February  2012, 5(1): 127-146. doi: 10.3934/dcdss.2012.5.127

Global solvability of a model for grain boundary motion with constraint

1. 

Department of Electronic Engineering and Computer, Science School of Engineering, Kinki University, Takayaumenobe, Higashihiroshimashi, Hiroshima, 739-2116

2. 

Department of Education, School of Education, Bukkyo University, 96 Kitahananobo-cho, Murasakino, Kita-ku, Kyoto, 603-8301, Japan

3. 

Department of Mathematics, Faculty of Engineering, Kanagawa University, 3-27-1 Rokkakubashi, Kanagawa-ku, 221-8686, Japan

Received  June 2009 Revised  December 2009 Published  February 2011

We consider a model for grain boundary motion with constraint. In composite material science it is very important to investigate the grain boundary formation and its dynamics. In this paper we study a phase-filed model of grain boundaries, which is a modified version of the one proposed by R. Kobayashi, J.A. Warren and W.C. Carter [18]. The model is described as a system of a nonlinear parabolic partial differential equation and a nonlinear parabolic variational inequality. The main objective of this paper is to show the global existence of a solution for our model, employing some subdifferential techniques in the convex analysis.
Citation: Akio Ito, Nobuyuki Kenmochi, Noriaki Yamazaki. Global solvability of a model for grain boundary motion with constraint. Discrete and Continuous Dynamical Systems - S, 2012, 5 (1) : 127-146. doi: 10.3934/dcdss.2012.5.127
References:
[1]

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.

[2]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear equation: The parabolic case, Arch. Ration. Mech. Anal., 176 (2005), 415-453. doi: 10.1007/s00205-005-0358-5.

[3]

H. Attouch, "Variational Convergence for Functions and Operators," Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984.

[4]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei Republicii Socialiste Romania, Bucharest, Noordhoff International Publishing, Leiden, 1976.

[5]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.

[6]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973.

[7]

J. W. Cahn, P. Fife and O. Penrose, A phase-field model for diffusion-induced grain-boundary motion, Acta Mater., 45 (1997), 4397-4413. doi: 10.1016/S1359-6454(97)00074-8.

[8]

L. Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater Res., 32 (2002), 113-140. doi: 10.1146/annurev.matsci.32.112001.132041.

[9]

K. Deckelnick and C. M. Elliott, An existence and uniqueness result for a phase-field model of diffusion-induced grain-boundary motion, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1323-1344. doi: 10.1017/S0308210500001414.

[10]

M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, Proc. Taniguchi Conf. on Math., Advanced Studies in Pure Math., 31 (2001), 93-125.

[11]

M. E. Gurtin and M. T. Lusk, Sharp interface and phase-field theories of recrystallization in the plane, Phys. D, 130 (1999), 133-154. doi: 10.1016/S0167-2789(98)00323-6.

[12]

A. Ito, M. Gokieli, M. Niezgódka and M. Szpindler, Mathematical analysis of approximate system for one-dimensional grain boundary motion of Kobayashi-Warren-Carter type,, submitted., (). 

[13]

A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433-454. doi: 10.1007/s10492-008-0035-8.

[14]

A. Ito, N. Kenmochi and N. Yamazaki, Weak solutions of grain boundary motion model with singularity, Rend. Mat. Appl. (7), 29 (2009), 51-63.

[15]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.

[16]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in "Handbook of Differential Equations, Stationary Partial Differential Equations," (ed. M. Chipot), Vol. 4, North Holland, Amsterdam, (2007), 203-298.

[17]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372.

[18]

R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundaries, Phys. D, 140 (2000), 141-150. doi: 10.1016/S0167-2789(00)00023-3.

[19]

R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, in "Free boundary problems: Theory and applications, II (Chiba, 1999)," 283-294, GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakko-tosho, Tokyo, 2000.

[20]

A. E. Lobkovsky and J. A. Warren, Phase field model of premelting of grain boundaries, Phys. D, 164 (2002), 202-212.

[21]

M. T. Lusk, A phase field paradigm for grain growth and recrystallization, Proc. R. Soc. London A, 455 (1999), 677-700.

[22]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46 (1982), 268-299.

