Metastable Bubble Solutions for the Allen-Cahn Equation with Mass Conservation
Metastable Bubble Solutions for the Allen-Cahn Equation with Mass Conservation
复制标题
具有质量守恒的 Allen-Cahn 方程的亚稳态气泡解
DOI：
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发表时间：
1996
影响因子：
1.9
通讯作者：
M. Ward
中科院分区：
文献类型：
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作者：
M. Ward
在多维域中，对于质量守恒的非局域Allen-Cahn方程，分析了具有球面界面的内层解的慢运动行为，称为气泡解。这个问题代表了存在质量约束的二元混合物相分离的最简单模型。气泡以指数方式缓慢地漂移穿过区域，形状不变，朝向区域边界上最近的点。通过将作者以前发展的处理一维空间亚稳态问题的渐近投影法推广到多维环境，导出了气泡中心运动的显式常微分方程组。根据初始接触点的边界曲率半径，导出了气泡对区域边界崩塌时间的渐近公式。将慢泡运动与经典的出射探测进行类比。
In a multidimensional domain, the slow motion behavior of internal layer solutions with spherical interfaces, referred to as bubble solutions, is analyzed for the nonlocal Allen–Cahn equation with mass conservation. This problem represents the simplest model for the phase separation of a binary mixture in the presence of a mass constraint. The bubble is shown to drift exponentially slowly across the domain, without change of shape, toward the closest point on the boundary of the domain. An explicit ordinary differential equation for the motion of the center of the bubble is derived by extending, to a multidimensional setting, the asymptotic projection method developed previously by the author to treat metastable problems in one spatial dimension. An asymptotic formula for the time of collapse of the bubble against the boundary of the domain is derived in terms of the principal radii of curvature of the boundary at the initial contact point. An analogy between slow bubble motion and the classical exit prob...