A Blow-up Criterion for the Curve Diffusion Flow with a Contact Angle

2018/10/16 by Abels, Helmut, Butz, Julia
#35K35 #35K55 #53C44 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.07257

Abstract

We prove a blow-up criterion in terms of an L₂-bound of the curvature for solutions to the curve diffusion flow if the maximal time of existence is finite. In our setting, we consider an evolving family of curves driven by curve diffusion flow, which has free boundary points supported on a line. The evolving curve has fixed contact angle α∈ (0, π) with that line and satisfies a no-flux condition. The proof is led by contradiction: A compactness argument combined with the short time existence result enables us to extend the flow, which contradicts the maximality of the solution.

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