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

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

Ken Shirakawa 1,

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.
Keywords: Stability analysis, steady-state pattern., crystalline-type setting.
Mathematics Subject Classification: Primary: 35B35, 74A50; Secondary: 82B2.
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), 91-133.  Google Scholar

[3]

F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.  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), 347-403. 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), 109-152. 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, Amsterdam, 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), 803-832. 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, CRC Press, Inc., Boca Raton, 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), 109-147. doi: 10.1007/s12220-007-9004-9.  Google Scholar

[11]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser, 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), 233-253. doi: 10.1007/BF03167521.  Google Scholar

[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.  Google Scholar

[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. 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), 177-218. 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), 1-17. doi: 10.1142/S0218202505000571.  Google Scholar

[18]

A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. 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 "Nonlinear Partial Differential Equations and Their Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkōtosho, (2004), 289-304.  Google Scholar

[20]

K. Shirakawa, Large-time behavior for a phase field system associated with total variation energy, Adv. Math. Sci. Appl., 15 (2005), 1-27.  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), 1215-1236. doi: 10.3934/dcds.2006.15.1215.  Google Scholar

[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.  Google Scholar

[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.  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), 257-282.  Google Scholar

[25]

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]

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), 91-133.  Google Scholar

[3]

F. Andreu, C. Ballester, V. Caselles and J. M. Mazón, Minimizing total variation flow, Differential Integral Equations, 14 (2001), 321-360.  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), 347-403. 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), 109-152. 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, Amsterdam, 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), 803-832. 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, CRC Press, Inc., Boca Raton, 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), 109-147. doi: 10.1007/s12220-007-9004-9.  Google Scholar

[11]

E. Giusti, "Minimal Surfaces and Functions of Bounded Variation," Monographs in Mathematics, 80, Birkhäuser, 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), 233-253. doi: 10.1007/BF03167521.  Google Scholar

[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.  Google Scholar

[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. 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), 177-218. 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), 1-17. doi: 10.1142/S0218202505000571.  Google Scholar

[18]

A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc., 48 (1942), 883-890. 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 "Nonlinear Partial Differential Equations and Their Applications," GAKUTO Internat. Ser. Math. Sci. Appl., 20, Gakkōtosho, (2004), 289-304.  Google Scholar

[20]

K. Shirakawa, Large-time behavior for a phase field system associated with total variation energy, Adv. Math. Sci. Appl., 15 (2005), 1-27.  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), 1215-1236. doi: 10.3934/dcds.2006.15.1215.  Google Scholar

[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.  Google Scholar

[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.  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), 257-282.  Google Scholar

[25]

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

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