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Absence of singular stretching of interacting vortex filaments

Date: 
Friday 27th July 2012 - 14:45 to 15:05
Venue: 
INI Seminar Room 1
Session Chair: 
Charlie Doering
Abstract: 
A promising mechanism for generating a finite-time singularity in the incompressible Euler equations is the stretching of vortex filaments. Here, we argue that interacting vortex filaments cannot generate a singularity by analyzing the asymptotic dynamics of their collapse. We use the separation of the dynamics of the filament shape, from that of its core to derive constraints that must be satisfied for a singular solution to remain self consistent uniformly in time. Our only assumption is that the length scales characterizing filament shape obey scaling laws set by the dimension of circulation as the singularity is approached. The core radius necessarily evolves on a different length scale. We show that a self similar ansatz for the filament shapes cannot induce singular stretching, due to the logarithmic prefactor in the self interaction term for the filaments. More generally, there is an antagonistic relationship between the stretching rate of the filaments and the requ irement that the radius of curvature of filament shape obeys the dimensional scaling laws. This suggests that it is unlikely that solutions in which the core radii vanish sufficiently fast to maintain the filament approximation exist.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons