American Institute of Mathematical Sciences

April  2011, 4(2): 483-504. doi: 10.3934/dcdss.2011.4.483

Stability analysis for phase field systems associated with crystalline-type energies

 1 Department of Mathematics, Department of Mathematics Faculty of Education, Chiba University, 1-33 Yayoichō, Inage, Chiba, 263-8522, Japan

Received  February 2009 Revised  October 2009 Published  November 2010

In this paper, a mathematical model, to represent the dynamics of two-dimensional solid-liquid phase transition, is considered. This mathematical model is formulated as a coupled system of a heat equation with a time-relaxation diffusion, and an Allen-Cahn equation such that the two-dimensional norm, of crystalline-type, is adopted as the mathematical expression of the anisotropy. Through the structural observations for steady-state solutions, some geometric conditions to guarantee their stability will be presented in the main theorem of this paper.
Citation: Ken Shirakawa. Stability analysis for phase field systems associated with crystalline-type energies. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 483-504. doi: 10.3934/dcdss.2011.4.483
References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Science Publications, (2000).   Google Scholar [2] M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91.   Google Scholar [3] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow,, Differential Integral Equations, 14 (2001), 321.   Google Scholar [4] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The dirichlet problem for the total variation flow,, J. Funct. Anal., 180 (2001), 347.  doi: 10.1006/jfan.2000.3698.  Google Scholar [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff International Publishing, (1976).   Google Scholar [6] G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets,, Arch. Rat. Mech. Anal., 179 (2006), 109.  doi: 10.1007/s00205-005-0387-0.  Google Scholar [7] H. Brézis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espace de Hilbert,", North-Holland, (1973).   Google Scholar [8] V. Caselles, A. Chambolle, S. Moll and M. Novaga, A characterization of convex calibrable sets in $\mathbbR^N$ with respect to anisotropic norms,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 803.  doi: 10.1016/j.anihpc.2008.04.003.  Google Scholar [9] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions",, Studies in Advanced Mathematics, (1992).   Google Scholar [10] Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane,, J. Geom. Anal., 18 (2008), 109.  doi: 10.1007/s12220-007-9004-9.  Google Scholar [11] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monographs in Mathematics, 80 (1984).   Google Scholar [12] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena,, Japan J. Indust. Appl. Math., 25 (2008), 233.  doi: 10.1007/BF03167521.  Google Scholar [13] N. Kenmochi, Systems of nonlinear PDEs arising from dynamical phase transitions,, in, 1584 (1994), 39.   Google Scholar [14] N. Kenmochi, Y. Mizuta and T. Nagai, Projections onto convex sets, convex functions and their subdifferentials,, Bull. Fac. Edu., 29 (1980), 11.   Google Scholar [15] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications Vol. I,", Springer-Verlag, (1972).   Google Scholar [16] J. S. Moll, The anisotropic total variation flow,, Math. Annalen., 332 (2005), 177.  doi: 10.1007/s00208-004-0624-0.  Google Scholar [17] M. Novaga and E. Paolini, Stability of crystalline evolutions,, Math. Mod. Meth. Appl. Sci., 15 (2005), 1.  doi: 10.1142/S0218202505000571.  Google Scholar [18] A. Sard, The measure of the critical values of differentiable maps,, Bull. Amer. Math. Soc., 48 (1942), 883.  doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar [19] K. Shirakawa, Stability for steady-state solutions of a nonisothermal Allen-Cahn equation generated by a total variation energy,, in, 20 (2004), 289.   Google Scholar [20] K. Shirakawa, Large-time behavior for a phase field system associated with total variation energy,, Adv. Math. Sci. Appl., 15 (2005), 1.   Google Scholar [21] K. Shirakawa, Stability for steady-state patterns in phase field dynamics associated with total variation energies,, Discrete Contin. Dyn. Syst., 15 (2006), 1215.  doi: 10.3934/dcds.2006.15.1215.  Google Scholar [22] K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients,, in, 71 (2006), 269.   Google Scholar [23] K. Shirakawa, Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies,, Discrete Contin. Dyn. Syst. 2009, (2009), 697.   Google Scholar [24] K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy,, Nonlinear Anal., 60 (2005), 257.   Google Scholar [25] A. Visintin, "Models of Phase Transitions,", Progress in Nonlinear Differential Equations and Their Applications, 28 (1996).   Google Scholar

