A phase-field model of grain boundary motion

Akio Ito; Nobuyuki Kenmochi; Noriaki Yamazaki

Applications of Mathematics (2008)

  • Volume: 53, Issue: 5, page 433-454
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Ito, Akio, Kenmochi, Nobuyuki, and Yamazaki, Noriaki. "A phase-field model of grain boundary motion." Applications of Mathematics 53.5 (2008): 433-454. <http://eudml.org/doc/37794>.

@article{Ito2008,
abstract = {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.},
author = {Ito, Akio, Kenmochi, Nobuyuki, Yamazaki, Noriaki},
journal = {Applications of Mathematics},
keywords = {grain boundary motion; singular diffusion; subdifferential; grain boundary motion; singular diffusion; subdifferential},
language = {eng},
number = {5},
pages = {433-454},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A phase-field model of grain boundary motion},
url = {http://eudml.org/doc/37794},
volume = {53},
year = {2008},
}

TY - JOUR
AU - Ito, Akio
AU - Kenmochi, Nobuyuki
AU - Yamazaki, Noriaki
TI - A phase-field model of grain boundary motion
JO - Applications of Mathematics
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 53
IS - 5
SP - 433
EP - 454
AB - 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.
LA - eng
KW - grain boundary motion; singular diffusion; subdifferential; grain boundary motion; singular diffusion; subdifferential
UR - http://eudml.org/doc/37794
ER -

References

top
  1. Andreu, F., Ballester, C., Caselles, V., Mazón, J. M., 10.1006/jfan.2000.3698, J. Funct. Anal. 180 (2001), 347-403. (2001) Zbl0973.35109MR1814993DOI10.1006/jfan.2000.3698
  2. Andreu, F., Caselles, V., Díaz, J. I., Mazón, J. M., 10.1006/jfan.2001.3829, J. Funct. Anal. 188 (2002), 516-547. (2002) Zbl1042.35018MR1883415DOI10.1006/jfan.2001.3829
  3. Andreu, F., Caselles, V., Mazón, J. M., 10.1007/s00205-005-0358-5, Arch. Ration. Mech. Anal. 176 (2005), 415-453. (2005) Zbl1112.35111MR2185664DOI10.1007/s00205-005-0358-5
  4. Attouch, H., Variational Convergence for Functions and Operators, Pitman Advanced Publishing Program Boston-London-Melbourne (1984). (1984) Zbl0561.49012MR0773850
  5. Barbu, V., Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei Republicii Socialiste România, Bucharest Noordhoff International Publishing Leiden (1976). (1976) Zbl0328.47035MR0390843
  6. Bellettini, G., Caselles, V., Novaga, M., 10.1006/jdeq.2001.4150, J. Differ. Equations 184 (2002), 475-525. (2002) Zbl1036.35099MR1929886DOI10.1006/jdeq.2001.4150
  7. Brézis, H., Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland Amsterdam (1973), French. (1973) MR0348562
  8. Chen, L. Q., 10.1146/annurev.matsci.32.112001.132041, Ann. Rev. Mater. Res. 32 (2002), 113-140. (2002) DOI10.1146/annurev.matsci.32.112001.132041
  9. Clarke, F. H., Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley &amp; Sons, Inc. New York (1983). (1983) MR0709590
  10. Friedman, A., Partial Differential Equations of Parabolic Type, Prentice-Hall Englewood Cliffs (1964). (1964) Zbl0144.34903MR0181836
  11. Giga, M.-H., Giga, Y., Kobayashi, R., 10.2969/aspm/03110093, Adv. Stud. Pure Math. 31 (2001), 93-125. (2001) MR1865089DOI10.2969/aspm/03110093
  12. Gurtin, M. E., Lusk, M. T., 10.1016/S0167-2789(98)00323-6, Physica D 130 (1999), 133-154. (1999) Zbl0948.74042MR1694730DOI10.1016/S0167-2789(98)00323-6
  13. Ito, A., Gokieli, M., Niezgódka, M., Szpindler, M., Mathematical analysis of approximate system for one-dimensional grain boundary motion of Kobayashi-Warren-Carter type, Submitted. 
  14. Kenmochi, N., Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ. Vol. 30 (1981), 1-87. (1981) Zbl0662.35054
  15. Kenmochi, N., Monotonicity and compactness methods for nonlinear variational inequalities, Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 4 M. Chipot North Holland Amsterdam (2007), 203-298. (2007) Zbl1192.35083MR2569333
  16. Kenmochi, N., Niezgódka, M., 10.1016/0362-546X(94)90235-6, Nonlinear Anal., Theory Methods Appl. 22 (1994), 1163-1180. (1994) MR1279139DOI10.1016/0362-546X(94)90235-6
  17. Kobayashi, R., Giga, Y., 10.1023/A:1004570921372, J. Statist. Phys. 95 (1999), 1187-1220. (1999) Zbl0952.74014MR1712447DOI10.1023/A:1004570921372
  18. Kobayashi, R., Warren, J. A., Carter, W. C., 10.1016/S0167-2789(00)00023-3, Physica D 140 (2000), 141-150. (2000) Zbl0956.35123MR1752970DOI10.1016/S0167-2789(00)00023-3
  19. Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars Paris (1969), French. (1969) Zbl0189.40603MR0259693
  20. Lusk, M. T., A phase field paradigm for grain growth and recrystallization, Proc. R. Soc. London A 455 (1999), 677-700. (1999) Zbl0933.74016MR1700887
  21. Ôtani, M., 10.1016/0022-0396(82)90119-X, J. Differ. Equations 46 (1982), 268-299. (1982) MR0675911DOI10.1016/0022-0396(82)90119-X
  22. Visintin, A., Models of Phase Transitions. Progress in Nonlinear Differential Equations and their Applications, Vol. 28, Birkhäuser-Verlag Boston (1996). (1996) MR1423808

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.