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 parameter \(\varepsilon >0\) that is proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, as \(\varepsilon \) tends 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.

我们通过严格证明其尖锐的界面限制来研究 Willmore 流的相场近似。这是一个四阶扩散方程，其参数\(\varepsilon >0\)与扩散界面的厚度成正比。我们严格地证明，对于准备好的初始数据，如\(\varepsilon \)趋向于零，只要威尔莫尔流的经典解存在，解的水平集就会收敛到威尔莫尔流的运动。这是通过通过匹配的渐近展开从极限流构造近似解，然后估计其与实际解的差异来完成的。关键的一步是证明线性化算子在最优剖面上的谱不等式，它是一个四阶算子，写成 Allen-Cahn 算子的平方加上奇异扰动。