The Blow Up Method for Brakke Flows: Networks Near Triple Junctions

Mean curvature flow of networks models a physical system of grain boundaries in an annealing pure metal such as aluminum. Under the assumption that the surface tension drives the motion of grain boundaries, the simplest motion law dictates that the velocity of boundaries is proportional to the mean curvature, that is, the more convex that portion of boundaries is, the faster it moves. Idealized planar grain boundaries are typically consisted of smooth curves with their endpoints joined by triple junctions of 120 degree angles, which one may call a classical network flow. In the dynamical process of motion of grain boundaries, these triple junctions collide with each other, and various topological changes of networks take place due to the elimination of grains. The same can be said about the physical three dimensional case, where far more complex singularity patterns of grain boundaries as well as grain shapes may be expected.

Motivated by such a physical system, Brakke proposed in 1978 a novel notion of mean curvature flow, now called Brakke flow, which sets a rigorous mathematical framework for mean curvature flow in the presence of space-time singularities such as triple junctions and their collisions. The usual notion of curve or surface is replaced by a generalized geometric object called varifold to allow singularities, and the motion law is formulated using an integral inequality in a suitable distributional sense. There have been a great deal of studies on the equilibrium counterpart, minimal surfaces, in the framework of varifold since the inception of varifold in the 1960’s. The particularly well-studied aspects are the existence and regularity questions as well as the asymptotic behaviors of singularities of minimal surfaces. In comparison, little has been known about Brakke flow and a number of fundamental questions remain unsolved still now.

As one of natural questions on Brakke flow, one expects that any planar Brakke flow, whose definition does not guarantee any a priori structural regularity, should be a classical network flow for most of the time. For this type of questions, Professor Yoshihiro Tonegawa from Tokyo Institute of Technology and Professor Neshan Wickramasekera from University of Cambridge introduced a parabolic blow up method to study the singularities of Brakke flow, and proved in particular that any Brakke flow which is sufficiently close in an appropriately weak sense to a regular triple junction is indeed a classical network flow, that is, the flow locally consists of three smooth curves joined at a triple junction of 120 degree angle moving smoothly. The research work can be found in the peer-reviewed journal, Archive for Rational Mechanics and Analysis.

Under a physically motivated assumption that there exist no static tangent flows of density greater than 3/2, which is expected naturally for grain boundary motion, their work also proves that any Brakke flow under this assumption is a classical network flow except for the space-time singularities of parabolic dimension at most 1. This regularity result conforms a feature of network flow which one observes in the numerical simulations: there occasionally are collisions of triple junctions and sudden vanishing of curves, but such events happen and end instantly and flows are mostly classical.

Since the introduction of Brakke flow in 1978, there had been a significant lack of knowledge on the existence, regularity and structure of singularities of Brakke flow. Their research work revealed a deep hidden mathematical structure of Brakke flow of networks and made a substantial advance in the understanding of mean curvature flow.