# 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 & 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., (2000). Google Scholar [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, (2007). Google Scholar [3] S. Arnrich, Lower semicontinuity of the surface energy functional-an alternative proof,, DFG Priority Programme 1095 Analysis, 148 (1095). Google Scholar [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. Google Scholar [5] P. Blanchard and E. Brüning, "Variational Methods in Mathematical Physics. A Unified Approach,", Springer, (1992). Google Scholar [6] P. Blanchard and E. Brüning, "Mathematical Methods in Physics,", Birkh\, (2003). Google Scholar [7] J. M. Borwein and A. S. Lewis, "Convex Analysis and Nonlinear Optimization. Theory and Examples,", (CMS Books in Mathematics), (2000). Google Scholar [8] M. Brokate and J.Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). Google Scholar [9] P. G. Ciarlet, "Mathematical Elasticity,", North Holland, (1988). Google Scholar [10] B. Dacorogna, "Direct Methods in the Calculus of Variations,", Springer, (1989). Google Scholar [11] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Property of Functions,", CRC Press, (1992). Google Scholar [12] H. Garcke and T. Sturzenhecker, The degenerate multi-phase Stefan problem with Gibbs-Thomson law,, Adv. Math. Sci. Appl., 8 (1998), 929. Google Scholar [13] M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations. I,", Springer, (1998). Google Scholar [14] H.O. Georgi, O. Häggström and C. Maess, The random geometry of equilibrium phases,, in, 18 (2001), 1. Google Scholar [15] S. R. de Groot and P. Mazur, "Non-Equillibrium Thermodynamics,", Dover Publications, (1984). Google Scholar [16] M. E. Gurtin, "An Introduction to Continuum Mechanics,", Academic Press, (1981). Google Scholar [17] G. A. Holzapfel, "Nonlinear Solid Mechanics,", Wiley, (2000). Google Scholar [18] A. Khachaturyan, Theory of structural transformations in solids,, Manuscripta Mathematica, 43 (1983), 261. Google Scholar [19] J. S. Kirkaldy and D.J. Young, "Diffusion in the Condensed State,", London: The Institute of Metals, (1987). Google Scholar [20] D. Kondepudi and I. Prigogine, "Modern Thermodynamics,", John Wiley & Sons Ltd, (1998). Google Scholar [21] S. Luckhaus, Solidification of alloys and the Gibbs-Thomson law,, Bonn: SFB 256, 335 (1994). Google Scholar [22] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation,, Calc. Var., 3 (1995), 253. doi: 10.1007/BF01205007. Google Scholar [23] S. Müller, "Variational Models for Microstructure and Phase Transitions,", Lecture notes no.: 2 des Max-Planck-Instituts f\, (1998). Google Scholar [24] R. W. Ogden, "Non-linear Elastic Deformations,", John Wiley \verb|&| Sons, (1984). Google Scholar [25] L. Onsager, Reciprocal relations in irreversible processes I,, Phys. Rev., 37 (1931), 405. doi: 10.1103/PhysRev.37.405. Google Scholar [26] L. Onsager, Reciprocal relations in irreversible processes II,, Phys. Rev., 38 (1931), 2265. doi: 10.1103/PhysRev.38.2265. Google Scholar [27] P. Pedregal, Optimization, relaxatian and Young measures,, Bulletin (New Series) of the American Mathematical Society, 36 (1999), 27. Google Scholar [28] M. Slemrod and V. Roytburd, Measure-valued solutions to a problem in dynamic phase transitions,, in, 60 (1987), 115. Google Scholar [29] A. Visintin, "Models of Phase Transitions,", Birkh\, (1996). Google Scholar [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. doi: 10.1016/0167-2789(93)90189-8. Google Scholar [31] L. C. Young, Generalized curves and the existence of an attainment absolute minimum in the calculus of variations,, Comptes Rendus de la Soci$\rm {\acute e}$t$\rm {\acute e}$ des Sciences et des Lettres de Varsovie, 30 (1937), 212. Google Scholar [32] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variations,", Springer, (1989). Google Scholar

show all references

##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Math. Monogr., (2000). Google Scholar [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, (2007). Google Scholar [3] S. Arnrich, Lower semicontinuity of the surface energy functional-an alternative proof,, DFG Priority Programme 1095 Analysis, 148 (1095). Google Scholar [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. Google Scholar [5] P. Blanchard and E. Brüning, "Variational Methods in Mathematical Physics. A Unified Approach,", Springer, (1992). Google Scholar [6] P. Blanchard and E. Brüning, "Mathematical Methods in Physics,", Birkh\, (2003). Google Scholar [7] J. M. Borwein and A. S. Lewis, "Convex Analysis and Nonlinear Optimization. Theory and Examples,", (CMS Books in Mathematics), (2000). Google Scholar [8] M. Brokate and J.Sprekels, "Hysteresis and Phase Transitions,", Springer, (1996). Google Scholar [9] P. G. Ciarlet, "Mathematical Elasticity,", North Holland, (1988). Google Scholar [10] B. Dacorogna, "Direct Methods in the Calculus of Variations,", Springer, (1989). Google Scholar [11] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Property of Functions,", CRC Press, (1992). Google Scholar [12] H. Garcke and T. Sturzenhecker, The degenerate multi-phase Stefan problem with Gibbs-Thomson law,, Adv. Math. Sci. Appl., 8 (1998), 929. Google Scholar [13] M. Giaquinta, G. Modica and J. Soucek, "Cartesian Currents in the Calculus of Variations. I,", Springer, (1998). Google Scholar [14] H.O. Georgi, O. Häggström and C. Maess, The random geometry of equilibrium phases,, in, 18 (2001), 1. Google Scholar [15] S. R. de Groot and P. Mazur, "Non-Equillibrium Thermodynamics,", Dover Publications, (1984). Google Scholar [16] M. E. Gurtin, "An Introduction to Continuum Mechanics,", Academic Press, (1981). Google Scholar [17] G. A. Holzapfel, "Nonlinear Solid Mechanics,", Wiley, (2000). Google Scholar [18] A. Khachaturyan, Theory of structural transformations in solids,, Manuscripta Mathematica, 43 (1983), 261. Google Scholar [19] J. S. Kirkaldy and D.J. Young, "Diffusion in the Condensed State,", London: The Institute of Metals, (1987). Google Scholar [20] D. Kondepudi and I. Prigogine, "Modern Thermodynamics,", John Wiley & Sons Ltd, (1998). Google Scholar [21] S. Luckhaus, Solidification of alloys and the Gibbs-Thomson law,, Bonn: SFB 256, 335 (1994). Google Scholar [22] S. Luckhaus and T. Sturzenhecker, Implicit time discretization for the mean curvature flow equation,, Calc. Var., 3 (1995), 253. doi: 10.1007/BF01205007. Google Scholar [23] S. Müller, "Variational Models for Microstructure and Phase Transitions,", Lecture notes no.: 2 des Max-Planck-Instituts f\, (1998). Google Scholar [24] R. W. Ogden, "Non-linear Elastic Deformations,", John Wiley \verb|&| Sons, (1984). Google Scholar [25] L. Onsager, Reciprocal relations in irreversible processes I,, Phys. Rev., 37 (1931), 405. doi: 10.1103/PhysRev.37.405. Google Scholar [26] L. Onsager, Reciprocal relations in irreversible processes II,, Phys. Rev., 38 (1931), 2265. doi: 10.1103/PhysRev.38.2265. Google Scholar [27] P. Pedregal, Optimization, relaxatian and Young measures,, Bulletin (New Series) of the American Mathematical Society, 36 (1999), 27. Google Scholar [28] M. Slemrod and V. Roytburd, Measure-valued solutions to a problem in dynamic phase transitions,, in, 60 (1987), 115. Google Scholar [29] A. Visintin, "Models of Phase Transitions,", Birkh\, (1996). Google Scholar [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. doi: 10.1016/0167-2789(93)90189-8. Google Scholar [31] L. C. Young, Generalized curves and the existence of an attainment absolute minimum in the calculus of variations,, Comptes Rendus de la Soci$\rm {\acute e}$t$\rm {\acute e}$ des Sciences et des Lettres de Varsovie, 30 (1937), 212. Google Scholar [32] W. P. Ziemer, "Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variations,", Springer, (1989). Google Scholar
 [1] Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407 [2] Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012 [3] Armin Lechleiter, Marcel Rennoch. Non-linear Tikhonov regularization in Banach spaces for inverse scattering from anisotropic penetrable media. Inverse Problems & Imaging, 2017, 11 (1) : 151-176. doi: 10.3934/ipi.2017008 [4] Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu, Kok Lay Teo. A numerical approach to infinite-dimensional linear programming in $L_1$ spaces. Journal of Industrial & Management Optimization, 2010, 6 (1) : 15-28. doi: 10.3934/jimo.2010.6.15 [5] Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations & Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207 [6] Der-Chen Chang, Jie Xiao. $L^q$-Extensions of $L^p$-spaces by fractional diffusion equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1905-1920. doi: 10.3934/dcds.2015.35.1905 [7] Kurt Falk, Marc Kesseböhmer, Tobias Henrik Oertel-Jäger, Jens D. M. Rademacher, Tony Samuel. Preface: Diffusion on fractals and non-linear dynamics. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : ⅰ-ⅳ. doi: 10.3934/dcdss.201702i [8] Samir Adly, Ba Khiet Le. Unbounded state-dependent sweeping processes with perturbations in uniformly convex and q-uniformly smooth Banach spaces. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 81-95. doi: 10.3934/naco.2018005 [9] Ulisse Stefanelli. Analysis of a variable time-step discretization for a phase transition model with micro-movements. Communications on Pure & Applied Analysis, 2006, 5 (3) : 659-673. doi: 10.3934/cpaa.2006.5.659 [10] G. Dal Maso, Antonio DeSimone, M. G. Mora, M. Morini. Time-dependent systems of generalized Young measures. Networks & Heterogeneous Media, 2007, 2 (1) : 1-36. doi: 10.3934/nhm.2007.2.1 [11] W. Cary Huffman. On the theory of $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes. Advances in Mathematics of Communications, 2013, 7 (3) : 349-378. doi: 10.3934/amc.2013.7.349 [12] Christoph Walker. Age-dependent equations with non-linear diffusion. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 691-712. doi: 10.3934/dcds.2010.26.691 [13] Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281 [14] J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731 [15] Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure & Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587 [16] A. Procacci, Benedetto Scoppola. Convergent expansions for random cluster model with $q>0$ on infinite graphs. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1145-1178. doi: 10.3934/cpaa.2008.7.1145 [17] Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473 [18] W. Cary Huffman. Self-dual $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes with an automorphism of prime order. Advances in Mathematics of Communications, 2013, 7 (1) : 57-90. doi: 10.3934/amc.2013.7.57 [19] Adrien Blanchet, Philippe Laurençot. Finite mass self-similar blowing-up solutions of a chemotaxis system with non-linear diffusion. Communications on Pure & Applied Analysis, 2012, 11 (1) : 47-60. doi: 10.3934/cpaa.2012.11.47 [20] Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517

2018 Impact Factor: 0.545