The binormal flow: a toy model for turbulence

These notes summarize the content of a one-week course given by Luis Vega on the binormal flow model applied to describe the dynamics of three-dimensional vortex filaments in incompressible fluids following the Euler equations. This topic touches connections between the binormal flow, the Schrödinger map and the 1d cubic nonlinear Schrödinger equation, providing a rich casuistry linked to the physical and numerical phenomenology. This course was part of the $2^{\text{nd}}$ Advanced Summer School held at the Institut d’Etudes Scientifiques de Cargèse (IESC), at Cargèse (Corsica), from ${20}^{\text{th}}$ to ${30}^{\text{th}}$ June 2023. These notes have been written and completed by two of its attendees, Elena Danesi and Pablo Merino.