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References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems,", Oxford Science Publications, (2000).   Google Scholar [2] M. Amar and G. Bellettini, A notion of total variation depending on a metric with discontinuous coefficients,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 91.   Google Scholar [3] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow,, Differential Integral Equations, 14 (2001), 321.   Google Scholar [4] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The dirichlet problem for the total variation flow,, J. Funct. Anal., 180 (2001), 347.  doi: 10.1006/jfan.2000.3698.  Google Scholar [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Noordhoff International Publishing, (1976).   Google Scholar [6] G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets,, Arch. Rat. Mech. Anal., 179 (2006), 109.  doi: 10.1007/s00205-005-0387-0.  Google Scholar [7] H. Brézis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espace de Hilbert,", North-Holland, (1973).   Google Scholar [8] V. Caselles, A. Chambolle, S. Moll and M. Novaga, A characterization of convex calibrable sets in $\mathbbR^N$ with respect to anisotropic norms,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 803.  doi: 10.1016/j.anihpc.2008.04.003.  Google Scholar [9] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions",, Studies in Advanced Mathematics, (1992).   Google Scholar [10] Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane,, J. Geom. Anal., 18 (2008), 109.  doi: 10.1007/s12220-007-9004-9.  Google Scholar [11] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation,", Monographs in Mathematics, 80 (1984).   Google Scholar [12] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena,, Japan J. Indust. Appl. Math., 25 (2008), 233.  doi: 10.1007/BF03167521.  Google Scholar [13] N. Kenmochi, Systems of nonlinear PDEs arising from dynamical phase transitions,, in, 1584 (1994), 39.   Google Scholar [14] N. Kenmochi, Y. Mizuta and T. Nagai, Projections onto convex sets, convex functions and their subdifferentials,, Bull. Fac. Edu., 29 (1980), 11.   Google Scholar [15] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications Vol. I,", Springer-Verlag, (1972).   Google Scholar [16] J. S. Moll, The anisotropic total variation flow,, Math. Annalen., 332 (2005), 177.  doi: 10.1007/s00208-004-0624-0.  Google Scholar [17] M. Novaga and E. Paolini, Stability of crystalline evolutions,, Math. Mod. Meth. Appl. Sci., 15 (2005), 1.  doi: 10.1142/S0218202505000571.  Google Scholar [18] A. Sard, The measure of the critical values of differentiable maps,, Bull. Amer. Math. Soc., 48 (1942), 883.  doi: 10.1090/S0002-9904-1942-07811-6.  Google Scholar [19] K. Shirakawa, Stability for steady-state solutions of a nonisothermal Allen-Cahn equation generated by a total variation energy,, in, 20 (2004), 289.   Google Scholar [20] K. Shirakawa, Large-time behavior for a phase field system associated with total variation energy,, Adv. Math. Sci. Appl., 15 (2005), 1.   Google Scholar [21] K. Shirakawa, Stability for steady-state patterns in phase field dynamics associated with total variation energies,, Discrete Contin. Dyn. Syst., 15 (2006), 1215.  doi: 10.3934/dcds.2006.15.1215.  Google Scholar [22] K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients,, in, 71 (2006), 269.   Google Scholar [23] K. Shirakawa, Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies,, Discrete Contin. Dyn. Syst. 2009, (2009), 697.   Google Scholar [24] K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy,, Nonlinear Anal., 60 (2005), 257.   Google Scholar [25] A. Visintin, "Models of Phase Transitions,", Progress in Nonlinear Differential Equations and Their Applications, 28 (1996).   Google Scholar
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