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We consider the binormal flow equation, which is a model for the dynamics ofvortex filaments in Euler equations. Geometrically it is a flow of curves inthree dimensions, explicitly connected to the 1-D Schr\"odinger map with valueson the 2-D sphere, and to the 1-D cubic Schr\"odinger equation. Although theseequations are completely integrable we show the existence of an unboundedgrowth of the energy density. The density is given by the amplitude of the highfrequencies of the derivative of the tangent vectors of the curves, thus givinginformation of the oscillation at small scales. In the setting of vortexfilaments the variation of the tangent vectors is related to the derivative ofthe direction of the vorticity, that according to theConstantin-Fefferman-Majda criterion plays a relevant role in the possibledevelopment of singularities for Euler equations.

Unbounded growth of the energy density associated to the Schr\"odinger map and the binormal flow