American Institute of Mathematical Sciences

Advanced
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

Jan Prüss 1, , Yoshihiro Shibata 2, , Senjo Shimizu 3, and  Gieri Simonett 4,

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.
Keywords: phase transitions, time weights., entropy, surface tension, Two-phase Navier-Stokes equations, well-posedness.
Mathematics Subject Classification: Primary: 35R35, Secondary: 35Q30, 76D45, 76T05, 80A.
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), 379-421. 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), 323-370. doi: 10.1017/S0022112007005587.  Google Scholar

[3]

E. DiBenedetto and A. Friedman, Conduction-convection problems with change of phase, J. Differential Equations, 62 (1986), 129-185.  Google Scholar

[4]

E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems with change of phase, Arch. Rational Mech. Anal., 123 (1993), 99-116. 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, Providence, R.I., 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), 193-224. 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), 173-183.  Google Scholar

[8]

K.-H. Hoffmann and V. N. Starovoitov, Phase transitions of liquid-liquid type with convection, Adv. Math. Sci. Appl., 8 (1998), 185-198.  Google Scholar

[9]

M. Ishii, "Thermo-Fluid Dynamics of Two-Phase Flow," Collection de la Direction des Etudes et Recherches d'Electricte de France, Paris, 1975.  Google Scholar

[10]

M. Ishii and H. Takashi, "Thermo-Fluid Dynamics of Two-Phase Flow," Springer, New York, 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), 665-683.  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), 584-602 (electronic). 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), 365-391. 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), 1-39.  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), 415-431.  Google Scholar

[19]

J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675-698. 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), 311-345.  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), 507-540.  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), 1-41.  Google Scholar

[25]

H. Triebel, "Theory of Function Spaces II," Birkhäuser Verlag, Basel, 1992.  Google Scholar

show all references

References:
[1]

D. Bothe, J. Prüss, $L_p$-theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421. 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), 323-370. doi: 10.1017/S0022112007005587.  Google Scholar

[3]

E. DiBenedetto and A. Friedman, Conduction-convection problems with change of phase, J. Differential Equations, 62 (1986), 129-185.  Google Scholar

[4]

E. DiBenedetto and M. O'Leary, Three-dimensional conduction-convection problems with change of phase, Arch. Rational Mech. Anal., 123 (1993), 99-116. 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, Providence, R.I., 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), 193-224. 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), 173-183.  Google Scholar

[8]

K.-H. Hoffmann and V. N. Starovoitov, Phase transitions of liquid-liquid type with convection, Adv. Math. Sci. Appl., 8 (1998), 185-198.  Google Scholar

[9]

M. Ishii, "Thermo-Fluid Dynamics of Two-Phase Flow," Collection de la Direction des Etudes et Recherches d'Electricte de France, Paris, 1975.  Google Scholar

[10]

M. Ishii and H. Takashi, "Thermo-Fluid Dynamics of Two-Phase Flow," Springer, New York, 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), 665-683.  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), 584-602 (electronic). 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), 365-391. 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), 1-39.  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), 415-431.  Google Scholar

[19]

J. Prüss and G. Simonett, Stability of equilibria for the Stefan problem with surface tension, SIAM J. Math. Anal., 40 (2008), 675-698. 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), 311-345.  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), 507-540.  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), 1-41.  Google Scholar

[25]

H. Triebel, "Theory of Function Spaces II," Birkhäuser Verlag, Basel, 1992.  Google Scholar

[1]

Changyan Li, Hui Li. Well-posedness of the two-phase flow problem in incompressible MHD. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5609-5632. doi: 10.3934/dcds.2021090
[2]

José Luiz Boldrini, Luís H. de Miranda, Gabriela Planas. On singular Navier-Stokes equations and irreversible phase transitions. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2055-2078. doi: 10.3934/cpaa.2012.11.2055
[3]

Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315
[4]

Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
[5]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
[6]

Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143
[7]

K. Domelevo. Well-posedness of a kinetic model of dispersed two-phase flow with point-particles and stability of travelling waves. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 591-607. doi: 10.3934/dcdsb.2002.2.591
[8]

Pavel Krejčí, Elisabetta Rocca. Well-posedness of an extended model for water-ice phase transitions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 439-460. doi: 10.3934/dcdss.2013.6.439
[9]

Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121
[10]

Haibo Cui, Qunyi Bie, Zheng-An Yao. Well-posedness in critical spaces for a multi-dimensional compressible viscous liquid-gas two-phase flow model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1395-1410. doi: 10.3934/dcdsb.2018156
[11]

Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[12]

Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142
[13]

Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
[14]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[15]

Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158
[16]

Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845
[17]

Giovanna Bonfanti, Fabio Luterotti. A well-posedness result for irreversible phase transitions with a nonlinear heat flux law. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 331-351. doi: 10.3934/dcdss.2013.6.331
[18]

Jingjing Zhang, Ting Zhang. Local well-posedness of perturbed Navier-Stokes system around Landau solutions. Electronic Research Archive, 2021, 29 (4) : 2719-2739. doi: 10.3934/era.2021010
[19]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437
[20]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (11) : 3973-3987. doi: 10.3934/dcdss.2020467

2020 Impact Factor: 1.081

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (18)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy