2015/04/30 by Yoshihiro Tonegawa, Neshan Wickramasekera
Mathematics · #Closed set #Dimension (graph theory) #Flow (mathematics) #Geometric Analysis and Curvature Flows #Hausdorff dimension #Hausdorff distance #Mathematical Dynamics and Fractals #Planar #Point (geometry) #Stochastic processes and statistical mechanics #Tangent #math.AP
paper · pdf · doi:10.1007/s00205-016-0981-3
published as Archive for Rational Mechanics and Analysis 221, (2016) no. 3, 1161-1222 · 51 pages
arxiv created 2016/02/09 · openalex publication_date 2016/02/23 · arxiv updated 2016/06/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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 ≥2, there exists a closed subset Σ⊂ \mathcal R of parabolic Hausdorff dimension at most 1 such that the flow is classical in \mathcal R ∖ Σ, i.e. near every point in \mathcal R ∖ Σ, 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 (1)/(2).