  In this article we consider the initial value problem of the binormal flow
with initial data given by curves that are regular except at one point where
they have a corner. We prove that under suitable conditions on the initial data
a unique regular solution exists for strictly positive and strictly negative
times. Moreover, this solution satisfies a weak version of the equation for all
times and can be seen as a perturbation of a suitably chosen self-similar
solution. Conversely, we also prove that if at time t = 1 a small regular
perturbation of a self-similar solution is taken as initial condition then
there exists a unique solution that at time t = 0 is regular except at a point
where it has a corner with the same angle as the one of the self-similar
solution. This solution can be extended for negative times. The proof uses the
full strength of the previous papers [9], [2], [3] and [4] on the study of
small perturbations of self-similar solutions. A compactness argument is used
to avoid the weighted conditions we needed in [4], as well as a more refined
analysis of the asymptotic in time and in space of the tangent and normal
vectors.
