Unbounded growth of the energy density associated to the Schrödinger map and the binormal flow
Valeria Banica (Sorbonne University, Laboratoire Jacques-Louis Lions)
In this talk I shall 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 1-D Schrödinger map with values on the 2-D sphere, and to the 1-D 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 of the oscillation at small scales. In the setting of vortex filaments the variation of the tangent vectors is related to the one of the direction of the vorticity, that according to the Constantin-Fefferman-Majda criterion plays a relevant role in the possible development of singularities for Euler equations. This is a joint work with Luis Vega.
A 3-D fluid displays a vortex filament if its vorticity is highly concentrated around a curve in space. Understanding the evolution of vortex filaments is a natural but challenging question, as this situation does not enter the framework of the general results available for 3-D incompressible Euler. It is conjectured that the binormal flow has an important role in the dynamics of one or several vortex filaments. This model was derived formally at the beginning of the last century and is at the heart of recent researches. In this lecture I will talk about the state of the art of deriving models for vortex filaments dynamics, as well as about mathematical methods and results for the binormal flow and some related models.