Luis VEGA - Remarks about the self - similar solutions of the Vortex Filament Equation
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2016-06-03

I  will  review  some  of  the  properties  of  the  self-similar  solutions  of  the  Vortex Filament  Equation.  This  equation  is  also  known  as  either  the  Localized  Induction Equation or the binormal flow and is related to the 1d Schrodinger map and the 1d  cubic  non-linear  Schrodinger  equation.  After  looking  at  the  uniqueness  and asymptotic  behavior  of  these  solutions,  I  will  recall  the  method  developed  with  V. Banica  to  continue  the  solution once  the  singularity  (a  corner)  is  created.  Issues 
concerning the lack of the preservation of linear momentum and 
the no-continuity of some critical Besov norms will be considered. Finally I will mention some recent work done with F. De La Hoz about the evolution of a regular polygon