Phase-Field Approximation of the Willmore Flow
Phase-Field Approximation of the Willmore Flow
复制标题
威尔莫尔流的相场近似
DOI：
10.1007/s00205-021-01678-9
复制
发表时间：
2019
影响因子：
2.5
通讯作者：
Yuning Liu
中科院分区：
文献类型：
--
作者：
Mingwen Fei;Yuning Liu
我们调查相场近似的Willmore流严格证明其尖锐的界面限制。这是一个四阶扩散方程，其参数与扩散界面的厚度成正比。我们严格表明，对于精心准备的初始数据，astends到零的水平集的解决方案将收敛到运动的Willmore流，只要经典的解决方案，后者存在。这是通过构造一个近似的解决方案，从极限流通过匹配渐近展开，然后估计其差异与真实的解决方案。关键的一步是证明线性化算子在最佳轮廓处的谱不等式，该线性化算子是一个四阶算子，写为Allen-Cahn算子的平方加上一个奇异摄动。
We investigate the phase-field approximation of the Willmore flow by rigorously justifying its sharp interface limit. This is a fourth-order diffusion equation with a parameterthat is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, astends to zero the level-set of the solution will converge to the motion by Willmore flow as long as the classical solution to the latter exists. This is done by constructing an approximate solution from the limiting flow via matched asymptotic expansions, and then estimating its difference with the real solution. The crucial step is to prove a spectrum inequality of the linearized operator at the optimal profile, which is a fourth-order operator written as the square of the Allen–Cahn operator plus a singular perturbation.