Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation
Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation
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DOI：
10.1137/s0036141094279279
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发表时间：
1997-07-01
影响因子：
2
通讯作者：
Stoth, B
中科院分区：
文献类型：
--
作者：
Bronsard, L;Stoth, B
我们研究 R-n 有界球对称域 Omega 子集中非局部方程 epsilon phi(t) - epsilon Delta phi + 1/epsilon W'(phi) - lambda(epsilon)(t) = 0 的径向对称解的渐近行为，其中 lambda(epsilon)(t) = 1/epsilon f(Omega)W'(phi) dx，其中诺伊曼边界条件。该分析基于“能量方法”与一些先验估计相结合，后者用于通过渐近展开的前两项来近似解。我们只需要假设初始数据及其能量是有界的。我们证明，在 epsilon --> 0 的极限下，界面通过非局部平均曲率流移动，从而保留了质量。作为我们分析的副产品，我们获得了“拉格朗日乘数”lambda(epsilon)(t) 的 L-2 估计，该估计在非径向情况下也成立。此外，我们严格地（在一般几何中）证明非局部 Ginzburg-Landau 方程和 Cahn-Hilliard 方程作为粘性 Cahn-Hilliard 方程的特殊简并极限而出现。
We study the asymptotic behavior of radially symmetric solutions of the nonlocal equationepsilon phi(t) - epsilon Delta phi + 1/epsilon W'(phi) - lambda(epsilon)(t) = 0in a bounded spherically symmetric domain Omega subset of R-n, where lambda(epsilon)(t) = 1/epsilon f(Omega)W'(phi) dx, with a Neumann boundary condition. The analysis is based on ''energy methods'' combined with some a priori estimates, the latter being used to approximate the solution by the first two terms of an asymptotic expansion. We only need to assume that the initial data as well as their energy are bounded. We show that, in the limit as epsilon --> 0, the interfaces move by a nonlocal mean curvature flow, which preserves mass. As a by-product of our analysis, we obtain an L-2 estimate on the ''Lagrange multiplier'' lambda(epsilon)(t), which holds in the nonradial case as well. In addition, we show rigorously (in general geometry) that the nonlocal Ginzburg-Landau equation and the Cahn-Hilliard equation occur as special degenerate limits of a viscous Cahn-Hilliard equation.