[23]

A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and their Applications, Vol. 28, Birkhäser, Boston, 1996.

show all references

References:
[1]

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.

[2]

F. Andreu, V. Caselles and J. M. Mazón, A strongly degenerate quasilinear equation: The parabolic case, Arch. Ration. Mech. Anal., 176 (2005), 415-453. doi: 10.1007/s00205-005-0358-5.

[3]

H. Attouch, "Variational Convergence for Functions and Operators," Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984.

[4]

V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces," Editura Academiei Republicii Socialiste Romania, Bucharest, Noordhoff International Publishing, Leiden, 1976.

[5]

G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN J. Differential Equations, 184 (2002), 475-525. doi: 10.1006/jdeq.2001.4150.

[6]

H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," North-Holland, Amsterdam, 1973.

[7]

J. W. Cahn, P. Fife and O. Penrose, A phase-field model for diffusion-induced grain-boundary motion, Acta Mater., 45 (1997), 4397-4413. doi: 10.1016/S1359-6454(97)00074-8.

[8]

L. Q. Chen, Phase-field models for microstructure evolution, Annu. Rev. Mater Res., 32 (2002), 113-140. doi: 10.1146/annurev.matsci.32.112001.132041.

[9]

K. Deckelnick and C. M. Elliott, An existence and uniqueness result for a phase-field model of diffusion-induced grain-boundary motion, Proc. Roy. Soc. Edinburgh Sect. A, 131 (2001), 1323-1344. doi: 10.1017/S0308210500001414.

[10]

M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations, Proc. Taniguchi Conf. on Math., Advanced Studies in Pure Math., 31 (2001), 93-125.

[11]

M. E. Gurtin and M. T. Lusk, Sharp interface and phase-field theories of recrystallization in the plane, Phys. D, 130 (1999), 133-154. doi: 10.1016/S0167-2789(98)00323-6.

[12]

A. Ito, M. Gokieli, M. Niezgódka and M. Szpindler, Mathematical analysis of approximate system for one-dimensional grain boundary motion of Kobayashi-Warren-Carter type,, submitted., (). 

[13]

A. Ito, N. Kenmochi and N. Yamazaki, A phase-field model of grain boundary motion, Appl. Math., 53 (2008), 433-454. doi: 10.1007/s10492-008-0035-8.

[14]

A. Ito, N. Kenmochi and N. Yamazaki, Weak solutions of grain boundary motion model with singularity, Rend. Mat. Appl. (7), 29 (2009), 51-63.

[15]

N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87.

[16]

N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities, in "Handbook of Differential Equations, Stationary Partial Differential Equations," (ed. M. Chipot), Vol. 4, North Holland, Amsterdam, (2007), 203-298.

[17]

R. Kobayashi and Y. Giga, Equations with singular diffusivity, J. Statist. Phys., 95 (1999), 1187-1220. doi: 10.1023/A:1004570921372.

[18]

R. Kobayashi, J. A. Warren and W. C. Carter, A continuum model of grain boundaries, Phys. D, 140 (2000), 141-150. doi: 10.1016/S0167-2789(00)00023-3.

[19]

R. Kobayashi, J. A. Warren and W. C. Carter, Grain boundary model and singular diffusivity, in "Free boundary problems: Theory and applications, II (Chiba, 1999)," 283-294, GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakko-tosho, Tokyo, 2000.

[20]

A. E. Lobkovsky and J. A. Warren, Phase field model of premelting of grain boundaries, Phys. D, 164 (2002), 202-212.

[21]

M. T. Lusk, A phase field paradigm for grain growth and recrystallization, Proc. R. Soc. London A, 455 (1999), 677-700.

[22]

M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems, J. Differential Equations, 46 (1982), 268-299.

[23]

A. Visintin, "Models of Phase Transitions," Progress in Nonlinear Differential Equations and their Applications, Vol. 28, Birkhäser, Boston, 1996.

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