A New Approach to Front Propagation Problems: Theory and Applications
A New Approach to Front Propagation Problems: Theory and Applications
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DOI：
10.1007/s002050050077
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发表时间：
1998-03
影响因子：
2.5
通讯作者：
G. Barles;P. Souganidis
中科院分区：
文献类型：
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作者：
G. Barles;P. Souganidis
在这篇文章中，我们给出了一个新的定义，定义了具有规定的法向速度的锋面(超曲面、边界)的全局时间传播(运动)，超过了它们第一次发展的奇点。我们证明，如果这种传播满足几何最大值原理(包含-回避)型性质，则法向速度只取决于锋面的位置及其法向和主曲率。这种新的方法更具几何性，并被证明相当于水平集方法，然后被用来开发一种非常通用和简单的方法来严格验证一般演化系统(如相互作用粒子和反应扩散方程)在渐近极限处运动界面的出现。最后，我们给出了一些新的渐近结果。其中包括(I)具有快速振荡系数的反应扩散方程，(Ii)完全非线性的非局域(积分-微分)方程和(Iii)具有长程各向异性相互作用和一般自旋反转动力学的随机伊辛模型的渐近性。
In this paper we present a new definition for the global in time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion‐avoidance)‐type property, then the normal velocity must depend only on the position of the front and its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level‐set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interfaces at the asymptotic limit of general evolving systems like interacting particles and reaction‐diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction‐diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations and (iii) stochastic Ising models with long-range anisotropic interactions and general spin flip dynamics.