Stability of the selfsimilar dynamics of a vortex filament

Details Stability of the selfsimilar dynamics of a vortex filament Stability of the selfsimilar dynamics of a vortex filament

2012-02-06

arxiv.org/abs/1202.1106v1

Mathematics

Analysis of PDEs

30 pages

Scientific paper

In this paper we continue our investigation about selfsimilar solutions of the vortex filament equation, also known as the binormal flow (BF) or the localized induction equation (LIE). Our main result is the stability of the selfsimilar dynamics of small pertubations of a given selfsimilar solution. The proof relies on finding precise asymptotics in space and time for the tangent and the normal vectors of the perturbations. A main ingredient in the proof is the control of the evolution of weighted norms for a cubic 1-D Schr\"odinger equation, connected to the binormal flow by Hasimoto's transform.

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