Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-16T17:01:21.824Z Has data issue: false hasContentIssue false
  • English
  • Français

Mean curvature flow by the Allen–Cahn equation

Published online by Cambridge University Press:  13 May 2015

D. S. LEE  and
J. S. KIM
D. S. LEE
Affiliation:
Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 689-798, Republic of Korea
J. S. KIM
Affiliation:
Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea email: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we investigate motion by mean curvature using the Allen–Cahn (AC) equation in two and three space dimensions. We use an unconditionally stable hybrid numerical scheme to solve the equation. Numerical experiments demonstrate that we can use the AC equation for applications to motion by mean curvature. We also study the curve-shortening flow with a prescribed contact angle condition.

Keywords

motion by mean curvatureAllen-Cahn equationcontact angle condition
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Issue 4 , August 2015 , pp. 535 - 559
Copyright
Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Allen, S. M. & Cahn, J. W. (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27 (6), 1085–1095.CrossRefGoogle Scholar
[2]Andrews, B. & Baker, C. (2010) Mean curvature flow of pinched submanifolds to spheres. J. Differ. Geom. 85 (3), 357–395.CrossRefGoogle Scholar
[3]Angenent, S. & Gurtin, M. E. (1989) Multiphase thermomechanics with interfacial structure 2. Evolution of an isothermal interface. Arch. Ration. Mech. Anal. 108 (3), 323–391.CrossRefGoogle Scholar
[4]Arezzo, C. & Sun, J. (2013) Self-shrinkers for the mean curvature flow in arbitrary codimension. Math. Z. 274, 993–1027.CrossRefGoogle Scholar
[5]Altschuler, S. J. & Wu, L. F. (1993) Convergence to translating solutions for a class of quasilinear parabolic boundary problems. Math. Ann. 295 (1), 761–765.CrossRefGoogle Scholar
[6]Altschuler, S. J. & Wu, L. F. (1994) Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc. Var. 2 (1), 101–111.CrossRefGoogle Scholar
[7]Barles, G., Biton, S. & Ley, O. (2002) A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations. Arch. Ration. Mech. Anal. 162, 287–325.CrossRefGoogle Scholar
[8]Barles, G., Soner, H. M. & Souganidis, P. E. (1993) Front propagation and phase field theory. SIAM J. Control. Optim. 31 (2), 439–469.CrossRefGoogle Scholar
[9]Barrett, J. W., Garcke, H. & Nürnberg, R. (2010) Parametric approximation of surface clusters driven by isotropic and anisotropic surface energies. Interface Free Bound. 12, 187–234.CrossRefGoogle Scholar
[10]Barrett, J. W., Garcke, H. & Nürnberg, R. (2010) Finite-element approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering. Eur. J. Appl. Math. 21, 519–556.CrossRefGoogle Scholar
[11]Barrett, J. W., Garcke, H. & Nürnberg, R. (2011) The approximation of planar curve evolutions by stable fully implicit finite element schemes that equidistribute. Numer. Methods Part. D. E. 27, 1–30.CrossRefGoogle Scholar
[12]Bartels, S & Müller, R (2010) Quasi-optimal and robust a posteriori error estimates in L ∞(L 2) for the approximation of Allen–Cahn equations past singularities. Math. Comput. 80 (274), 761–780.CrossRefGoogle Scholar
[13]Bellettini, G. & Novaga, M. (2011) Curvature evolution of nonconvex lens-shaped domains. J. Reine Angew. Math. 2011, 17–46.CrossRefGoogle Scholar
