Research article Special Issues

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

  • Received: 22 October 2016 Accepted: 16 January 2017 Published: 08 March 2017
  • 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.

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

    Related Papers:

  • 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.


    加载中
    [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 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 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 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 Lp(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.
  • Reader Comments
  • © 2017 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2926) PDF downloads(834) Cited by(1)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog