The volume-preserving motion by mean curvature as an asymptotic limit of reaction-diffusion equations

Author:
Dmitry Golovaty

Journal:
Quart. Appl. Math. **55** (1997), 243-298

MSC:
Primary 35K57; Secondary 35B25, 35Q55

DOI:
https://doi.org/10.1090/qam/1447577

MathSciNet review:
MR1447577

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Abstract | References | Similar Articles | Additional Information

Abstract: The asymptotic behavior of a nonlocal Ginzburg-Landau equation

**[1]**J. Rubinstein and P. Sternberg,*Nonlocal reaction-diffusion equations and nucleation*, IMA J. Appl. Math.**48**, 249-264 (1992) MR**1167735****[2]**A. Novick-Cohen,*On the viscous Cahn-Hilliard equation*, Material Instabilities in Continuum Mechanics and Related Mathematical Problems (J. Ball, ed.), Clarendon Press, 1988, pp. 329-342 MR**970531****[3]**G. Barles, H. M. Soner, and P. E. Souganidis,*Front propagation and phase field theory*, SIAM J. Control Optim., March 1993, issue dedicated to W. H. Fleming MR**1205984****[4]**L. Bronsard and B. Stoth,*Volume preserving mean curvature flow as a limit of nonlocal Ginzburg-Landau equation*(1994) (to appear) MR**1453306****[5]**E. Giusti,*Minimal Surfaces and Functions of Bounded Variation*, Birkhaüser, Boston, 1984 MR**775682****[6]**L. Bronsard and R. Kohn,*Motion by mean curvature as the singular limit of the Ginzburg-Landau model*, J. Differential Equations**90**, 211-237 (1991) MR**1101239****[7]**L. Modica,*Gradient theory of phase transitions and the minimal interface criteria*, Arch. Rat. Mech. Anal.**98**, 123-142 (1987) MR**866718****[8]**P. Sternberg,*The effect of a singular perturbation on nonconvex variational problems*, Arch. Rat. Mech. Anal.**101**, 209-260 (1988) MR**930124****[9]**I. Fonseca and L. Tartar,*The gradient theory of phase transitions for systems with two potential wells*, Proc. Royal Soc. Edinburgh Sect. A**111**, 89-102 (1989) MR**985992****[10]**S. B. Angenent,*The zero set of a solution of a parabolic equation*, J. für die reine und angewandte Math.**390**, 79-96 (1988) MR**953678****[11]**M. Protter and H. Weinberger,*Maximum Principles in Differential Equations*, Prentice-Hall, Englewood Cliffs, N. J., 1967 MR**0219861****[12]**B. Stoth,*The Stefan problem with the Gibbs-Thompson law as singular limit of phase-field equations in the radial case*, European J. Appl. Math. (1992) (to appear)**[13]**H. M. Soner,*Ginzburg-Landau equation and motion by mean curvature*, I:*Convergence*, preprint, 1993 MR**1674799****[14]**L. Evans and J. Spruck,*Motion of the level sets by mean curvature*III, J. Geom. Analysis**2**, 121-150 (1992) MR**1151756****[15]**T. Ilmanen,*Convergence of the Allen-Cahn equation to the Brakke's motion by mean curvature*, preprint, 1991 MR**1237490****[16]**K. A. Brakke,*The Motion of the Surface by its Mean Curvature*, Princeton University Press, Princeton, N.J., 1978 MR**485012****[17]**I. P. Natanson,*Theory of functions of a real variable*, vol. 2, Ungar Publishing Company, New York, NY, 1955 MR**0067952****[18]**D. Hilhorst, E. Logak, and Y. Nishiura,*Singular limit for an Allen-Cahn equation with a nonlocal term*, preprint, 1994 MR**1277394****[19]**L. Evans and J. Spruck,*Motion of the level sets by mean curvature*I, J. Differential Geom.**33**, 635-681 (1991) MR**1100206****[20]**Y. G. Chen, Y. Giga, and S. Goto,*Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations*, J. Differential Geom.**33**, 749-786 (1991) MR**1100211****[21]**M. Crandall, H. Ishii, and P.-L. Lions,*User's guide to viscosity solutions of second order partial differential equations*, Bull. Amer. Math. Soc.**27**, 1-67 (1992) MR**1118699****[22]**G. Caginalp,*An analysis of a phase field model of a free boundary*, Arch. Rat. Mech. Anal.**92**, 205-245 (1986) MR**816623****[23]**E. Fried and M. Gurtin,*Continuum phase transitions with an order parameter; accretion and heat conduction*, preprint, 1992**[24]**O. Penrose and P. Fife,*Thermodynamically consistent models for the kinetics of phase transitions*, Physica D**43**, 44-62 (1990) MR**1060043****[25]**X. Chen,*Hele-Shaw problem and area-preserving, curve shortening motion*, Arch. Rat. Mech. Anal.**123**, 117-151 (1993) MR**1219420****[26]**X. Chen, D. Hilhorst, and E. Logak,*Asymptotic behavior of an Allen-Cahn equation with a nonlocal term*, preprint, 1994 MR**1422816****[27]**L. Evans and R. Gariepy,*Measure Theory and Fine Properties of Functions*, CRC Press, Boca Raton, 1992 MR**1158660**

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DOI:
https://doi.org/10.1090/qam/1447577

Article copyright:
© Copyright 1997
American Mathematical Society