The blow up method for Brakke flows: networks near triple junctions

TL;DR

This paper develops a parabolic blow-up method to analyze the asymptotic behavior of planar network flows near triple junctions, proving regularity and smoothness properties in regions without static tangent flows of density ≥ 2.

Contribution

Introduces a novel blow-up technique for Brakke flows of networks, establishing regularity and smoothness near triple junctions and characterizing flow behavior outside a small singular set.

Findings

Flow is regular and smoothly close to a triple junction in smaller regions.

Flow consists of smooth curves or triple junctions outside a small singular set.

Flow is classical at almost all times, except a set of Hausdorff dimension at most 1/2.

Abstract

We introduce a parabolic blow-up method to study the asymptotic behavior of an integral Brakke flow of planar networks (i.e. a 1-dimensional integral Brakke flow in a two dimensional region) weakly close in a space-time region to a static multiplicity 1 triple junction $J$. We show that such a network flow is regular in a smaller space-time region, in the sense that it consists of three curves coming smoothly together at a single point at 120 degree angles, staying smoothly close to $J$ and moving smoothly. Using this result and White's stratification theorem, we deduce that whenever an integral Brakke flow of networks in a space-time region ${\mathcal R}$ has no static tangent flow with density $\geq2$, there exists a closed subset $\Sigma \subset {\mathcal R}$ of parabolic Hausdorff dimension at most 1 such that the flow is classical in ${\mathcal R} \setminus \Sigma$, i.e. near every point in ${\mathcal R} \setminus \Sigma$, the flow, if non-empty, consists of either an embedded curve moving smoothly or three embedded curves meeting smoothly at a single point at 120 degree angles and moving smoothly. In particular, such a flow is classical at all times except for a closed set of times of ordinary Hausdorff dimension at most $\frac{1}{2}$.