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

Phase separation in a gravity field

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.
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
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show all references

References:
[1]

Appl. Math. Sci., 121, Springer, New York, 1996.  Google Scholar

[2]

Quart. Appl. Math., 48 (1990), 31-47.  Google Scholar

[3]

Discrete Contin. Dyn. Syst., 25 (2009), 63-81. doi: 10.3934/dcds.2009.25.63.  Google Scholar

[4]

Springer-Verlag, Berlin, 2002.  Google Scholar

[5]

Math. Models Methods Appl. Sci., 16 (2006), 559-586. doi: 10.1142/S0218202506001261.  Google Scholar

[6]

Q. Appl. Math., 66 (2008), 609-632.  Google Scholar

[7]

Springer-Verlag, Berlin, 1986.  Google Scholar

[8]

Comment. Math. Univ. Carolinae, 33 (1989), 525-536.  Google Scholar

[9]

SIAM J. Math. Anal., 41 (2009), 1851-1873. doi: 10.1137/09075086X.  Google Scholar

[10]

Mathematica Bohemica, to appear (2010). Google Scholar

[11]

SIAM J. Math. Anal., 34 (2002), 409-434. doi: 10.1137/S0036141001387604.  Google Scholar

[12]

Adv. Math. Sci. Appl., 13 (2003), 695-712.  Google Scholar

[13]

J. Differential Equations, 245 (2008), 3327-3375. doi: 10.1016/j.jde.2008.02.006.  Google Scholar

[14]

Appl. Math., 53 (2008), 485-520. doi: 10.1007/s10492-008-0038-5.  Google Scholar

[15]

Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser Boston, 1996.  Google Scholar

[16]

Blackwell Publishing, John Wiley & Sons, 2006. Google Scholar

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