Applications of Mathematics

, 53:433 | Cite as

A phase-field model of grain boundary motion

  • Akio ItoEmail author
  • Nobuyuki Kenmochi
  • Noriaki Yamazaki


We consider a phase-field model of grain structure evolution, which appears in materials sciences. In this paper we study the grain boundary motion model of Kobayashi-Warren-Carter type, which contains a singular diffusivity. The main objective of this paper is to show the existence of solutions in a generalized sense. Moreover, we show the uniqueness of solutions for the model in one-dimensional space.


grain boundary motion singular diffusion subdifferential 


  1. [1]
    F. Andreu, C. Ballester, V. Caselles, J. M. Mazón: The Dirichlet problem for the total variation flow. J. Funct. Anal. 180 (2001), 347–403.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    F. Andreu, V. Caselles, J. I. Díaz, J. M. Mazón: Some qualitative properties for the total variation flow. J. Funct. Anal. 188 (2002), 516–547.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    F. Andreu, V. Caselles, J. M. Mazón: A strongly degenerate quasilinear equation: the parabolic case. Arch. Ration. Mech. Anal. 176 (2005), 415–453.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    H. Attouch: Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program, Boston-London-Melbourne, 1984.zbMATHGoogle Scholar
  5. [5]
    V. Barbu: Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Republicii Socialiste România, Bucharest. Noordhoff International Publishing, Leiden, 1976.zbMATHGoogle Scholar
  6. [6]
    G. Bellettini, V. Caselles, M. Novaga: The total variation flow in ℝN. J. Differ. Equations 184 (2002), 475–525.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Brézis: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam, 1973. (In French.)zbMATHGoogle Scholar
  8. [8]
    L. Q. Chen: Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 32 (2002), 113–140.CrossRefGoogle Scholar
  9. [9]
    F. H. Clarke: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1983.Google Scholar
  10. [10]
    A. Friedman: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, 1964.zbMATHGoogle Scholar
  11. [11]
    M.-H. Giga, Y. Giga, R. Kobayashi: Very singular diffusion equations. Proc. Taniguchi Conf. Math. Adv. Stud. Pure Math. 31 (2001), 93–125.MathSciNetGoogle Scholar
  12. [12]
    M. E. Gurtin, M. T. Lusk: Sharp interface and phase-field theories of recrystallization in the plane. Physica D 130 (1999), 133–154.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. Ito, M. Gokieli, M. Niezgódka, M. Szpindler: Mathematical analysis of approximate system for one-dimensional grain boundary motion of Kobayashi-Warren-Carter type. Submitted.Google Scholar
  14. [14]
    N. Kenmochi: Solvability of nonlinear evolution equations with time-dependent constraints and applications. Bull. Fac. Education, Chiba Univ. Vol. 30. 1981, pp. 1–87.Google Scholar
  15. [15]
    N. Kenmochi: Monotonicity and compactness methods for nonlinear variational inequalities. Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 4 (M. Chipot, ed.). North Holland, Amsterdam, 2007, pp. 203–298.Google Scholar
  16. [16]
    N. Kenmochi, M. Niezgódka: Evolution systems of nonlinear variational inequalities arising from phase change problems. Nonlinear Anal., Theory Methods Appl. 22 (1994), 1163–1180.zbMATHCrossRefGoogle Scholar
  17. [17]
    R. Kobayashi, Y. Giga: Equations with singular diffusivity. J. Statist. Phys. 95 (1999), 1187–1220.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    R. Kobayashi, J. A. Warren, W. C. Carter: A continuum model of grain boundaries. Physica D 140 (2000), 141–150.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    J.-L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars, Paris, 1969. (In French.)zbMATHGoogle Scholar
  20. [20]
    M. T. Lusk: A phase field paradigm for grain growth and recrystallization. Proc. R. Soc. London A 455 (1999), 677–700.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [21]
    M. Ôtani: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators. Cauchy problems. J. Differ. Equations 46 (1982), 268–299.zbMATHCrossRefGoogle Scholar
  22. [22]
    A. Visintin: Models of Phase Transitions. Progress in Nonlinear Differential Equations and their Applications, Vol. 28. Birkhäuser-Verlag, Boston, 1996.zbMATHGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2008

Authors and Affiliations

  1. 1.Department of Electronic Engineering and Computer Science, School of EngineeringKinki UniversityHiroshimaJapan
  2. 2.Department of Mathematics, Faculty of EducationChiba UniversityChibaJapan
  3. 3.Department of Mathematical Science, Common Subject DivisionMuroran Institute of TechnologyMuroranJapan

Personalised recommendations