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On wellposedness of incompressible twophase flows with phase transitions: The case of equal densities
1.  Institut für Mathematik, MartinLutherUniversität HalleWittenberg, D60120 Halle, Germany 
2.  Department of Mathematics and Research Institute of Science and Engineering, JST CREST, Waseda University, Ohkubo 341, Shinjukuku, Tokyo 1698555, Japan 
3.  Department of Mathematics, Shizuoka University, Shizuoka 4228529, Japan 
4.  Department of Mathematics, Vanderbilt University, Nashville, TN 37240 
References:
[1] 
D. Bothe, J. Prüss, $L_p$theory for a class of nonNewtonian fluids, SIAM J. Math. Anal., 39 (2007), 379421. doi: 10.1137/060663635. 
[2] 
D. M. Anderson, P. Cermelli, E. Fried, M. E. Gurtin and G. B. McFadden, General dynamical sharpinterface conditions for phase transformations in viscous heatconducting fluids, J. Fluid Mech., 581 (2007), 323370. doi: 10.1017/S0022112007005587. 
[3] 
E. DiBenedetto and A. Friedman, Conductionconvection problems with change of phase, J. Differential Equations, 62 (1986), 129185. 
[4] 
E. DiBenedetto and M. O'Leary, Threedimensional conductionconvection problems with change of phase, Arch. Rational Mech. Anal., 123 (1993), 99116. doi: 10.1007/BF00695273. 
[5] 
R. Denk, M. Hieber and J. Prüss, "$\mathcal R$boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type," AMS Memoirs, 788, Providence, R.I., 2003. 
[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), 193224. doi: 10.1007/s0020900701209. 
[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), 173183. 
[8] 
K.H. Hoffmann and V. N. Starovoitov, Phase transitions of liquidliquid type with convection, Adv. Math. Sci. Appl., 8 (1998), 185198. 
[9] 
M. Ishii, "ThermoFluid Dynamics of TwoPhase Flow," Collection de la Direction des Etudes et Recherches d'Electricte de France, Paris, 1975. 
[10] 
M. Ishii and H. Takashi, "ThermoFluid Dynamics of TwoPhase Flow," Springer, New York, 2006. 
[11] 
M. Köhne, J. Prüss and M. Wilke, Qualitative behaviour of solutions for the twophase NavierStokes equations with surface tension, Math. Annalen, to appear. 
[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), 665683. 
[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), 584602 (electronic). doi: 10.1137/S0036141098334936. 
[14] 
Y. Kusaka and A. Tani, Classical solvability of the twophase Stefan problem in a viscous incompressible fluid flow, Math. Models Methods Appl. Sci., 12 (2002), 365391. doi: 10.1142/S0218202502001696. 
[15] 
M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions, Math. Nachr., to appear. 
[16] 
J. Prüss, Maximal regularity for evolution equations in $L_p$spacess, Conf. Sem. Mat. Univ. Bari, 285 (2003), 139. 
[17] 
J. Prüss and S. Shimizu, Incompressible twophase flows with phase transition: Nonequal densities, submitted. 
[18] 
J. Prüss and G. Simonett, Maximal regularity for evolution equations in weighted $L_p$spaces, Archiv Math., 82 (2004), 415431. 
[19] 
J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675698. doi: 10.1137/070700632. 
[20] 
J. Prüss and G. Simonett, On the twophase NavierStokes equations with surface tension, Interfaces & Free Bound, 12 (2010), 311345. 
[21] 
J. Prüss and G. Simonett, Analytic solutions for the twophase NavierStokes equations with surface tension, Progr. Nonlin. Diff. Eqns. Appl., 80 (2011), 507540. 
[22] 
J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension, arXiv:1101.3763, submitted. 
[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. 
[24] 
N. Tanaka, Twophase free boundary problem for viscous incompressible thermocapillary convection, Japan J. Mech., 21 (1995), 141. 
[25] 
H. Triebel, "Theory of Function Spaces II," Birkhäuser Verlag, Basel, 1992. 
show all references
References:
[1] 
D. Bothe, J. Prüss, $L_p$theory for a class of nonNewtonian fluids, SIAM J. Math. Anal., 39 (2007), 379421. doi: 10.1137/060663635. 
[2] 
D. M. Anderson, P. Cermelli, E. Fried, M. E. Gurtin and G. B. McFadden, General dynamical sharpinterface conditions for phase transformations in viscous heatconducting fluids, J. Fluid Mech., 581 (2007), 323370. doi: 10.1017/S0022112007005587. 
[3] 
E. DiBenedetto and A. Friedman, Conductionconvection problems with change of phase, J. Differential Equations, 62 (1986), 129185. 
[4] 
E. DiBenedetto and M. O'Leary, Threedimensional conductionconvection problems with change of phase, Arch. Rational Mech. Anal., 123 (1993), 99116. doi: 10.1007/BF00695273. 
[5] 
R. Denk, M. Hieber and J. Prüss, "$\mathcal R$boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type," AMS Memoirs, 788, Providence, R.I., 2003. 
[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), 193224. doi: 10.1007/s0020900701209. 
[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), 173183. 
[8] 
K.H. Hoffmann and V. N. Starovoitov, Phase transitions of liquidliquid type with convection, Adv. Math. Sci. Appl., 8 (1998), 185198. 
[9] 
M. Ishii, "ThermoFluid Dynamics of TwoPhase Flow," Collection de la Direction des Etudes et Recherches d'Electricte de France, Paris, 1975. 
[10] 
M. Ishii and H. Takashi, "ThermoFluid Dynamics of TwoPhase Flow," Springer, New York, 2006. 
[11] 
M. Köhne, J. Prüss and M. Wilke, Qualitative behaviour of solutions for the twophase NavierStokes equations with surface tension, Math. Annalen, to appear. 
[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), 665683. 
[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), 584602 (electronic). doi: 10.1137/S0036141098334936. 
[14] 
Y. Kusaka and A. Tani, Classical solvability of the twophase Stefan problem in a viscous incompressible fluid flow, Math. Models Methods Appl. Sci., 12 (2002), 365391. doi: 10.1142/S0218202502001696. 
[15] 
M. Meyries and R. Schnaubelt, Maximal regularity with temporal weights for parabolic problems with inhomogeneous boundary conditions, Math. Nachr., to appear. 
[16] 
J. Prüss, Maximal regularity for evolution equations in $L_p$spacess, Conf. Sem. Mat. Univ. Bari, 285 (2003), 139. 
[17] 
J. Prüss and S. Shimizu, Incompressible twophase flows with phase transition: Nonequal densities, submitted. 
[18] 
J. Prüss and G. Simonett, Maximal regularity for evolution equations in weighted $L_p$spaces, Archiv Math., 82 (2004), 415431. 
[19] 
J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675698. doi: 10.1137/070700632. 
[20] 
J. Prüss and G. Simonett, On the twophase NavierStokes equations with surface tension, Interfaces & Free Bound, 12 (2010), 311345. 
[21] 
J. Prüss and G. Simonett, Analytic solutions for the twophase NavierStokes equations with surface tension, Progr. Nonlin. Diff. Eqns. Appl., 80 (2011), 507540. 
[22] 
J. Prüss, G. Simonett and R. Zacher, Qualitative behavior of solutions for thermodynamically consistent Stefan problems with surface tension, arXiv:1101.3763, submitted. 
[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. 
[24] 
N. Tanaka, Twophase free boundary problem for viscous incompressible thermocapillary convection, Japan J. Mech., 21 (1995), 141. 
[25] 
H. Triebel, "Theory of Function Spaces II," Birkhäuser Verlag, Basel, 1992. 
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