# American Institute of Mathematical Sciences

2015, 2015(special): 1009-1018. doi: 10.3934/proc.2015.1009

## Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint

 1 Department of Mathematics, Faculty of Education, Chiba University, 1-33 Yayoi-cho, Inage-ku, Chiba, 263-8522 2 Department of General Education, Salesian Polytechnic, 4-6-8 Oyamagaoka, Machida-city, Tokyo, 194-0215

Received  September 2014 Revised  January 2015 Published  November 2015

In this paper, a system of parabolic initial-boundary value problems is considered as a possible PDE model of isothermal grain boundary motion. The solvability of this system was proved in [preprint, arXiv:1408.4204., by means of the notion of weighted total variation. In this light, we set our goal to prove two main theorems, which are concerned with the $\Gamma$-convergence for time-dependent versions of the weighted total variations, and the large-time behavior of solution.
Citation: Ken Shirakawa, Hiroshi Watanabe. Large-time behavior for a PDE model of isothermal grain boundary motion with a constraint. Conference Publications, 2015, 2015 (special) : 1009-1018. doi: 10.3934/proc.2015.1009
##### References:
 [1] 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. Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford Science Publications, (2000). Google Scholar [3] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces,, Applications to PDEs and Optimization, (2001). Google Scholar [4] G. Dal Maso, An introduction to $\Gamma$-convergence,, Progress in Nonlinear Differential Equations and their Applications, 8 (1993). Google Scholar [5] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [6] E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monographs in Mathematics, 80 (1984). Google Scholar [7] R. Kobayashi, Modeling of grain structure evolution,, Variational Problems and Related Topics, 1210 (2001), 68. Google Scholar [8] J. S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi-Warren-Carter system,, Calc. Var. Partial Differential Equations, 51 (2014), 3. Google Scholar [9] J. S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the Kobayashi-Warren-Carter system,, In preparation., (). Google Scholar [10] Ken Shirakawa and H. Watanabe, Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion,, Discrete Conin. Dyn. Syst. Ser. S, 7 (2014), 139. Google Scholar [11] K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion,, Math. Ann., 356 (2013), 301. Google Scholar [12] K. Shirakawa, H. Watanabe and N. Yamazaki, Existence for a PDE-model of a grain boundary motion involving solidification effect,, New Role of the Theory of Abstract Evolution Equations, 1892 (2014), 52. Google Scholar [13] K. Shirakawa, H. Watanabe and N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications,, Adv. Math. Sci. Appl. (to appear)., (). Google Scholar [14] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. Google Scholar [15] H. Watanabe and K. Shirakawa, Qualitative properties of a one-dimensional phase-field system associated with grain boundary,, GAKUTO Internat. Ser. Math. Sci. Appl., 36 (2013), 301. Google Scholar [16] H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system,, Math. Bohem., 139 (2014), 381. Google Scholar

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##### References:
 [1] 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. Google Scholar [2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems,, Oxford Science Publications, (2000). Google Scholar [3] H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces,, Applications to PDEs and Optimization, (2001). Google Scholar [4] G. Dal Maso, An introduction to $\Gamma$-convergence,, Progress in Nonlinear Differential Equations and their Applications, 8 (1993). Google Scholar [5] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992). Google Scholar [6] E. Giusti, Minimal Surfaces and Functions of Bounded Variation,, Monographs in Mathematics, 80 (1984). Google Scholar [7] R. Kobayashi, Modeling of grain structure evolution,, Variational Problems and Related Topics, 1210 (2001), 68. Google Scholar [8] J. S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi-Warren-Carter system,, Calc. Var. Partial Differential Equations, 51 (2014), 3. Google Scholar [9] J. S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the Kobayashi-Warren-Carter system,, In preparation., (). Google Scholar [10] Ken Shirakawa and H. Watanabe, Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion,, Discrete Conin. Dyn. Syst. Ser. S, 7 (2014), 139. Google Scholar [11] K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion,, Math. Ann., 356 (2013), 301. Google Scholar [12] K. Shirakawa, H. Watanabe and N. Yamazaki, Existence for a PDE-model of a grain boundary motion involving solidification effect,, New Role of the Theory of Abstract Evolution Equations, 1892 (2014), 52. Google Scholar [13] K. Shirakawa, H. Watanabe and N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications,, Adv. Math. Sci. Appl. (to appear)., (). Google Scholar [14] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65. Google Scholar [15] H. Watanabe and K. Shirakawa, Qualitative properties of a one-dimensional phase-field system associated with grain boundary,, GAKUTO Internat. Ser. Math. Sci. Appl., 36 (2013), 301. Google Scholar [16] H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system,, Math. Bohem., 139 (2014), 381. Google Scholar
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