## H. T. Nguyen - S. Wang

# Brakke Regularity for the Allen-Cahn Flow

created by wang on 13 Nov 2020

[

BibTeX]

*preprint*

**Inserted:** 13 nov 2020

**Year:** 2020

**Abstract:**

In this paper we prove an analogue of the Brakke's $\varepsilon$-regularity
theorem for the parabolic Allen-Cahn equation. In particular, we show uniform
$C^{2,\alpha}$ regularity for the transition layers converging to smooth mean
curvature flows as $\varepsilon\rightarrow0$. The proof utilises Allen-Cahn
versions of the monotonicity formula, parabolic Lipschitz approximation and
blowups. A corresponding gap theorem for entire eternal solutions of the
parabolic Allen-Cahn is also obtained. As an application of the regularity
theorem, we give an affirmative answer to a question of Ilmanen that there is
no cancellation in $\mathbf {BV}$ convergence in the mean convex setting.