Curve Diffusion Flow with Boundary on Skew Lines

Abstract We consider the curve diffusion flow for open planar curves with boundary on two skew lines. For each angle $$\theta \in (0, \pi )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>π</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> of the two skew lines, we prove the existence of global-in-time solutions under a suitable initial condition. Furthermore, we show the full limit convergence of solutions to the arc of the sector with the central angle $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> and the same area as that of the initial curve.