# American Institute of Mathematical Sciences

February  2012, 5(1): 29-48. doi: 10.3934/dcdss.2012.5.29

## Modelling phase transitions via Young measures

 1 Mathcces, Department of Mathematics RWTH Aachen University, Pauwelsstrasse 19, D-52074 Aachen, Germany

Received  April 2009 Revised  December 2009 Published  February 2011

We consider the elastic theory of single crystals at constant temperature where the free energy density depends on the local concentration of one or more species of particles in such a way that for a given local concentration vector certain lattice geometries (phases) are preferred. Furthermore we consider possible large deformations of the crystal lattice. After deriving the physical model, we indicate by means of a suitable implicite time discretization an existence result for measure-valued solutions that relies on a new existence theorem for Young measures in infinite settings. This article is an overview of [2].
Citation: Steffen Arnrich. Modelling phase transitions via Young measures. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 29-48. doi: 10.3934/dcdss.2012.5.29
##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Math. Monogr., Oxford University Press, New York, 2000. [2] S. Arnrich, "Maßwertige Lösungen für ein Gleichungssystem zur Beschreibung von Phasenübergängen in Kristallen," (German) [Measure valued solutions to a system of equations describing phase transitions in crystals], Ph.D thesis, University of Leipzig, 2007, http://www.mathcces.rwth-aachen.de/doku.php/staff/arnrich. [3] S. Arnrich, Lower semicontinuity of the surface energy functional-an alternative proof, DFG Priority Programme 1095 Analysis, Modelling and Simulation of Multiscale Problems, preprint 148, 2004. [4] K. Bente and Th. Doering, Solid state diffusion in sphalerites: an experimental verification of the chalcopyrite disease, European Journal of Mineralogy, 5 (1993), 465-478. [5] P. Blanchard and E. Brüning, "Variational Methods in Mathematical Physics. A Unified Approach," Springer, Berlin, 1992. [6] P. Blanchard and E. Brüning, "Mathematical Methods in Physics," Birkhäuser, Boston, 2003. [7] J. M. Borwein and A. S. Lewis, "Convex Analysis and Nonlinear Optimization. Theory and Examples," (CMS Books in Mathematics), Springer, New York, 2000. [8] M. Brokate and J.Sprekels, "Hysteresis and Phase Transitions," Springer, Berlin, 1996. [9] P. G. Ciarlet, "Mathematical Elasticity," North Holland, Amsterdam, 1988. [10] B. Dacorogna, "Direct Methods in the Calculus of Variations," Springer, Berlin, 1989. [11] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Property of Functions," CRC Press, Inc., London, 1992. [12] H. Garcke and T. Sturzenhecker, The degenerate multi-phase Stefan problem with Gibbs-Thomson law, Adv. Math. Sci. Appl., 8 (1998), 929-941. [13] M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations. I," Springer, Berlin, 1998. [14] H.O. Georgi, O. Häggström and C. Maess, The random geometry of equilibrium phases, in "Phase Transitions and Critical Phenomena, 18" (eds. C. Domb and J. L. Lebowitz), Academic Press, London, (2001), 1-142. arXiv:math/9905031v1 [15] S. R. de Groot and P. Mazur, "Non-Equillibrium Thermodynamics," Dover Publications, Inc., New York, 1984. [16] M. E. Gurtin, "An Introduction to Continuum Mechanics," Academic Press, INC., San Diego, California, 1981. [17] G. A. Holzapfel, "Nonlinear Solid Mechanics," Wiley, New York, 2000. [18] A. Khachaturyan, Theory of structural transformations in solids, Manuscripta Mathematica, 43 (1983), 261-288. [19] J. S. Kirkaldy and D.J. Young, "Diffusion in the Condensed State," London: The Institute of Metals, London, 1987. [20] D. Kondepudi and I. Prigogine, "Modern Thermodynamics," John Wiley & Sons Ltd, Cichester, England, 1998. [21] S. Luckhaus, Solidification of alloys and the Gibbs-Thomson law, Bonn: SFB 256, preprint 335, 1994. [22] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var., 3 (1995), 253-271. doi: 10.1007/BF01205007. [23] S. Müller, "Variational Models for Microstructure and Phase Transitions," Lecture notes no.: 2 des Max-Planck-Instituts für Mathematik in den Naturwissenschaften zu Leipzig, Leipzig, 1998, http://www.mis.mpg.de/de/publications/andere-reihen/ln/lecturenote-0298.html. [24] R. W. Ogden, "Non-linear Elastic Deformations," John Wiley & Sons, Inc., New York, 1984. [25] L. Onsager, Reciprocal relations in irreversible processes I, Phys. Rev., 37 (1931), 405-426. doi: 10.1103/PhysRev.37.405. [26] L. Onsager, Reciprocal relations in irreversible processes II, Phys. Rev., 38 (1931), 2265-2279. doi: 10.1103/PhysRev.38.2265. [27] P. Pedregal, Optimization, relaxatian and Young measures, Bulletin (New Series) of the American Mathematical Society, 36 (1999), 27-58. [28] M. Slemrod and V. Roytburd, Measure-valued solutions to a problem in dynamic phase transitions, in "Nonstrictly Hyperbolic Conservation Laws (Anaheim, Calif., 1985),'' Contemp. Math., 60, Amer. Math. Soc., Providence, RI, (1987), 115-124. [29] A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996. [30] S. Wang, R. Sekerka, A. Wheeler, B. Murray, C. Coriell, R. Braun and G. McFadden, Thermodynamically consistent phase field models for solid solidification, Physica D, 69 (1993), 189-200. doi: 10.1016/0167-2789(93)90189-8. [31] L. C. Young, Generalized curves and the existence of an attainment absolute minimum in the calculus of variations, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, classe III 30 (1937), 212-234. [32] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variations," Springer, New York, 1989.

