I will present work about self-similar solutions of the binormal flow.
This is a flow in three dimensions of curves that move in the direction of their
binormal with a speed proportional to their curvature. The flow was
obtained in 1906 by Da Rios as an approximation of the evolution of a
vortex filament. I will first justify the validity and limitations of
this approximation to the Euler equations. Then, I will characterize
the self-similar solutions (joint work with S. Gutierrez) from a
geometric point of view. Finally I will give some results obtained
with V. Banica about the stability of these solutions. These results
are based on the connection of this equation with the one-dimenstional cubic non-linear Schrödinger equation. As a byproduct of our analysis we obtain some new scenarios of dispersive break down.
Refreshments preceding the lecture, 3:30-4:00pm, POT745.