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

## TL;DR

This paper studies the long-term evolution of a generalized curve flow in the plane, proving that under certain conditions, the flow causes curves to become circular exponentially fast.

## Contribution

It demonstrates exponential convergence of the $H^{-m}$-gradient flow of length for closed curves to a circle, extending previous results on curve diffusion flow.

## Key findings

- The flow converges exponentially to a circle.
- Interpolation inequalities are key to the analysis.
- The results assume global existence of the flow.

## Abstract

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