Unbounded growth of the energy density associated to the Schrödinger map and the binormal flow

Abstract

We consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically, it is a flow of curves in three dimensions, explicitly connected to the one-dimensional Schrödinger map with values on the two-dimensional sphere, and to the one-dimensional cubic Schrödinger equation. Although these equations are completely integrable, we show the existence of an unbounded growth of the energy density. The density is given by the amplitude of the high frequencies of the derivative of the tangent vectors of the curves, thus giving information about oscillation at small scales. In the setting of vortex filaments, the variation of the tangent vectors is related to the derivative of the direction of the vorticity, which according to the Constantin–Fefferman–Majda criterion is relevant in the possible development of singularities for the Euler equations.