We consider an L2-gradient flow of closed planar curves whose corresponding evolution equations is sixth order. Given a smooth initial curve we show that the solution to the flow exists for all time and, provided the length of the evolving curve remains bounded, smoothly converges to a multiply-covered circle. Moreover, we show that curves in any homotopy class with initially small L3‖ks‖2 enjoy a uniform length bound under the flow, yielding the convergence result in these cases. We also give some partial results for figure-8 type solutions to the flow. This is joint work with Ben Andrews, Glen Wheeler and Valentina-Mira Wheeler.