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April  2013, 6(2): 439-460. doi: 10.3934/dcdss.2013.6.439

Well-posedness of an extended model for water-ice phase transitions

Pavel Krejčí 1, and  Elisabetta Rocca 2,

1. 

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

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

Received  July 2011 Revised  December 2011 Published  November 2012

We propose an improved model explaining the occurrence of high stresses due to the difference in specific volumes during phase transitions between water and ice. The unknowns of the resulting evolution problem are the absolute temperature, the volume increment, and the liquid fraction. The main novelty here consists in including the dependence of the specific heat and of the speed of sound upon the phase. These additional nonlinearities bring new mathematical difficulties which require new estimation techniques based on Moser iteration. We establish the existence of a global solution to the corresponding initial-boundary value problem, as well as lower and upper bounds for the absolute temperature. Assuming constant heat conductivity, we also prove uniqueness and continuous data dependence of the solution.
Keywords: existence and uniqueness, Phase transitions, global bounds., nonlocal problems.
Mathematics Subject Classification: 82B26, 34B10, 74G25, 74G30, 74G4.
Citation: 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
References:
[1]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44(119) (1994), 109-140.  Google Scholar

[2]

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

[3]

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

[4]

P. Colli, P. Krejčí, E. Rocca and J. Sprekels, A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity, J. Differ. Equations, 251 (2011), 1354-1387.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

G. Joos, "Lehrbuch der Theoretischen Physik," Akademische Verlagsgesellschaft, Leipzig 1939 (In German). Google Scholar

[10]

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

[11]

P. Krejčí, E. Rocca and J. Sprekels, Phase separation in a gravity field, Discrete Contin. Dyn. Syst. Ser. S, 4, (2011), 391-407. doi: 10.3934/dcdss.2011.4.391.  Google Scholar

[12]

P. Krejčí, E. Rocca and J. Sprekels, Liquid-solid phase transitions in a deformable container, Contribution to the book "Continuous Media with Microstructure'' on the occasion of Krzysztof Wilmanski's 70th birthday, Springer, (2010), 285-300. Google Scholar

[13]

E. Madelung, "Die mathematischen Hilfsmittel des Physikers," Sixth Edition, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957. (In German).  Google Scholar

[14]

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

show all references

References:
[1]

C. Amrouche and V. Girault, Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J. 44(119) (1994), 109-140.  Google Scholar

[2]

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

[3]

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

[4]

P. Colli, P. Krejčí, E. Rocca and J. Sprekels, A nonlocal quasilinear multi-phase system with nonconstant specific heat and heat conductivity, J. Differ. Equations, 251 (2011), 1354-1387.  Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

G. Joos, "Lehrbuch der Theoretischen Physik," Akademische Verlagsgesellschaft, Leipzig 1939 (In German). Google Scholar

[10]

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

[11]

P. Krejčí, E. Rocca and J. Sprekels, Phase separation in a gravity field, Discrete Contin. Dyn. Syst. Ser. S, 4, (2011), 391-407. doi: 10.3934/dcdss.2011.4.391.  Google Scholar

[12]

P. Krejčí, E. Rocca and J. Sprekels, Liquid-solid phase transitions in a deformable container, Contribution to the book "Continuous Media with Microstructure'' on the occasion of Krzysztof Wilmanski's 70th birthday, Springer, (2010), 285-300. Google Scholar

[13]

E. Madelung, "Die mathematischen Hilfsmittel des Physikers," Sixth Edition, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1957. (In German).  Google Scholar

[14]

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

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