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##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Math. Monogr., Oxford University Press, New York, 2000. [2] S. Arnrich, "Maßwertige Lösungen für ein Gleichungssystem zur Beschreibung von Phasenübergängen in Kristallen," (German) [Measure valued solutions to a system of equations describing phase transitions in crystals], Ph.D thesis, University of Leipzig, 2007, http://www.mathcces.rwth-aachen.de/doku.php/staff/arnrich. [3] S. Arnrich, Lower semicontinuity of the surface energy functional-an alternative proof, DFG Priority Programme 1095 Analysis, Modelling and Simulation of Multiscale Problems, preprint 148, 2004. [4] K. Bente and Th. Doering, Solid state diffusion in sphalerites: an experimental verification of the chalcopyrite disease, European Journal of Mineralogy, 5 (1993), 465-478. [5] P. Blanchard and E. Brüning, "Variational Methods in Mathematical Physics. A Unified Approach," Springer, Berlin, 1992. [6] P. Blanchard and E. Brüning, "Mathematical Methods in Physics," Birkhäuser, Boston, 2003. [7] J. M. Borwein and A. S. Lewis, "Convex Analysis and Nonlinear Optimization. Theory and Examples," (CMS Books in Mathematics), Springer, New York, 2000. [8] M. Brokate and J.Sprekels, "Hysteresis and Phase Transitions," Springer, Berlin, 1996. [9] P. G. Ciarlet, "Mathematical Elasticity," North Holland, Amsterdam, 1988. [10] B. Dacorogna, "Direct Methods in the Calculus of Variations," Springer, Berlin, 1989. [11] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Property of Functions," CRC Press, Inc., London, 1992. [12] H. Garcke and T. Sturzenhecker, The degenerate multi-phase Stefan problem with Gibbs-Thomson law, Adv. Math. Sci. Appl., 8 (1998), 929-941. [13] M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations. I," Springer, Berlin, 1998. [14] H.O. Georgi, O. Häggström and C. Maess, The random geometry of equilibrium phases, in "Phase Transitions and Critical Phenomena, 18" (eds. C. Domb and J. L. Lebowitz), Academic Press, London, (2001), 1-142. arXiv:math/9905031v1 [15] S. R. de Groot and P. Mazur, "Non-Equillibrium Thermodynamics," Dover Publications, Inc., New York, 1984. [16] M. E. Gurtin, "An Introduction to Continuum Mechanics," Academic Press, INC., San Diego, California, 1981. [17] G. A. Holzapfel, "Nonlinear Solid Mechanics," Wiley, New York, 2000. [18] A. Khachaturyan, Theory of structural transformations in solids, Manuscripta Mathematica, 43 (1983), 261-288. [19] J. S. Kirkaldy and D.J. Young, "Diffusion in the Condensed State," London: The Institute of Metals, London, 1987. [20] D. Kondepudi and I. Prigogine, "Modern Thermodynamics," John Wiley & Sons Ltd, Cichester, England, 1998. [21] S. Luckhaus, Solidification of alloys and the Gibbs-Thomson law, Bonn: SFB 256, preprint 335, 1994. [22] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation, Calc. Var., 3 (1995), 253-271. doi: 10.1007/BF01205007. [23] S. Müller, "Variational Models for Microstructure and Phase Transitions," Lecture notes no.: 2 des Max-Planck-Instituts für Mathematik in den Naturwissenschaften zu Leipzig, Leipzig, 1998, http://www.mis.mpg.de/de/publications/andere-reihen/ln/lecturenote-0298.html. [24] R. W. Ogden, "Non-linear Elastic Deformations," John Wiley & Sons, Inc., New York, 1984. [25] L. Onsager, Reciprocal relations in irreversible processes I, Phys. Rev., 37 (1931), 405-426. doi: 10.1103/PhysRev.37.405. [26] L. Onsager, Reciprocal relations in irreversible processes II, Phys. Rev., 38 (1931), 2265-2279. doi: 10.1103/PhysRev.38.2265. [27] P. Pedregal, Optimization, relaxatian and Young measures, Bulletin (New Series) of the American Mathematical Society, 36 (1999), 27-58. [28] M. Slemrod and V. Roytburd, Measure-valued solutions to a problem in dynamic phase transitions, in "Nonstrictly Hyperbolic Conservation Laws (Anaheim, Calif., 1985),'' Contemp. Math., 60, Amer. Math. Soc., Providence, RI, (1987), 115-124. [29] A. Visintin, "Models of Phase Transitions," Birkhäuser, Boston, 1996. [30] S. Wang, R. Sekerka, A. Wheeler, B. Murray, C. Coriell, R. Braun and G. McFadden, Thermodynamically consistent phase field models for solid solidification, Physica D, 69 (1993), 189-200. doi: 10.1016/0167-2789(93)90189-8. [31] L. C. Young, Generalized curves and the existence of an attainment absolute minimum in the calculus of variations, Comptes Rendus de la Société des Sciences et des Lettres de Varsovie, classe III 30 (1937), 212-234. [32] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variations," Springer, New York, 1989.
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