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: The asymptotic behavior of a nonlocal Ginzburg-Landau equation \[ u_t^\varepsilon = \Delta {u^\varepsilon } - \frac {1}{{{\varepsilon ^2}}}W’\left ( {{u^\varepsilon }} \right ) + \frac {1}{\varepsilon }g\left ( {{u^\varepsilon }} \right ){\lambda ^\varepsilon }\] is studied when the small parameter $\varepsilon$ tends to zero. Here a Lagrange multiplier ${\lambda ^\varepsilon }$ is introduced into the equation to enforce the conservation of mass. An energy-estimates approach is used to show that a limiting solution can be characterized by moving interfaces. It is further shown that the asymptotic limit of solutions of the nonlocal Ginzburg-Landau equation is a weak solution of the nonlocal, mass-preserving mean curvature flow. The weak solutions are constructed within a framework of the theory of viscosity solutions. In addition, the results describing interactions between the interfaces are obtained.
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J. Rubinstein and P. Sternberg, Nonlocal reaction-diffusion equations and nucleation, IMA J. Appl. Math. 48, 249–264 (1992)
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
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
L. Bronsard and B. Stoth, Volume preserving mean curvature flow as a limit of nonlocal Ginzburg-Landau equation (1994) (to appear)
E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhaüser, Boston, 1984
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)
L. Modica, Gradient theory of phase transitions and the minimal interface criteria, Arch. Rat. Mech. Anal. 98, 123–142 (1987)
P. Sternberg, The effect of a singular perturbation on nonconvex variational problems, Arch. Rat. Mech. Anal. 101, 209–260 (1988)
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)
S. B. Angenent, The zero set of a solution of a parabolic equation, J. für die reine und angewandte Math. 390, 79–96 (1988)
M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, N. J., 1967
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)
H. M. Soner, Ginzburg-Landau equation and motion by mean curvature, I: Convergence, preprint, 1993
L. Evans and J. Spruck, Motion of the level sets by mean curvature III, J. Geom. Analysis 2, 121–150 (1992)
T. Ilmanen, Convergence of the Allen-Cahn equation to the Brakke’s motion by mean curvature, preprint, 1991
K. A. Brakke, The Motion of the Surface by its Mean Curvature, Princeton University Press, Princeton, N.J., 1978
I. P. Natanson, Theory of functions of a real variable, vol. 2, Ungar Publishing Company, New York, NY, 1955
D. Hilhorst, E. Logak, and Y. Nishiura, Singular limit for an Allen-Cahn equation with a nonlocal term, preprint, 1994
L. Evans and J. Spruck, Motion of the level sets by mean curvature I, J. Differential Geom. 33, 635–681 (1991)
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)
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)
G. Caginalp, An analysis of a phase field model of a free boundary, Arch. Rat. Mech. Anal. 92, 205–245 (1986)
E. Fried and M. Gurtin, Continuum phase transitions with an order parameter; accretion and heat conduction, preprint, 1992
O. Penrose and P. Fife, Thermodynamically consistent models for the kinetics of phase transitions, Physica D 43, 44–62 (1990)
X. Chen, Hele-Shaw problem and area-preserving, curve shortening motion, Arch. Rat. Mech. Anal. 123, 117–151 (1993)
X. Chen, D. Hilhorst, and E. Logak, Asymptotic behavior of an Allen-Cahn equation with a nonlocal term, preprint, 1994
L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992
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Article copyright:
© Copyright 1997
American Mathematical Society
