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

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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

Abstract
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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 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 $\mathbb{R}^N2$ 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 $\mathbb{R}^N2$ 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.

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