On the existence of canonical multi-phase Brakke flows

Abstract

This paper establishes the global-in-time existence of a multi-phase mean curvature flow, evolving from an arbitrary closed rectifiable initial datum, which is a Brakke flow and a BV solution at the same time. In particular, we prove the validity of an explicit identity concerning the change of volume of the evolving grains, showing that their boundaries move according to the generalized mean curvature vector of the Brakke flow. Under suitable assumptions on the initial datum, such additional property resolves the non-uniqueness issue of Brakke flows. Contents 1. Introduction 2. Notation and main results 2.1. Basic notation 2.2. Three weak notions of MCF 2.3. Main results 2.4. Outline of the proofs and plan of the paper 2.5. Further notation 3. Existence of multi-phase Brakke flow 3.1. Volume-controlled Lipschitz deformations 3.2. The constructive scheme 4. BV flow: proof of Theorem 2.11(6) 4.1. Characterization as L 2 flow 4.2. Existence of measure-theoretic velocities 4.3. Boundaries move by their mean curvature 5. Ilmanen's density lower bound 6. Two-sidedness at unit density point 7. Final remarks 7.1. On generalized BV solutions 7.2. MCF with fixed boundary conditions 7.3. A lower bound estimate for extinction time Appendix A. The existence theorem of [18] revisited A.1. Construction of approximate flows A.2. Proofs of rectifiability and integrality A.3. Volume change of grains A.4. Changes in [19] References

Łojasiewicz–Simon inequalities for minimal networks: stability and convergence

“…As we said above there is an extensive amount of literature concerning the analysis of the flow in the framework of classical PDEs (see [43] and references within). There are also several generalized weak notions of the flow, for instance [3,21,32,35,59]. Recently interesting pro-gresses has been made both in the direction of proving regularity of weak solution [32,33] and in establishing the so-called weak-strong uniqueness theorem [23,27].…”

Łojasiewicz-Simon inequalities for minimal networks: stability and convergence

“…As we said above there is an extensive amount of literature concerning the analysis of the flow in the framework of classical PDEs (see [39] and references within). There are also several generalized weak notions of the flow, for instance [3,19,29,32,52]. Recently interesting progresses has been made both in the direction of proving regularity of weak solution [29,30] and in establishing the so-called weak-strong uniqueness theorem [21,24].…”