Curve diffusion and straightening flows on parallel lines

Abstract: In this paper, we study families of immersed curves $\gamma:(-1,1)\times[0,T)\rightarrow\mathbb{R}^2$ with free boundary supported on parallel lines ${\eta_1, \eta_2}:\mathbb{R}\rightarrow\mathbb{R}^2$ evolving by the curve diffusion flow and the curve straightening flow. The evolving curves are orthogonal to the boundary and satisfy a no-flux condition. We give estimates and monotonicity on the normalised oscillation of curvature, yielding global results for the flows.