The binormal flow with initial data being polygonal lines and non-uniqueness

Ruobing Bai, Yifei Wu

Published 2020-02-17Version 1

In this paper, we consider the evolution of a curve $\chi\in \mathbb R^3$ by the binormal flow: $$ \chi_t=\chi_x\land \chi_{xx}. $$ We give a new construction for the solution of the binormal flow with initial data being the polygonal lines, which was previously obtained by V. Benica and L. Vega. Our construction is based on the linear Schr\"odinger equation at fractional time. This gives a simple and direct proof of the existence result. Since it does not rely on the solvability of the cubic nonlinear Schr\"odinger equation, as an improvement, we do not require the corners of the polygonal line to be located in integer numbers and are able to obtain the convergence $Lipschitz$ in arclength. The solutions with the same initial datum constructed by Benica and Vega are different from the ones given in this paper, hence we have the non-uniqueness of the solution for the binormal flow generated by the polygonal lines. In particular, the convergence rate of solutions, as $t$ goes to 0, can be attained as $|t|^\beta$ at corners for each $\beta \in (\frac35,1)$.