM J. Math. Anal., 39 (2007), 379.  doi: 10.1137/060663635.  Google Scholar [2] D. M. Anderson, P. Cermelli, E. Fried, M. E. Gurtin and G. B. McFadden, General dynamical sharp-interface conditions for phase transformations in viscous heat-conducting fluids,, J. Fluid Mech., 581 (2007), 323.  doi: 10.1017/S0022112007005587.  Google Scholar [3] E. DiBenedetto and A. Friedman, Conduction-convection problems with change of phase,, J. Differential Equations, 62 (1986), 129.   Google Scholar [4] E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems with change of phase,, Arch. Rational Mech. Anal., 123 (1993), 99.  doi: 10.1007/BF00695273.  Google Scholar [5] R. Denk, M. Hieber and J. Prüss, "$\mathcal R$-boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type,", AMS Memoirs, 788 (2003).   Google Scholar [6] R. Denk, M. Hieber and J. Prüss, Optimal $L^ p$-$L^ q$-estimates for parabolic boundary value problems with inhomogeneous data,, Math. Z., 257 (2007), 193.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [7] K.-H. Hoffmann and V. N. Starovoitov, The Stefan problem with surface tension and convection in Stokes fluid,, Adv. Math. Sci. Appl., 8 (1998), 173.   Google Scholar [8] K.-H. Hoffmann and V. N. Starovoitov, Phase transitions of liquid-liquid type with convection,, Adv. Math. Sci. Appl., 8 (1998), 185.   Google Scholar [9] M. Ishii, "Thermo-Fluid Dynamics of Two-Phase Flow,", Collection de la Direction des Etudes et Recherches d'Electricte de France, (1975).   Google Scholar [10] M. Ishii and H. Takashi, "Thermo-Fluid Dynamics of Two-Phase Flow,", Springer, (2006).   Google Scholar [11] M. Köhne, J. Prüss and M. Wilke, Qualitative behaviour of solutions for the two-phase Navier-Stokes equations with surface tension,, Math. Annalen, ().   Google Scholar [12] Y. Kusaka, On a limit problem of the Stefan problem with surface tension in a viscous incompressible fluid flow,, Adv. Math. Sci. Appl., 12 (2002), 665.   Google Scholar [13] Y. Kusaka and A. Tani, On the classical solvability of the Stefan problem in a viscous incompressible fluid flow,, SIAM J. Math. Anal., 30 (1999), 584.  doi: 10.1137/S0036141098334936.  Google Scholar [14] Y. Kusaka and A. Tani, Classical solvability of the two-phase Stefan problem in a viscous incompressible fluid flow,, Math. Models Methods Appl. Sci., 12 (2002), 365.  doi: 10.1142/S0218202502001696.  Google Scholar [15] M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions,, Math. Nachr., ().   Google Scholar [16] J. Prüss, Maximal regularity for evolution equations in $L_p$-spacess,, Conf. Sem. Mat. Univ. Bari, 285 (2003), 1.   Google Scholar [17] J. Prüss and S. Shimizu, Incompressible two-phase flows with phase transition: Non-equal densities,, submitted., ().   Google Scholar [18] J. Prüss and G. Simonett, Maximal regularity for evolution equations in weighted $L_p$-spaces,, Archiv Math., 82 (2004), 415.   Google Scholar [19] J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension,, SIAM J. Math. Anal., 40 (2008), 675.  doi: 10.1137/070700632.  Google Scholar [20] J. Prüss and G. Simonett, On the two-phase Navier-Stokes equations with surface tension,, Interfaces & Free Bound, 12 (2010), 311.   Google Scholar [21] J. Prüss and G. Simonett, Analytic solutions for the two-phase Navier-Stokes equations with surface tension,, Progr. Nonlin. Diff. Eqns. Appl., 80 (2011), 507.   Google Scholar [22] J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension,, \arXiv{1101.3763}, ().   Google Scholar [23] Y. Shibata and S. Shimizu, Resolvent estimates and maximal regularity of the interface problem for the Stokes system in a bounded domain,, preprint, (2009).   Google Scholar [24] N. Tanaka, Two-phase free boundary problem for viscous incompressible thermo-capillary convection,, Japan J. Mech., 21 (1995), 1.   Google Scholar [25] H. Triebel, "Theory of Function Spaces II,", Birkhäuser Verlag, (1992).   Google Scholar
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