We investigate the phase-field approximation of the Willmore flow which is a fourth-order diffusion equation with a small parameter ε>0 proportional to the thickness of the diffuse interface. We show rigorously that for well-prepared initial data, the level-set of the solution will converge to motion by the Willmore flow as ε trends to zero before the singularity of the latter occurs. This is done by constructing an approximate solution via matched asymptotic expansions, and then estimating its difference with the real solution by showing a spectrum condition of the linearized operator at the optimal profile. This is a joint work with Yuning Liu in NYU Shanghai.

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