For small-energy initial regular planar curves with generalised Neumann boundary conditions, we consider the steepest-descent gradient flow for the $L^2$-norm of the derivative of curvature with respect to arc length. We show that such curves between parallel lines converge exponentially in the $C^\infty$ topology in infinite time to straight lines.
"A sixth order flow of plane curves with boundary conditions." Tohoku Math. J. (2) 72 (3) 379 - 393, 2020. https://doi.org/10.2748/tmj/1601085621