# American Institute of Mathematical Sciences

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 & 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. 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. doi: 10.1007/s00205-005-0358-5. [3] H. Attouch, "Variational Convergence for Functions and Operators,", Pitman Advanced Publishing Program, (1984). [4] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste Romania, (1976). [5] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN, J. Differential Equations, 184 (2002), 475. doi: 10.1006/jdeq.2001.4150. [6] H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (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. 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. 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. doi: 10.1017/S0308210500001414. [10] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations,, Proc. Taniguchi Conf. on Math., 31 (2001), 93. [11] M. E. Gurtin and M. T. Lusk, Sharp interface and phase-field theories of recrystallization in the plane,, Phys. D, 130 (1999), 133. 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. 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. [15] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. [16] N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, in, 4 (2007), 203. [17] R. Kobayashi and Y. Giga, Equations with singular diffusivity,, J. Statist. Phys., 95 (1999), 1187. 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. 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, 14 (1999), 283. [20] A. E. Lobkovsky and J. A. Warren, Phase field model of premelting of grain boundaries,, Phys. D, 164 (2002), 202. [21] M. T. Lusk, A phase field paradigm for grain growth and recrystallization,, Proc. R. Soc. London A, 455 (1999), 677. [22] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems,, J. Differential Equations, 46 (1982), 268. [23] A. Visintin, "Models of Phase Transitions,", Progress in Nonlinear Differential Equations and their Applications, 28 (1996).

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##### 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. 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. doi: 10.1007/s00205-005-0358-5. [3] H. Attouch, "Variational Convergence for Functions and Operators,", Pitman Advanced Publishing Program, (1984). [4] V. Barbu, "Nonlinear Semigroups and Differential Equations in Banach Spaces,", Editura Academiei Republicii Socialiste Romania, (1976). [5] G. Bellettini, V. Caselles and M. Novaga, The total variation flow in RN, J. Differential Equations, 184 (2002), 475. doi: 10.1006/jdeq.2001.4150. [6] H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert,", North-Holland, (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. 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. 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. doi: 10.1017/S0308210500001414. [10] M.-H. Giga, Y. Giga and R. Kobayashi, Very singular diffusion equations,, Proc. Taniguchi Conf. on Math., 31 (2001), 93. [11] M. E. Gurtin and M. T. Lusk, Sharp interface and phase-field theories of recrystallization in the plane,, Phys. D, 130 (1999), 133. 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. 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. [15] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications,, Bull. Fac. Education, 30 (1981), 1. [16] N. Kenmochi, Monotonicity and compactness methods for nonlinear variational inequalities,, in, 4 (2007), 203. [17] R. Kobayashi and Y. Giga, Equations with singular diffusivity,, J. Statist. Phys., 95 (1999), 1187. 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. 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, 14 (1999), 283. [20] A. E. Lobkovsky and J. A. Warren, Phase field model of premelting of grain boundaries,, Phys. D, 164 (2002), 202. [21] M. T. Lusk, A phase field paradigm for grain growth and recrystallization,, Proc. R. Soc. London A, 455 (1999), 677. [22] M. Ôtani, Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems,, J. Differential Equations, 46 (1982), 268. [23] A. Visintin, "Models of Phase Transitions,", Progress in Nonlinear Differential Equations and their Applications, 28 (1996).
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