| Title:
|
A phase-field model of grain boundary motion (English) |
| Author:
|
Ito, Akio |
| Author:
|
Kenmochi, Nobuyuki |
| Author:
|
Yamazaki, Noriaki |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
53 |
| Issue:
|
5 |
| Year:
|
2008 |
| Pages:
|
433-454 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
grain boundary motion |
| Keyword:
|
singular diffusion |
| Keyword:
|
subdifferential |
| MSC:
|
35D05 |
| MSC:
|
35K45 |
| MSC:
|
35K50 |
| MSC:
|
35K55 |
| MSC:
|
35R35 |
| idZBL:
|
Zbl 1199.35138 |
| idMR:
|
MR2469586 |
| DOI:
|
10.1007/s10492-008-0035-8 |
| . |
| Date available:
|
2010-07-20T12:32:46Z |
| Last updated:
|
2020-07-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140333 |
| . |
| Reference:
|
[1] Andreu, F., Ballester, C., Caselles, V., Mazón, J. M.: The Dirichlet problem for the total variation flow.J. Funct. Anal. 180 (2001), 347-403. Zbl 0973.35109, MR 1814993, 10.1006/jfan.2000.3698 |
| Reference:
|
[2] Andreu, F., Caselles, V., Díaz, J. I., Mazón, J. M.: Some qualitative properties for the total variation flow.J. Funct. Anal. 188 (2002), 516-547. Zbl 1042.35018, MR 1883415, 10.1006/jfan.2001.3829 |
| Reference:
|
[3] Andreu, F., Caselles, V., Mazón, J. M.: A strongly degenerate quasilinear equation: the parabolic case.Arch. Ration. Mech. Anal. 176 (2005), 415-453. Zbl 1112.35111, MR 2185664, 10.1007/s00205-005-0358-5 |
| Reference:
|
[4] Attouch, H.: Variational Convergence for Functions and Operators.Pitman Advanced Publishing Program Boston-London-Melbourne (1984). Zbl 0561.49012, MR 0773850 |
| Reference:
|
[5] Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces.Editura Academiei Republicii Socialiste România, Bucharest Noordhoff International Publishing Leiden (1976). Zbl 0328.47035, MR 0390843 |
| Reference:
|
[6] Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in ${\Bbb R}^N$.J. Differ. Equations 184 (2002), 475-525. Zbl 1036.35099, MR 1929886, 10.1006/jdeq.2001.4150 |
| Reference:
|
[7] Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert.North-Holland Amsterdam (1973), French. MR 0348562 |
| Reference:
|
[8] Chen, L. Q.: Phase-field models for microstructure evolution.Ann. Rev. Mater. Res. 32 (2002), 113-140. 10.1146/annurev.matsci.32.112001.132041 |
| Reference:
|
[9] Clarke, F. H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts.John Wiley & Sons, Inc. New York (1983). MR 0709590 |
| Reference:
|
[10] Friedman, A.: Partial Differential Equations of Parabolic Type.Prentice-Hall Englewood Cliffs (1964). Zbl 0144.34903, MR 0181836 |
| Reference:
|
[11] Giga, M.-H., Giga, Y., Kobayashi, R.: Very singular diffusion equations. Proc. Taniguchi Conf. Math.Adv. Stud. Pure Math. 31 (2001), 93-125. MR 1865089, 10.2969/aspm/03110093 |
| Reference:
|
[12] Gurtin, M. E., Lusk, M. T.: Sharp interface and phase-field theories of recrystallization in the plane.Physica D 130 (1999), 133-154. Zbl 0948.74042, MR 1694730, 10.1016/S0167-2789(98)00323-6 |
| Reference:
|
[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. |
| Reference:
|
[14] Kenmochi, N.: Solvability of nonlinear evolution equations with time-dependent constraints and applications.Bull. Fac. Education, Chiba Univ. Vol. 30 (1981), 1-87. Zbl 0662.35054 |
| Reference:
|
[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. Zbl 1192.35083, MR 2569333 |
| Reference:
|
[16] Kenmochi, N., Niezgódka, M.: Evolution systems of nonlinear variational inequalities arising from phase change problems.Nonlinear Anal., Theory Methods Appl. 22 (1994), 1163-1180. MR 1279139, 10.1016/0362-546X(94)90235-6 |
| Reference:
|
[17] Kobayashi, R., Giga, Y.: Equations with singular diffusivity.J. Statist. Phys. 95 (1999), 1187-1220. Zbl 0952.74014, MR 1712447, 10.1023/A:1004570921372 |
| Reference:
|
[18] Kobayashi, R., Warren, J. A., Carter, W. C.: A continuum model of grain boundaries.Physica D 140 (2000), 141-150. Zbl 0956.35123, MR 1752970, 10.1016/S0167-2789(00)00023-3 |
| Reference:
|
[19] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires.Dunod, Gauthier-Villars Paris (1969), French. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[20] Lusk, M. T.: A phase field paradigm for grain growth and recrystallization.Proc. R. Soc. London A 455 (1999), 677-700. Zbl 0933.74016, MR 1700887 |
| Reference:
|
[21] Ôtani, M.: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators. Cauchy problems.J. Differ. Equations 46 (1982), 268-299. MR 0675911, 10.1016/0022-0396(82)90119-X |
| Reference:
|
[22] Visintin, A.: Models of Phase Transitions. Progress in Nonlinear Differential Equations and their Applications, Vol. 28.Birkhäuser-Verlag Boston (1996). MR 1423808 |
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