[14]Beneš, M. (2003) Diffuse-interface treatment of the anisotropic mean-curvature flow. Appl. Math. 48 (6), 437–453.CrossRefGoogle Scholar
[15]Bobenko, A. I. & Springborn, B. A. (2007) A discrete Laplace–Beltrami operator for simplicial surfaces. Discrete Comput. Geom. 38 (4), 740–756.CrossRefGoogle Scholar
[16]Brakke, K. (1978) The Motion of a Surface by its Mean Curvature, Math Notes, Princeton University Press, Princeton, New York.Google Scholar
[17]Bronsard, L. & Kohn, R. V. (1991) Motion by mean curvature as the singular limit of Ginzburg–Landau dynamics. J. Differ. Eq. 90 (2), 211–237.CrossRefGoogle Scholar
[18]Caginalp, G. & Chen, X. (1998) Convergence of the phase field model to its sharp interface limits. Eur. J. Appl. Math. 9, 417–445.CrossRefGoogle Scholar
[19]Cao, H. D. & Li, H. (2013) A gap theorem for self-shrinkers of the mean curvature flow in arbitrary codimension. Calc. Var. 46, 879–889.CrossRefGoogle Scholar
[20]Chen, X. (1994) Spectrum for the Allen–Cahn, Cahn–Hilliard, and phase-field equations for generic interfaces. Commun. Part. Differ. Eq. 19 (7–8), 1371–1395.CrossRefGoogle Scholar
[21]Chen, Y. G., Giga, Y. & Goto, S. (1991) Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations. J. Differ. Geom. 33, 749–786.CrossRefGoogle Scholar
[22]Chen, X. & Guo, J. S. (2011) Motion by curvature of planar curves with end points moving freely on a line. Math. Ann. 350 (2), 277–311.CrossRefGoogle Scholar
[23]Chopp, D. L. (1993) Computing minimal surfaces via level set curvature. J. Comput. Phys. 106, 79–91.CrossRefGoogle Scholar
[24]Chopp, D. L. (1994) Computation of self-similar solutions for mean curvature flow. Exp. Math. 3 (2), 1–15.CrossRefGoogle Scholar
[25]Chopp, D. L. & Sethian, J. A. (1993) Flow under curvature: singularity formation, minimal surfaces, and geodesics. Exp. Math. 2 (4), 235–255.CrossRefGoogle Scholar
[26]Deckelnick, K., Dziuk, G. & Elliott, C. M. (2005) Computation of geometric partial differential equations and mean curvature flow. Acta Numer. 14, 139–232.CrossRefGoogle Scholar
[27]Dirr, N., Karali, G. & Yip, N. K. (2008) Pulsating wave for mean curvature flow in inhomogeneous medium. Eur. J. Appl. Math. 19, 661–699.CrossRefGoogle Scholar
[28]Dziuk, G. & Elliott, C. M. (2012) L 2-estimates for the evolving surface finite element method. Math. Comput. 82 (281), 1–24.CrossRefGoogle Scholar
[29]Ecker, K. (2004) Progress in Nonlinear Differential Equations and their Applications, Regularity Theory for Mean Curvature Flow, Birkhäuser, Boston.CrossRefGoogle Scholar
[30]Elsey, M. & Esedoglu, S. (2009) Analogue of the total variation denoising model in the context of geometry processing. Multiscale Model. Simul. 7 (4), 1549–1573.CrossRefGoogle Scholar
[31]Epstein, C. L. & Gage, M. (1987) The Curve Shortening Flow. Wave Motion: Theory, Modelling, and Computation, Springer, New York.Google Scholar
[32]Evans, L. C., Soner, H. M. & Souganidis, P. E. (1992) Phase transitions and generalized motion by mean curvature. Commun. Pure Appl. Math. 45 (9), 1097–1123.CrossRefGoogle Scholar
[33]Evans, L. C. & Spruck, J. (1991) Motion of level sets by mean curvature I. J. Differ. Geom. 33 (3), 635–681.CrossRefGoogle Scholar
[34]Evans, L. C. & Spruck, J. (1992) Motion of level sets by mean curvature II. Trans. Am. Math. Soc. 1 (1), 321–332.CrossRefGoogle Scholar
