# 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 and 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. [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), 91-133. [3] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360. [4] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), 347-403. doi: 10.1006/jfan.2000.3698. [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff International Publishing, 1976. [6] G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Rat. Mech. Anal., 179 (2006), 109-152. doi: 10.1007/s00205-005-0387-0. [7] H. Brézis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espace de Hilbert," North-Holland, Amsterdam, 1973. [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), 803-832. doi: 10.1016/j.anihpc.2008.04.003. [9] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions", Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992. [10] Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane, J. Geom. Anal., 18 (2008), 109-147. doi: 10.1007/s12220-007-9004-9. [11] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser, 1984. [12] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233-253. doi: 10.1007/BF03167521. [13] N. Kenmochi, Systems of nonlinear PDEs arising from dynamical phase transitions, in "Phase Transitions and Hysteresis (Montecatini Terme, 1993)," Lecture Notes in Math., 1584, Springer, Berlin, (1994), 39-86. [14] N. Kenmochi, Y. Mizuta and T. Nagai, Projections onto convex sets, convex functions and their subdifferentials, Bull. Fac. Edu., Chiba Univ., 29 (1980), 11-22. [15] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications Vol. I," Springer-Verlag, 1972. [16] J. S. Moll, The anisotropic total variation flow, Math. Annalen., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0. [17] M. Novaga and E. Paolini, Stability of crystalline evolutions, Math. Mod. Meth. Appl. Sci., 15 (2005), 1-17. doi: 10.1142/S0218202505000571. [18] A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6. [19] K. Shirakawa, Stability for steady-state solutions of a nonisothermal Allen-Cahn equation generated by a total variation energy, in "Nonlinear Partial Differential Equations and Their Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkōtosho, (2004), 289-304. [20] K. Shirakawa, Large-time behavior for a phase field system associated with total variation energy, Adv. Math. Sci. Appl., 15 (2005), 1-27. [21] K. Shirakawa, Stability for steady-state patterns in phase field dynamics associated with total variation energies, Discrete Contin. Dyn. Syst., 15 (2006), 1215-1236. doi: 10.3934/dcds.2006.15.1215. [22] K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions," Series on Advances in Mathematics for Applied Sciences, 71, World Scientific Publishing, (2006), 269-288. [23] K. Shirakawa, Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl., 697-707. [24] K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257-282. [25] A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser, Boston, 1996.

