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AIMS Mathematics, 2017, 2(1): 161-194. doi: 10.3934/Math.2017.1.161. Research article Special Issue

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Solvability for the non-isothermal Kobayashi–Warren–Carter system

1 Department of Mathematics, Faculty of Education, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba, 263-8522, Japan
2 Department of Computer Science and Intelligent Systems, Faculty of Engineering, Oita University, 700 Dannoharu, Oita, 870-1192, Japan

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Special Issue: Nonlinear Evolution PDEs, Interfaces and Applications

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In this paper, a system of parabolic type initial-boundary value problems are considered. The system (S)$_\nu$ is based on the non-isothermal model of grain boundary motion by ^[38], which was derived as an extending version of the ``Kobayashi--Warren--Carter model'' of grain boundary motion by ^[23]. Under suitable assumptions, the existence theorem of $ L^2 $-based solutions is concluded, as a versatile mathematical theory to analyze various Kobayashi--Warren--Carter type models.

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Keywords: Non-isothermal grain boundary motion; Kobayashi–Warren–Carter type model; existence of L2-based solution; weighted total variation; time-discretization

Citation: Ken Shirakawa, HiroshiWatanabe. Solvability for the non-isothermal Kobayashi–Warren–Carter system. AIMS Mathematics, 2017, 2(1): 161-194. doi: 10.3934/Math.2017.1.161

References:

