American Institute of Mathematical Sciences

Advanced
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

Ken Shirakawa 1, and  Hiroshi Watanabe 2,

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.
Keywords: PDE model, weighted total variation, Isothermal grain boundary motion, large-time behavior., subdiff erential.
Mathematics Subject Classification: Primary: 35B40, 35K87; Secondary: 35K6.
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), no. 1, 91-133.

[2]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications, (2000).

[3]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, MPS-SIAM Series on Optimization, SIAM and MPS (2001).

[4]

G. Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA (1993).

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton (1992).

[6]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser (1984).

[7]

R. Kobayashi, Modeling of grain structure evolution, Variational Problems and Related Topics, RIMS Kôkyûroku, 1210 (2001), 68-77.

[8]

J. S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi-Warren-Carter system, Calc. Var. Partial Differential Equations, 51 (2014), no. 3-4, 621-656.

[9]

J. S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the Kobayashi-Warren-Carter system, In preparation.

[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), no. 1, 139-159.

[11]

K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion, Math. Ann., 356 (2013), 301-330.

[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, RIMS KôKyûroku, 1892 (2014), 52-72.

[13]

K. Shirakawa, H. Watanabe and N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications, Adv. Math. Sci. Appl. (to appear).

[14]

J. Simon, Compact sets in the space $ L^p(0,T;B) $, Ann. Mat. Pura Appl. (4), 146, 65-96 (1987).

[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), 301-328.

[16]

H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system, Math. Bohem., 139 (2014), no. 2, 381-389.

show all references

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), no. 1, 91-133.

[2]

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications, (2000).

[3]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, MPS-SIAM Series on Optimization, SIAM and MPS (2001).

[4]

G. Dal Maso, An introduction to $\Gamma$-convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA (1993).

[5]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Inc., Boca Raton (1992).

[6]

E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser (1984).

[7]

R. Kobayashi, Modeling of grain structure evolution, Variational Problems and Related Topics, RIMS Kôkyûroku, 1210 (2001), 68-77.

[8]

J. S. Moll and K. Shirakawa, Existence of solutions to the Kobayashi-Warren-Carter system, Calc. Var. Partial Differential Equations, 51 (2014), no. 3-4, 621-656.

[9]

J. S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the Kobayashi-Warren-Carter system, In preparation.

[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), no. 1, 139-159.

[11]

K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion, Math. Ann., 356 (2013), 301-330.

[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, RIMS KôKyûroku, 1892 (2014), 52-72.

[13]

K. Shirakawa, H. Watanabe and N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications, Adv. Math. Sci. Appl. (to appear).

[14]

J. Simon, Compact sets in the space $ L^p(0,T;B) $, Ann. Mat. Pura Appl. (4), 146, 65-96 (1987).

[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), 301-328.

[16]

H. Watanabe and K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system, Math. Bohem., 139 (2014), no. 2, 381-389.

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