On the energy of critical solutions of the binormal flow

Abstract

The binormal flow is a model for the dynamics of a vortex filament in a 3-D inviscid incompressible fluid. The flow is also related with the classical continuous Heisenberg model in ferromagnetism, and the 1-D cubic Schrödinger equation. We consider a class of solutions at the critical level of regularity that generate singularities in finite time. One of our main results is to prove the existence of a natural energy associated to these solutions. This energy remains constant except at the time of the formation of the singularity when it has a jump discontinuity. When interpreting this conservation law in the framework of fluid mechanics, it involves the amplitude of the Fourier modes of the variation of the direction of the vorticity.

Locations

• Communications in Partial Differential Equations - View
• arXiv (Cornell University) - View - PDF
• BIRD (Basque Center for Applied Mathematics) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) - View - PDF