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We consider the $H^{-m}$-gradient flow of length for closed plane curves.This flow is a generalization of curve diffusion flow. We investigate thelarge-time behavior assuming the global existence of the flow. Then we showthat the evolving curve converges exponentially to a circle. To do this, we useinterpolation inequalities between the deviation of curvature and theisoperimetric ratio, recently established by Nagasawa and the author.

Large-time behavior of the $H^{-m}$-gradient flow of length for closed plane curves