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Keywords:
grain boundary motion; singular diffusion; subdifferential
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.
References:
[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. DOI 10.1006/jfan.2000.3698 | MR 1814993 | Zbl 0973.35109
[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. DOI 10.1006/jfan.2001.3829 | MR 1883415 | Zbl 1042.35018
[3] Andreu, F., Caselles, V., Mazón, J. M.: A strongly degenerate quasilinear equation: the parabolic case. Arch. Ration. Mech. Anal. 176 (2005), 415-453. DOI 10.1007/s00205-005-0358-5 | MR 2185664 | Zbl 1112.35111
[4] Attouch, H.: Variational Convergence for Functions and Operators. Pitman Advanced Publishing Program Boston-London-Melbourne (1984). MR 0773850 | Zbl 0561.49012
[5] Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. Editura Academiei Republicii Socialiste România, Bucharest Noordhoff International Publishing Leiden (1976). MR 0390843 | Zbl 0328.47035
[6] Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in ${\Bbb R}^N$. J. Differ. Equations 184 (2002), 475-525. DOI 10.1006/jdeq.2001.4150 | MR 1929886 | Zbl 1036.35099
[7] Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Amsterdam (1973), French. MR 0348562
[8] Chen, L. Q.: Phase-field models for microstructure evolution. Ann. Rev. Mater. Res. 32 (2002), 113-140. DOI 10.1146/annurev.matsci.32.112001.132041
[9] Clarke, F. H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc. New York (1983). MR 0709590
[10] Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall Englewood Cliffs (1964). MR 0181836 | Zbl 0144.34903
[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
[12] Gurtin, M. E., Lusk, M. T.: Sharp interface and phase-field theories of recrystallization in the plane. Physica D 130 (1999), 133-154. DOI 10.1016/S0167-2789(98)00323-6 | MR 1694730 | Zbl 0948.74042
[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. Zbl 0662.35054
[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. MR 2569333 | Zbl 1192.35083
[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. DOI 10.1016/0362-546X(94)90235-6 | MR 1279139
[17] Kobayashi, R., Giga, Y.: Equations with singular diffusivity. J. Statist. Phys. 95 (1999), 1187-1220. DOI 10.1023/A:1004570921372 | MR 1712447 | Zbl 0952.74014
[18] Kobayashi, R., Warren, J. A., Carter, W. C.: A continuum model of grain boundaries. Physica D 140 (2000), 141-150. DOI 10.1016/S0167-2789(00)00023-3 | MR 1752970 | Zbl 0956.35123
[19] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Gauthier-Villars Paris (1969), French. MR 0259693 | Zbl 0189.40603
[20] Lusk, M. T.: A phase field paradigm for grain growth and recrystallization. Proc. R. Soc. London A 455 (1999), 677-700. MR 1700887 | Zbl 0933.74016
[21] Ôtani, M.: Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators. Cauchy problems. J. Differ. Equations 46 (1982), 268-299. DOI 10.1016/0022-0396(82)90119-X | MR 0675911
[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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