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April  2011, 4(2): 391-407. doi: 10.3934/dcdss.2011.4.391

Phase separation in a gravity field

Pavel Krejčí 1, , Elisabetta Rocca 2, and  Jürgen Sprekels 3,

1. 

Institute of Mathematics, Czech Academy of Sciences, Žitná 25, CZ-11567 Praha 1, Czech Republic
2. 

WIAS Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany, Dipartimento di Matematica, Università di Milano, Via Saldini 50, 20133 Milano
3. 

Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D–10117 Berlin

Received  May 2009 Revised  August 2009 Published  November 2010

We prove here well-posedness and convergence to equilibria for the solution trajectories associated with a model for solidification of the liquid content of a rigid container in a gravity field. We observe that the gravity effects, which can be neglected without considerable changes of the process on finite time intervals, have a substantial influence on the long time behavior of the evolution system. Without gravity, we find a temperature interval, in which all phase distributions with a prescribed total liquid content are admissible equilibria, while, under the influence of gravity, the only equilibrium distribution in a connected container consists in two pure phases separated by one plane interface perpendicular to the gravity force.
Keywords: existence and uniqueness, Phase separation, long-time behavior of solutions..
Mathematics Subject Classification: Primary: 80A22; Secondary: 74C05, 35K5.
Citation: Pavel Krejčí, Elisabetta Rocca, Jürgen Sprekels. Phase separation in a gravity field. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 391-407. doi: 10.3934/dcdss.2011.4.391
References:
[1]

M. Brokate, J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer, New York, 1996.  Google Scholar

[2]

P. Colli, M. Frémond and A Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math., 48 (1990), 31-47.  Google Scholar

[3]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81. doi: 10.3934/dcds.2009.25.63.  Google Scholar

[4]

M. Frémond, "Non-Smooth Thermo-Mechanics," Springer-Verlag, Berlin, 2002.  Google Scholar

[5]

M. Frémond and E. Rocca, Well-posedness of a phase transition model with the possibility of voids, Math. Models Methods Appl. Sci., 16 (2006), 559-586. doi: 10.1142/S0218202506001261.  Google Scholar

[6]

M. Frémond and E. Rocca, Solid liquid phase changes with different densities, Q. Appl. Math., 66 (2008), 609-632.  Google Scholar

[7]

V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations," Springer-Verlag, Berlin, 1986.  Google Scholar

[8]

P. Krejčí, Hysteresis operators-A new approach to evolution differential inequalities, Comment. Math. Univ. Carolinae, 33 (1989), 525-536.  Google Scholar

[9]

P. Krejčí, E. Rocca and J. Sprekels, A bottle in a freezer, SIAM J. Math. Anal., 41 (2009), 1851-1873. doi: 10.1137/09075086X.  Google Scholar

[10]

P. Krejčí, Elastoplastic reaction of a container to water freezing, Mathematica Bohemica, to appear (2010). Google Scholar

[11]

P. Krejčí, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermoviscoplasticity, SIAM J. Math. Anal., 34 (2002), 409-434. doi: 10.1137/S0036141001387604.  Google Scholar

[12]

P. Krejčí, J. Sprekels and U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Adv. Math. Sci. Appl., 13 (2003), 695-712.  Google Scholar

[13]

E. Rocca and R. Rossi, A nonlinear degenerating PDE system modelling phase transitions in thermoviscoelastic materials, J. Differential Equations, 245 (2008), 3327-3375. doi: 10.1016/j.jde.2008.02.006.  Google Scholar

[14]

E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math., 53 (2008), 485-520. doi: 10.1007/s10492-008-0038-5.  Google Scholar

[15]

A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser Boston, 1996.  Google Scholar

[16]

H. Y. Erbil, "Surface Chemistry of Solid and Liquid Interfaces," Blackwell Publishing, John Wiley & Sons, 2006. Google Scholar

show all references

References:
[1]

M. Brokate, J. Sprekels, "Hysteresis and Phase Transitions," Appl. Math. Sci., 121, Springer, New York, 1996.  Google Scholar

[2]

P. Colli, M. Frémond and A Visintin, Thermo-mechanical evolution of shape memory alloys, Quart. Appl. Math., 48 (1990), 31-47.  Google Scholar

[3]

P. Colli, D. Hilhorst, F. Issard-Roch and G. Schimperna, Long time convergence for a class of variational phase field models, Discrete Contin. Dyn. Syst., 25 (2009), 63-81. doi: 10.3934/dcds.2009.25.63.  Google Scholar

[4]

M. Frémond, "Non-Smooth Thermo-Mechanics," Springer-Verlag, Berlin, 2002.  Google Scholar

[5]

M. Frémond and E. Rocca, Well-posedness of a phase transition model with the possibility of voids, Math. Models Methods Appl. Sci., 16 (2006), 559-586. doi: 10.1142/S0218202506001261.  Google Scholar

[6]

M. Frémond and E. Rocca, Solid liquid phase changes with different densities, Q. Appl. Math., 66 (2008), 609-632.  Google Scholar

[7]

V. Girault and P.-A. Raviart, "Finite Element Methods for Navier-Stokes Equations," Springer-Verlag, Berlin, 1986.  Google Scholar

[8]

P. Krejčí, Hysteresis operators-A new approach to evolution differential inequalities, Comment. Math. Univ. Carolinae, 33 (1989), 525-536.  Google Scholar

[9]

P. Krejčí, E. Rocca and J. Sprekels, A bottle in a freezer, SIAM J. Math. Anal., 41 (2009), 1851-1873. doi: 10.1137/09075086X.  Google Scholar

[10]

P. Krejčí, Elastoplastic reaction of a container to water freezing, Mathematica Bohemica, to appear (2010). Google Scholar

[11]

P. Krejčí, J. Sprekels and U. Stefanelli, Phase-field models with hysteresis in one-dimensional thermoviscoplasticity, SIAM J. Math. Anal., 34 (2002), 409-434. doi: 10.1137/S0036141001387604.  Google Scholar

[12]

P. Krejčí, J. Sprekels and U. Stefanelli, One-dimensional thermo-visco-plastic processes with hysteresis and phase transitions, Adv. Math. Sci. Appl., 13 (2003), 695-712.  Google Scholar

[13]

E. Rocca and R. Rossi, A nonlinear degenerating PDE system modelling phase transitions in thermoviscoelastic materials, J. Differential Equations, 245 (2008), 3327-3375. doi: 10.1016/j.jde.2008.02.006.  Google Scholar

[14]

E. Rocca and R. Rossi, Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials, Appl. Math., 53 (2008), 485-520. doi: 10.1007/s10492-008-0038-5.  Google Scholar

[15]

A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser Boston, 1996.  Google Scholar

[16]

H. Y. Erbil, "Surface Chemistry of Solid and Liquid Interfaces," Blackwell Publishing, John Wiley & Sons, 2006. Google Scholar

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