Previous |  Up |  Next


Title: Diffuse-interface treatment of the anisotropic mean-curvature flow (English)
Author: Beneš, Michal
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 48
Issue: 6
Year: 2003
Pages: 437-453
Summary lang: English
Category: math
Summary: We investigate the motion by mean curvature in relative geometry by means of the modified Allen-Cahn equation, where the anisotropy is incorporated. We obtain the existence result for the solution as well as a result concerning the asymptotical behaviour with respect to the thickness parameter. By means of a numerical scheme, we can approximate the original law, as shown in several computational examples. (English)
Keyword: mean-curvature flow
Keyword: phase-field method
Keyword: FDM
Keyword: Finsler geometry
MSC: 35A40
MSC: 53C44
MSC: 80A22
MSC: 82C26
idZBL: Zbl 1099.53044
idMR: MR2025297
DOI: 10.1023/B:APOM.0000024485.24886.b9
Date available: 2009-09-22T18:15:05Z
Last updated: 2020-07-02
Stable URL:
Reference: [1] G. Bellettini, M. Novaga, and M. Paolini: Characterization of facet-breaking for nonsmooth mean curvature flow in the convex case.Interfaces and Free Boundaries 3 (2001), 415–446. MR 1869587
Reference: [2] G. Bellettini, M. Paolini: Anisotropic motion by mean curvature in the context of Finsler geometry.Hokkaido Math.  J. 25 (1996), 537–566. MR 1416006, 10.14492/hokmj/1351516749
Reference: [3] M. Beneš: Anisotropic phase-field model with focused latent-heat release.In: Free Boundary Problems: Theory and Applications II, GAKUTO International Series Mathematical Sciences and Applications, Vol. 14, Chiba, Japan, 2000, pp. 18–30. MR 1793055
Reference: [4] M. Beneš: Mathematical analysis of phase-field equations with numerically efficient coupling terms.Interfaces and Free Boundaries 3 (2001), 201–221. Zbl 0986.35116, MR 1825658
Reference: [5] M. Beneš: Mathematical and computational aspects of solidification of pure substances.Acta Math. Univ. Comenian. 70 (2001), 123–152. Zbl 0990.80006, MR 1865364
Reference: [6] M. Beneš, K. Mikula: Simulation of anisotropic motion by mean curvature-comparison of phase-field and sharp-interface approaches.Acta Math. Univ. Comenian. 67 (1998), 17–42. MR 1660813
Reference: [7] L. Bronsard, R. Kohn: Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics.J. Differential Equations 90 (1991), 211–237. MR 1101239, 10.1016/0022-0396(91)90147-2
Reference: [8] G. Caginalp: An analysis of a phase field model of a free boundary.Arch. Rational Mech. Anal. 92 (1986), 205–245. Zbl 0608.35080, MR 0816623, 10.1007/BF00254827
Reference: [9] C. M.  Elliott, M. Paolini, and R. Schtzle: Interface estimates for the fully anisotropic Allen-Cahn equation and anisotropic mean curvature flow.Math. Models Methods Appl. Sci. 6 (1996), 1103–1118. MR 1428147
Reference: [10] L. C.  Evans, J. Spruck: Motion of level sets by mean curvature  I.J.  Differential Geom. 33 (1991), 635–681. MR 1100206, 10.4310/jdg/1214446559
Reference: [11] Y. Giga, M. Paolini, and P. Rybka: On the motion by singular interfacial energy.Japan J.  Indust. Appl. Math. 18 (2001), 47–64. MR 1842909
Reference: [12] M. Gurtin: On the two-phase Stefan problem with interfacial energy and entropy.Arch. Rational Mech. Anal. 96 (1986), 200–240. Zbl 0654.73008, MR 0855304
Reference: [13] M. Gurtin: Thermomechanics of Evolving Phase Boundaries in the Plane.Clarendon Press, Oxford, 1993. Zbl 0787.73004, MR 1402243
Reference: [14] D. A.  Kessler, J. Koplik, and H. Levine: Geometrical models of interface evolution. III. Theory of dendritic growth.Phys. Rev.  A 31 (1985), 1712–1717.
Reference: [15] J.-L.  Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires.Dunod Gauthier-Villars, Paris, 1969. Zbl 0189.40603, MR 0259693
Reference: [16] T. Ohta, M. Mimura, and R. Kobayashi: Higher-dimensional localized patterns in excitable media.Physica  D 34 (1989), 115–144. MR 0982383, 10.1016/0167-2789(89)90230-3
Reference: [17] J. A.  Sethian: Level Set Methods.Cambridge University Press, New York, 1996. Zbl 0868.65059, MR 1409367
Reference: [18] R. Temam: Navier-Stokes Equations, Theory and Numerical Analysis.North-Holland, Amsterdam, 1979. Zbl 0426.35003, MR 0603444
Reference: [19] A. Visintin: Models of Phase Transitions.Birkhäuser, Boston, 1996. Zbl 0882.35004, MR 1423808


Files Size Format View
AplMat_48-2003-6_5.pdf 819.1Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo