Lower estimate of blow-up time and blow-up rate for non-local curvature flows for plane curves

Abstract This study examines several non-local curvature flows for plane curves with general rotation number. The flows investigated here are the area-preserving flow, the length-preserving flow, and the gradient flow of isoperimetric deficit. In a previous study, sufficient conditions for finite-time blow-up have been presented, and an upper bound for the blow-up time have been proposed. Here, we give estimates for solutions including at least two important applications. First, a lower bound for the blow-up time for the blow-up solutions is derived. Second, our estimate improves the blow-up rate of the blow-up solutions.