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Largetime behavior for a PDE model of isothermal grain boundary motion with a constraint
1.  Department of Mathematics, Faculty of Education, Chiba University, 133 Yayoicho, Inageku, Chiba, 2638522 
2.  Department of General Education, Salesian Polytechnic, 468 Oyamagaoka, Machidacity, Tokyo, 1940215 
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, 91133. Google Scholar 
[2] 
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publications, (2000). Google Scholar 
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H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, Applications to PDEs and Optimization, MPSSIAM Series on Optimization, SIAM and MPS (2001). Google Scholar 
[4] 
G. Dal Maso, An introduction to $\Gamma$convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA (1993). Google Scholar 
[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). Google Scholar 
[6] 
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser (1984). Google Scholar 
[7] 
R. Kobayashi, Modeling of grain structure evolution, Variational Problems and Related Topics, RIMS Kôkyûroku, 1210 (2001), 6877. Google Scholar 
[8] 
J. S. Moll and K. Shirakawa, Existence of solutions to the KobayashiWarrenCarter system, Calc. Var. Partial Differential Equations, 51 (2014), no. 34, 621656. Google Scholar 
[9] 
J. S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the KobayashiWarrenCarter system,, In preparation., (). Google Scholar 
[10] 
Ken Shirakawa and H. Watanabe, Energydissipative solution to a onedimensional phase field model of grain boundary motion, Discrete Conin. Dyn. Syst. Ser. S, 7 (2014), no. 1, 139159. Google Scholar 
[11] 
K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of onedimensional phase field systems associated with grain boundary motion, Math. Ann., 356 (2013), 301330. Google Scholar 
[12] 
K. Shirakawa, H. Watanabe and N. Yamazaki, Existence for a PDEmodel of a grain boundary motion involving solidification effect, New Role of the Theory of Abstract Evolution Equations, RIMS KôKyûroku, 1892 (2014), 5272. Google Scholar 
[13] 
K. Shirakawa, H. Watanabe and N. Yamazaki, Phasefield 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, 6596 (1987). Google Scholar 
[15] 
H. Watanabe and K. Shirakawa, Qualitative properties of a onedimensional phasefield system associated with grain boundary, GAKUTO Internat. Ser. Math. Sci. Appl., 36 (2013), 301328. Google Scholar 
[16] 
H. Watanabe and K. Shirakawa, Stability for approximation methods of the onedimensional KobayashiWarrenCarter system, Math. Bohem., 139 (2014), no. 2, 381389. Google Scholar 
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, 91133. 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, MPSSIAM Series on Optimization, SIAM and MPS (2001). Google Scholar 
[4] 
G. Dal Maso, An introduction to $\Gamma$convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA (1993). Google Scholar 
[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). Google Scholar 
[6] 
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser (1984). Google Scholar 
[7] 
R. Kobayashi, Modeling of grain structure evolution, Variational Problems and Related Topics, RIMS Kôkyûroku, 1210 (2001), 6877. Google Scholar 
[8] 
J. S. Moll and K. Shirakawa, Existence of solutions to the KobayashiWarrenCarter system, Calc. Var. Partial Differential Equations, 51 (2014), no. 34, 621656. Google Scholar 
[9] 
J. S. Moll, K. Shirakawa and H. Watanabe, Energy dissipative solutions to the KobayashiWarrenCarter system,, In preparation., (). Google Scholar 
[10] 
Ken Shirakawa and H. Watanabe, Energydissipative solution to a onedimensional phase field model of grain boundary motion, Discrete Conin. Dyn. Syst. Ser. S, 7 (2014), no. 1, 139159. Google Scholar 
[11] 
K. Shirakawa, H. Watanabe and N. Yamazaki, Solvability of onedimensional phase field systems associated with grain boundary motion, Math. Ann., 356 (2013), 301330. Google Scholar 
[12] 
K. Shirakawa, H. Watanabe and N. Yamazaki, Existence for a PDEmodel of a grain boundary motion involving solidification effect, New Role of the Theory of Abstract Evolution Equations, RIMS KôKyûroku, 1892 (2014), 5272. Google Scholar 
[13] 
K. Shirakawa, H. Watanabe and N. Yamazaki, Phasefield 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, 6596 (1987). Google Scholar 
[15] 
H. Watanabe and K. Shirakawa, Qualitative properties of a onedimensional phasefield system associated with grain boundary, GAKUTO Internat. Ser. Math. Sci. Appl., 36 (2013), 301328. Google Scholar 
[16] 
H. Watanabe and K. Shirakawa, Stability for approximation methods of the onedimensional KobayashiWarrenCarter system, Math. Bohem., 139 (2014), no. 2, 381389. Google Scholar 
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