[35]Evans, L. C. & Spruck, J. (1992) Motion of level sets by mean curvature III. J. Geom. Anal. 2 (2), 121–150.CrossRefGoogle Scholar
[36]Evans, L. C. & Spruck, J. (1995) Motion of level sets by mean curvature IV. J. Geom. Anal. 1 (1), 77–114.CrossRefGoogle Scholar
[37]Feng, X., He, Y. & Liu, C. (2006) Analysis of finite element approximations of a phase field model for two-phase fluids. Math. Comput. 76 (258), 539–571.CrossRefGoogle Scholar
[38]Feng, X. & Prohl, A. (2003) Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Math. Comput. 73 (246), 541–567.CrossRefGoogle Scholar
[39]Feng, X. & Wu, H. J. (2005) A posteriori error estimates and an adaptive finite element method for the Allen–Cahn equation and the mean curvature flow. J. Sci. Comput. 24 (2), 121–146.CrossRefGoogle Scholar
[40]Gage, M. & Hamilton, R. S. (1986) The heat equation shrinking convex plane curves. J. Differ. Geom. 23 (1), 69–96.CrossRefGoogle Scholar
[41]Ghoussoub, N. & Gui, C. (2003) On De Giorgi's conjecture in dimensions 4 and 5. Ann. Math. 157 (1), 313–334.CrossRefGoogle Scholar
[42]Grayson, M. A. (1987) The heat equation shrinks embedded plane curves to points. J. Differ. Geom. 26 (2), 285–314.CrossRefGoogle Scholar
[43]Grayson, M. A. (1989) A short note on the evolution of a surface by its mean curvature. Duke Math. J. 58 (3), 555–558.CrossRefGoogle Scholar
[44]Grayson, M. A. (1989) Shortening embedded curves. Ann. Math. 129 (1), 71–111.CrossRefGoogle Scholar
[45]Guggenheumer, H. W. (1977) Differential Geometry, Dover Publications, New York.Google Scholar
[46]Gurtin, M. E. (1988) Multiphase thermomechanics with interfacial structure 1. Heat conduction and the capillary balance law. Arch. Ration. Mech. Anal. 104 (3), 195–221.CrossRefGoogle Scholar
[47]Handlovičová, A. & Mikula, K. (2008) Stability and consistency of the semi-implicit co-volume scheme for regularized mean curvature flow equation in level set formulation. Appl. Math. 53 (2), 105–129.CrossRefGoogle Scholar
[48]Heinemann, C. & Kraus, C. (2013) Existence results for diffuse interface models describing phase separation and damage. Eur. J. Appl. Math. 24, 179–211.CrossRefGoogle Scholar
[49]Huisken, G. (1984) Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20 (1), 237–266.CrossRefGoogle Scholar
[50]Huisken, G. (1998) A distance comparison principle for evolving curves. Asian J. Math. 2 (1), 127–134.CrossRefGoogle Scholar
[51]Huisken, G. (1989) Non-parametric mean curvature evolution with boundary conditions. J. Differ. Equ. 77 (2), 369–378.CrossRefGoogle Scholar
[52]Ilmanen, T. (1993) Convergence of the Allen–Cahn equation to Brakke's motion by mean curvature. J. Differ. Geom. 38 (2), 417–461.CrossRefGoogle Scholar
[53]Kohsaka, Y. (2001) Free boundary problem for quasilinear parabolic equation with fixed angle of contact to a boundary. Nonlinear Anal. 45 (7), 865–894.CrossRefGoogle Scholar
[54]Ladyenskaja, O. A., Solonnikov, V. A. & Ural'ceva, N. N. (1968) Linear and quasilinear equations of parabolic type. Am. Math. Soc. 23.Google Scholar
[55]Lee, H.-G. & Kim, J. (2011) Accurate contact angle boundary conditions for the Cahn–Hilliard equations. Comput. Fluids 44 (1), 178–168.CrossRefGoogle Scholar
[56]Li, Y., Lee, H-G., Jeong, D. & Kim, J. (2010) An unconditionally stable hybrid numerical method for solving the Allen–Cahn equation. Comput. Math. Appl. 60 (6), 1591–1606.CrossRefGoogle Scholar
