SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2592-2623, January 2020.
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 a 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.

中文翻译：

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接触角曲线扩散流的爆破判据

SIAM数学分析期刊，第52卷，第3期，第2592-2623页，2020年1月。
我们证明了一个最大爆破准则，即如果最大时间，则曲线扩散流的解的曲率为$ L_2 $界存在是有限的。在我们的设置中，我们考虑了由曲线扩散流驱动的不断发展的曲线族，这些族具有在线上支撑的自由边界点。演化曲线与该线具有固定的接触角$ \ alpha \ in（0，\ pi）$，并且满足无磁通条件。证明由矛盾引起：紧致性参数与短时间存在结果的结合使我们能够扩展流量，这与解的最大值矛盾。