On the energy of critical solutions of the binormal flow

Valeria Banica (1, 2, 3) , Luis Vega

1 SU - Sorbonne Université
2 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
3 IUF - Institut universitaire de France

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 Heisen-berg 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.

Domaines: Equations aux dérivées partielles [math.AP]

Valeria Banica, Luis Vega. On the energy of critical solutions of the binormal flow. Communications in Partial Differential Equations, 2020, 45 (7), pp.820-845. ⟨10.1080/03605302.2020.1738460⟩. ⟨hal-02917029⟩