[57]Lie, J., Lysaker, M. & Tai, X-C. (2006) A variant of the level set method and applications to image segmentation. Math. Comput. 75 (255), 1155–1174.CrossRefGoogle Scholar
[58]Mantegazza, C. (2011) Lecture notes on mean curvature flow, volume 290 of Progress in Mathematics. Birkhauser/Springer Basel AG, Basel.CrossRefGoogle Scholar
[59]Mantegazza, C., Novaga, M. & Tortorelli, V. N. (2004) Motion by curvature of planar networks. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 3, 235–324.Google Scholar
[60]Merriman, B., Bence, J. & Osher, S. (1992) Diffusion-generated motion by mean curvature. In: Taylor, E. (editor), Computational Crystal Growers Workshop, pp. 73–83.Google Scholar
[61]Mottoni, P. D. & Schatzman, M. (1995) Geometrical evolution of developed interfaces. Trans. Am. Math. Soc. 347 (5), 1533–1589.CrossRefGoogle Scholar
[62]Mullins, W. W. (1956) Two-dimensional motion of idealized grain boundaries. J. Appl. Phys. 27, 900–904.CrossRefGoogle Scholar
[63]Pan, Z. & Wetton, B. (2008) A numerical method for coupled surface and grain boundary motion. Eur. J. Appl. Math. 19, 311–327.CrossRefGoogle Scholar
[64]Pino, M., Kowalczyk, M. & Wei, J. (2011) On De Giorgi's conjecture in dimension N ⩾ 9. Ann. Math. 174 (3), 1485–1569.CrossRefGoogle Scholar
[65]Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2002) Numerical Recepes in C++, 2nd ed., Cambridge University, Cambridge.Google Scholar
[66]Ren, X. & Wei, J. (2009) On a phase-field problem driven by interface area and interface curvature. Eur. J. Appl. Math. 20, 531–556.CrossRefGoogle Scholar
[67]Rubinstein, J., Sternberg, P. & Keller, J. B. (1989) Fast reaction, slow diffusion, and curve shortening. SIAM J. Appl. Math. 49 (1), 116–133.CrossRefGoogle Scholar
[68]Ruuth, S. (1998) Efficient algorithms for diffusion-generated motion by mean curvature. J. Comput. Phys. 144, 603–625.CrossRefGoogle Scholar
[69]Sapiro, G. (2006) Geometric Partial Differential Equations and Image Analysis, Cambridge University, Cambridge.Google Scholar
[70]Schnärer, O. C., Azouani, A., Georgi, M., Hell, J., Jangle, N., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F. & Smith, B. (2011) Evolution of convex lens-shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363 (5), 2265–2294.CrossRefGoogle Scholar
[71]Smoczyk, K. (2012) Global Differential Geometry: Mean Curvature Flow in Higher Codimension: Introduction and Survey, Springer, Berlin Heidelberg.Google Scholar
[72]Soner, H. M. (1993) Motion of a set by the curvature of its boundary. J. Differ. Equ. 101 313–372.CrossRefGoogle Scholar
[73]Soner, H. M. (1997) Ginzburg-Landau equation and motion by mean curvature, I: Convergence. J. Geom. Anal. 7 (3), 437–475.CrossRefGoogle Scholar
[74]Soner, H. M. (1997) Ginzburg–Landau equation and motion by mean curvature, II: Development of the initial interface. J. Geom. Anal. 7 (3), 477–491.CrossRefGoogle Scholar
[75]Tso, K. (1985) Deforming a hypersurface by its Gauss-Kronecker curvature. Commun. Pure Appl. Math. 38 (6), 867–882.CrossRefGoogle Scholar
[76]Zhu, X.-P. (2002) Studies in Advanced Mathematics, Lectures on Mean Curvature Flows, AMS, International Press, Somerville.CrossRefGoogle Scholar
[77]Xu, G. (2004) Surface fairing and featuring by mean curvature motions. J. Comput. Appl. Math. 163, 295–309.CrossRefGoogle Scholar