1. M. Amar, G. Bellettini, A notion of total variation depending on a metric with discontinuous coeffcients. Ann. Inst. H. Poincaré Anal. Non Linéaire,11 (1994), no. 1, 91-133.
2. L. Ambrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. (2000).
3. H. Attouch, G. Buttazzo, G. Michaille, Variational Analysis in Sobolev and BV Spaces. Applications to PDEs and Optimization. MPS-SIAM Series on Optimization, 6. SIAM and MPS, (2006).
4. V. Barbu, Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Republicii Socialiste România, Noordhoff International Publishing, (1976).
5. G. Bellettini, G. Bouchitté, I. Fragalà, BV functions with respect to a measure and relaxation of metric integral functionals. J. Convex Anal., 6 (1999), no. 2, 349-366.
6. H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert. North-Holland Mathematics Studies,5. Notas de Matemática (50). North-Holland Publishing and American Elsevier Publishing, (1973).
7. P. Colli, P. Laurençot, Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws. Phys. D, 111 (1998), 311-334.
8. P. Colli, J. Sprekels, Glob al solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier law. Adv. Math. Sci. Appl., 9 (1999), no. 1, 383-391.
9. G. Dal Maso, An Introduction to Γ-convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, Ma, (1993).
10. I. Ekeland, R. Temam, Convex analysis and variational problems. Translated from the French. Corrected reprint of the 1976 English edition. Classics in Applied Mathematics, 28. SIAM, Philadelphia, (1999).
11. L. C. Evans, R. F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, (1992).
12. E. Giusti, Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, 80. Birkhäuser, (1984).
13. M.-H. Giga, Y. Giga, Very singular diffusion equations: second and fourth order problems. Jpn. J. Ind. Appl. Math., 27 (2010), no. 3, 323-345.
14. W. Horn, J. Sprekels, S. Zheng, Global existence of smooth solutions to the Penrose-Fife model for Ising ferromagnets. Adv. Math. Sci. Appl., 6 (1996), no. 1, 227-241.
15. A. Ito, N. Kenmochi, N. Yamazaki, A phase-field model of grain boundary motion. Appl. Math., 53 (2008), no. 5, 433-454.
16. A. Ito, N. Kenmochi, N. Yamazaki, Weak solutions of grain boundary motion model with singularity. Rend. Mat. Appl. (7), 29 (2009), no. 1, 51-63.
17. A. Ito, N.Kenmochi, N. Yamazaki, Global solvability of a model for grain boundary motion with constraint. Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), no. 1, 127-146.
18. N. Kenmochi, : Systems of nonlinear PDEs arising from dynamical phase transitions. In: Phase transitions and hysteresis (Montecatini Terme, 1993), pp. 39-86, Lecture Notes in Math., 1584, Springer, Berlin, (1994).
19. N. Kenmochi, M. Kubo, Weak solutions of nonlinear systems for non-isothermal phase transitions. Adv. Math. Sci. Appl., 9 (1999), no. 1, 499-521.
20. N. Kenmochi, N. Yamazaki, Large-time behavior of solutions to a phase-field model of grain boundary motion with constraint. In: Current advances in nonlinear analysis and related topics, pp. 389-403, GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkōtosho, Tokyo, (2010).
21. R. Kobayashi, Y. Giga, Equations with singular diffusivity. J. Statist. Phys., 95 (1999), 1187-1220.
22. R. Kobayashi, J. A. Warren, W. C. Carter, A continuum model of grain boundary. Phys. D, 140 (2000), no. 1-2, 141-150.
23. R. Kobayashi, J. A.Warren, W. C. Carter, Grain boundary model and singular diffusivity. In: Free Boundary Problems: Theory and Applications, pp. 283-294, GAKUTO Internat. Ser. Math. Sci. Appl., 14, Gakkōtosho, Tokyo, (2000).
24. J. L. Lions, E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol I. Springer-Verlag, New York-Heidelberg, (1972).
25. S. Moll, K. Shirakawa, Existence of solutions to the Kobayashi-Warren-Carter system. Calc. Var. Partial Differential Equations, 51 (2014), 621-656. DOI:10.1007/ s00526-013-0689-2
26. S. Moll, K. Shirakawa, H.Watanabe, Energy dissipative solutions to the Kobayashi-Warren-Carter system. submitted.
27. U. Mosco, Convergence of convex sets and of solutions of variational inequalities. Advances in Math., 3 (1969), 510-585.
28. K. Shirakawa, H. Watanabe, Energy-dissipative solution to a one-dimensional phase field model of grain boundary motion. Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), no. 1, 139-159. DOI:10.3934/dcdss.2014.7.139
29. K. Shirakawa, H. Watanabe, Large-time behavior of a PDE model of isothermal grain boundary motion with a constraint. Discrete Contin. Dyn. Syst. 2015, Dynamical systems, differential equations and applications. 10th AIMS Conference. Suppl., 1009-1018.
30. K. Shirakawa, H. Watanabe, N. Yamazaki, Solvability of one-dimensional phase field systems associated with grain boundary motion. Math. Ann., 356 (2013), 301-330. DOI:10.1007/s00208-012-0849-2
31. K. Shirakawa, H. Watanabe, N. Yamazaki, Phase-field systems for grain boundary motions under isothermal solidifications. Adv. Math. Sci. Appl., 24 (2014), 353-400.
32. K. Shirakawa, H. Watanabe, N. Yamazaki, Mathematical analysis for a Warren-Kobayashi- Lobkovsky-Carter type system. RIMS Kôkyûroku, 1997 (2016), 64-85.
33. J. Simon, Compact sets in the space L^p(0; T; B). Ann. Mat. Pura Appl. (4), 146 (1987), 65-96.
34. J. Sprekels, S. Zheng, Global existence and asymptotic behaviour for a nonlocal phase-field model for non-isothermal phase transitions. J. Math. Anal. Appl., 279 (2003), 97-110.
35. A. Visintin, Models of phase transitions. Progress in Nonlinear Differential Equations and their Applications, 28, Birkhäuser, Boston, (1996).
36. J. A. Warren, R. Kobayashi, A. E. Lobkovsky, W. C. Carter, Extending phase field models of solidification to polycrystalline materials. Acta Materialia, 51 (2003), 6035-6058.
37. H.Watanabe, K. Shirakawa, Qualitative properties of a one-dimensional phase-field system associated with grain boundary. In: Current Advances in Applied Nonlinear Analysis and Mathematical Modelling Issues, pp. 301-328, GAKUTO Internat. Ser. Math. Sci. Appl., 36, Gakkōtosho, Tokyo, (2013).
38. James A.Warren, Ryo Kobayashi, Alexander E. Lobkovsky,W. Craig Carter, Extending phase field models of solidification to polycrystalline materials. Acta Materialia, 51 (2003), 60356058.
39. H. Watanabe, K. Shirakawa, Stability for approximation methods of the one-dimensional Kobayashi-Warren-Carter system. Mathematica Bohemica, 139 (2014), special issue dedicated to Equadiff 13, no. 2, 381-389.
40. N. Yamazaki, Global attractors for non-autonomous phase-field systems of grain boundary motion with constraint. Adv. Math. Sci. Appl. 23 (2013), no. 1, 267-296.

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