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

TL;DR

This paper establishes a criterion based on curvature bounds that predicts finite-time blow-up for solutions to the curve diffusion flow with free boundary conditions and a fixed contact angle.

Contribution

It introduces a blow-up criterion involving an $L_2$-bound of curvature for curve diffusion flow with free boundary and fixed contact angle, extending understanding of solution singularities.

Findings

Finite-time blow-up occurs if the curvature bound is violated.

The proof uses a contradiction argument with compactness and short-time existence.

The result applies to curves with free boundary supported on a line with a fixed contact angle.

Abstract

We prove a blow-up criterion in terms of an $L_2$-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 $\alpha \in (0, \pi)$ 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.