Orateur
Description
The binormal flow is a geometric flow of curves in R^3, that models vortex filament dynamics in fluids and superfluids, and is also related to the continuous classical Heisenberg ferromagnet equation. The result presented in this talk is the construction of weak solutions for non-closed Lipschitz curves data, and it extends the result of Bob Jerrard and Didier Smets obtained in 2015 for closed curves. This allows to consider general data including the known examples of strong solutions that generate singularities in finite time. The descriptions of the latter are precise, based on the Hasimoto transform that links the binormal flow to the 1D cubic Schrödinger equation, and on the explicit expression of the self-similar solutions. In the present general low regularity context this approach is no longer available and geometric measure theory tools are used instead. More precisely, we introduce a notion of renormalized length for non-closed curves and work in the framework of locally integral currents. This is a joint work with Bob Jerrard and Didier Smets.
