Classical formal arguments in fluid dynamics suggest that in certain limits, the evolution of thin vortex filaments in an ideal incompressible fluid should roughly be governed by an equation called the binormal curvature flow, which is very closely related to 1d focussing cubic NLS. However, these classical arguments rely on assumptions that are so unrealistic that it would be hard even to extract from them a precise conjecture that admits any realistic possibility of a proof. We present a different approach to this question that yields a reasonable formulation of a conjecture and genuinely plausible supporting evidence, and that clarifies the very substantial obstacles to a full proof. Our approach relies on novel recent estimates for the binormal curvature flow.

Parts of the talk are based on joint work with Didier Smets and with Christian Seis.