Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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

$\displaystyle u_t^\varepsilon = \Delta {u^\varepsilon } - \frac{1}{{{\varepsilo... ...\frac{1}{\varepsilon }g\left( {{u^\varepsilon }} \right){\lambda ^\varepsilon }$

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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DOI: https://doi.org/10.1090/qam/1447577
Article copyright: © Copyright 1997 American Mathematical Society

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