Generation of Interface for Solutions of the Mass Conserved Allen--Cahn Equation
Generation of Interface for Solutions of the Mass Conserved Allen--Cahn Equation
复制标题
质量守恒艾伦-卡恩方程解的接口生成
DOI：
10.1137/18m1204747
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发表时间：
2020
影响因子：
2
通讯作者：
Hilhorst D
中科院分区：
文献类型：
--
作者：
Hilhorst D
本文研究了含有非定域积分项的质量守恒Allen—Cahn方程解的界面生成问题。我们证明，对于一类相当一般的独立于的初始函数，解通常在一定的阶时间内发展出一个陡峭的厚度的过渡层。在某些情况下，我们证明了界面的厚度是完全有序的，这是最优的厚度估计。我们注意到，由于非局部项，比较原理不适用于我们的方程，因此在标准Allen- Cahn方程的早期研究中使用的方法不起作用。因此，我们将采用一种不同的方法，该方法基于对相应的非局部ode的长期行为的精细分析和一些能量估计。
In this paper, we study thegeneration of interfacefor the solution of the mass conserved Allen--Cahn equation involving a nonlocal integral term. We show that, for a rather general class of initial functions that are independent of, the solution generally develops a steep transition layer of thicknessat a certain time of order. In some cases, we prove that the thickness of the interface is exactly of order, which is the optimal thickness estimate. We note that the comparison principle does not hold for our equation because of the nonlocal term so that the methods that were employed in the earlier studies of the standard Allen--Cahn equation do not work. We will therefore take a different approach, which is based on the fine analysis of the long-time behavior of the corresponding nonlocal ODEs and some energy estimates.