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.

Unbounded growth of the energy density associated to the Schrödinger map and the binormal flow

“…Nevertheless, this "conservation law" does not give any information about Tx , the Fourier transform of T x . We proved in [5] that (23) can be also understood as a kind of scattering energy of Tx for the solutions of ( 1) and ( 2) that we constructed in [4]. More precisely, if…”

“…It was also proved in [5] that there is a jump discontinuity of Ξ(T (t)) at t = 0. From (24) we can see | T x (t, ξ)| 2 dξ as an asymptotic energy density in phase space.…”

“…It is also easy to see that c 2 (t, x)dx is an energy density that from (3) describes the kinetic energy of the filament and from (4) the interaction energy of the chain. More precisely (5) |χ t (t, x)| 2 dx = |T x (t, x)| 2 dx = c 2 (t, x)dx, and for smooth solutions these quantities are conserved in time if they are finite.…”

“…Then, we use again (8) to integrate by parts, and a second sum appears with some new amplitudes A r and new quadratic phases. It was observed in [5] that a resonance can happen if j − r ∈ Z is properly chosen. It is easy to obtain a logarithmic upper bound for this resonance.…”

On the Evolution of the Vortex Filament Equation for regular $M$-polygons with nonzero torsion

“…Let ψ(s, 0) = k α k (0)δ(s−2πk/M ). Then, in [1], assuming some decay conditions of α k (0), the existence of a unique solution ψ(s, t) 2 . Assume now that α k+M = α k , for all k. Then, if the solution exists and is unique, α k+M (t) = α k (t), and…”