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

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Abstract

We introduce a parabolic blow-up method to study the asymptotic behavior of a Brakke flow of planar networks (that is a 1-dimensional 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\({^{\circ}}\) angles, staying smoothly close to J and moving smoothly. Using this result and White’s stratification theorem, we deduce that whenever a Brakke flow of networks in a space-time region \({{\mathcal {R}}}\) has no static tangent flow with density \({{\geqq}2}\), there exists a closed subset \({{\Sigma \subset {\mathcal {R}}}}\) of parabolic Hausdorff dimension at most 1 such that the flow is classical in \({{\mathcal {R}}\backslash\Sigma}\), that is near every point in \({{\mathcal {R}}\backslash\Sigma}\), 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\({^{\circ}}\) 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 \({\frac{1}{2}}\).

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan

    Yoshihiro Tonegawa

  2. DPMMS, University of Cambridge, Cambridge, CB3 0WB, UK

    Neshan Wickramasekera

Authors
  1. Yoshihiro Tonegawa
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  2. Neshan Wickramasekera
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Correspondence to Yoshihiro Tonegawa.

Additional information

Communicated by D. Kinderlehrer

Y. Tonegawa is partially supported by JSPS Grant-in-aid for scientific research (A) #25247008, (S) #21224001 and challenging exploratory research #23654057.

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Tonegawa, Y., Wickramasekera, N. The Blow Up Method for Brakke Flows: Networks Near Triple Junctions. Arch Rational Mech Anal 221, 1161–1222 (2016). https://doi.org/10.1007/s00205-016-0981-3

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  • DOI: https://doi.org/10.1007/s00205-016-0981-3

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