Some existence results for the modified binormal curvature flow equation

We establish some existence results for the modified binormal curvature flow equation from ($\mathbb{R}$ or $\mathbb{T}^l$ ) to $\mathbb{R}^3$ where the velocity of the curve depends not only on the binormal vector but the parametrization of the curve, the time and the position of the point in the space. We achieve our objective via the Schr\"{o}dinger map equation. A Local well-posedness result is proved for the Schr\"{o}dinger map equation in the space $L^\infty(0,T_1,H_{loc}^3(\mathbb{R})).$