We focus on a class of solutions of the binormal flow, model of the evolution of vortex filaments, that generate several corner singularities in finite time. This phenomenon has been studied in [1, 4] in the regular case, which in this context is in terms of the summability of the angles of the corners generated. Our goal here is to investigate the lower regularity case, using further the Hasimoto approach that allows to use the 1D cubic nonlinear Schrödinger to study the binormal flow. We first obtain a deterministic result by proving an existence result for general binormal flow solutions at low regularity. Then we obtain improved results on the above class of solutions by a suitable randomization of the curvature and torsion of the vortex filament. To do so, we prove a scattering result for a quasi-invariant measure associated with a suitable 1D cubic nonlinear Schrödinger equation that we consider of independent interest. An interesting feature of this result is that we are able to identify a limit measure, which is usually not possible when working on quasi-invariant Gaussian measures for Hamiltonian PDE’s on bounded domains.