show all references

##### References:
 [1] L. Ambrosio, N. Fusco and D. Pallara, "Functions of Bounded Variation and Free Discontinuity Problems," Oxford Science Publications, 2000. [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), 91-133. [3] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360. [4] F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, The dirichlet problem for the total variation flow, J. Funct. Anal., 180 (2001), 347-403. doi: 10.1006/jfan.2000.3698. [5] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Noordhoff International Publishing, 1976. [6] G. Bellettini, V. Caselles, A. Chambolle and M. Novaga, Crystalline mean curvature flow of convex sets, Arch. Rat. Mech. Anal., 179 (2006), 109-152. doi: 10.1007/s00205-005-0387-0. [7] H. Brézis, "Opérateurs Maximaux Monotones et Semigroupes de Contractions Dans les Espace de Hilbert," North-Holland, Amsterdam, 1973. [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), 803-832. doi: 10.1016/j.anihpc.2008.04.003. [9] L. C. Evans and R. F. Gariepy, "Measure Theory and Fine Properties of Functions", Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton, 1992. [10] Y. Giga and P. Rybka, Facet bending in the driven crystalline curvature flow in the plane, J. Geom. Anal., 18 (2008), 109-147. doi: 10.1007/s12220-007-9004-9. [11] E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser, 1984. [12] T. Ishiwata, Motion of non-convex polygons by crystalline curvature and almost convexity phenomena, Japan J. Indust. Appl. Math., 25 (2008), 233-253. doi: 10.1007/BF03167521. [13] N. Kenmochi, Systems of nonlinear PDEs arising from dynamical phase transitions, in "Phase Transitions and Hysteresis (Montecatini Terme, 1993)," Lecture Notes in Math., 1584, Springer, Berlin, (1994), 39-86. [14] N. Kenmochi, Y. Mizuta and T. Nagai, Projections onto convex sets, convex functions and their subdifferentials, Bull. Fac. Edu., Chiba Univ., 29 (1980), 11-22. [15] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Value Problems and Applications Vol. I," Springer-Verlag, 1972. [16] J. S. Moll, The anisotropic total variation flow, Math. Annalen., 332 (2005), 177-218. doi: 10.1007/s00208-004-0624-0. [17] M. Novaga and E. Paolini, Stability of crystalline evolutions, Math. Mod. Meth. Appl. Sci., 15 (2005), 1-17. doi: 10.1142/S0218202505000571. [18] A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. doi: 10.1090/S0002-9904-1942-07811-6. [19] K. Shirakawa, Stability for steady-state solutions of a nonisothermal Allen-Cahn equation generated by a total variation energy, in "Nonlinear Partial Differential Equations and Their Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkōtosho, (2004), 289-304. [20] K. Shirakawa, Large-time behavior for a phase field system associated with total variation energy, Adv. Math. Sci. Appl., 15 (2005), 1-27. [21] K. Shirakawa, Stability for steady-state patterns in phase field dynamics associated with total variation energies, Discrete Contin. Dyn. Syst., 15 (2006), 1215-1236. doi: 10.3934/dcds.2006.15.1215. [22] K. Shirakawa, Stability for phase field systems involving indefinite surface tension coefficients, in "Dissipative Phase Transitions," Series on Advances in Mathematics for Applied Sciences, 71, World Scientific Publishing, (2006), 269-288. [23] K. Shirakawa, Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems, Differential Equations and Applications. 7th AIMS Conference, suppl., 697-707. [24] K. Shirakawa and M. Kimura, Stability analysis for Allen-Cahn type equation associated with the total variation energy, Nonlinear Anal., 60 (2005), 257-282. [25] A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and Their Applications, 28, Birkhäuser, Boston, 1996.
 [1] Mei-hua Wei, Jianhua Wu, Yinnian He. Steady-state solutions and stability for a cubic autocatalysis model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1147-1167. doi: 10.3934/cpaa.2015.14.1147 [2] Zhenzhen Zheng, Ching-Shan Chou, Tau-Mu Yi, Qing Nie. Mathematical analysis of steady-state solutions in compartment and continuum models of cell polarization. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1135-1168. doi: 10.3934/mbe.2011.8.1135 [3] Ikuo Arizono, Yasuhiko Takemoto. Statistical mechanics approach for steady-state analysis in M/M/s queueing system with balking. Journal of Industrial and Management Optimization, 2022, 18 (1) : 25-44. doi: 10.3934/jimo.2020141 [4] Theodore Kolokolnikov, Michael J. Ward, Juncheng Wei. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete and Continuous Dynamical Systems - B, 2014, 19 (5) : 1373-1410. doi: 10.3934/dcdsb.2014.19.1373 [5] Ken Shirakawa. Stability for steady-state patterns in phase field dynamics associated with total variation energies. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1215-1236. doi: 10.3934/dcds.2006.15.1215 [6] Shihe Xu, Fangwei Zhang, Meng Bai. Stability of positive steady-state solutions to a time-delayed system with some applications. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021286 [7] Federica Di Michele, Bruno Rubino, Rosella Sampalmieri. A steady-state mathematical model for an EOS capacitor: The effect of the size exclusion. Networks and Heterogeneous Media, 2016, 11 (4) : 603-625. doi: 10.3934/nhm.2016011 [8] Ken Shirakawa. Stability analysis for two dimensional Allen-Cahn equations associated with crystalline type energies. Conference Publications, 2009, 2009 (Special) : 697-707. doi: 10.3934/proc.2009.2009.697 [9] Tomoyuki Miyaji, Yoshio Tsutsumi. Steady-state mode interactions of radially symmetric modes for the Lugiato-Lefever equation on a disk. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1633-1650. doi: 10.3934/cpaa.2018078 [10] Wing-Cheong Lo. Morphogen gradient with expansion-repression mechanism: Steady-state and robustness studies. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 775-787. doi: 10.3934/dcdsb.2014.19.775 [11] Li Ma, Youquan Luo. Dynamics of positive steady-state solutions of a nonlocal dispersal logistic model with nonlocal terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2555-2582. doi: 10.3934/dcdsb.2020022 [12] Yacine Chitour, Jean-Michel Coron, Mauro Garavello. On conditions that prevent steady-state controllability of certain linear partial differential equations. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 643-672. doi: 10.3934/dcds.2006.14.643 [13] Thomas Lepoutre, Salomé Martínez. Steady state analysis for a relaxed cross diffusion model. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 613-633. doi: 10.3934/dcds.2014.34.613 [14] Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691 [15] Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045 [16] Shin-Yi Lee, Shin-Hwa Wang, Chiou-Ping Ye. Explicit necessary and sufficient conditions for the existence of a dead core solution of a p-laplacian steady-state reaction-diffusion problem. Conference Publications, 2005, 2005 (Special) : 587-596. doi: 10.3934/proc.2005.2005.587 [17] Christos Sourdis. Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 1039-1059. doi: 10.3934/dcds.2017043 [18] La-Su Mai, Kaijun Zhang. Asymptotic stability of steady state solutions for the relativistic Euler-Poisson equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 981-1004. doi: 10.3934/dcds.2016.36.981 [19] Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $MAP/M/s+G$ queueing model with generally distributed patience times. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021078 [20] Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial and Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811

2020 Impact Factor: 2.425