Helicity is the only invariant of incompressible flows whose derivative is continuous in $C^1$-topology

TL;DR

This paper proves that helicity is the only invariant of incompressible flows with a continuous derivative in the $C^1$-topology, under certain conditions, highlighting its fundamental role in flow invariants.

Contribution

It establishes that any invariant functional with a regular, continuous $C^1$ derivative must be a function of helicity, demonstrating its uniqueness among flow invariants.

Findings

Helicity is the only invariant with a continuous $C^1$ derivative.

Invariant functionals are locally expressible in terms of helicity.

In certain cases, invariants are globally functions of helicity.

Abstract

Let $Q$ be a smooth compact orientable 3--manifold with smooth boundary $\partial Q$. Let $\mathcal{B}$ be the set of exact 2--forms $B\in\Omega^2(Q)$ such that $j_{\partial Q}^*B=0$, where $j_{\partial Q}:{\partial Q}\to Q$ is the inclusion map. The group $\mathcal{D}=\mathrm{Diff}_0(Q)$ of self-diffeomorphisms of $Q$ isotopic to the identity acts on the set $\mathcal{B}$ by $\mathcal{D}\times\mathcal{B}\to\mathcal{B}$, $(h,B)\mapsto h^*B$. Let $\mathcal{B}^\circ$ be the set of 2--forms $B\in\mathcal{B}$ without zeros. We prove that every $\mathcal{D}$--invariant functional $I:\mathcal{B}^\circ\to\mathbb{R}$ having a regular and continuous derivative with respect to the $C^1$--topology can be locally (and, if $Q=M\times S^1$ with $\partial Q\ne\varnothing$, globally on the set of all 2--forms $B\in\mathcal{B}^\circ$ admitting a cross-section isotopic to $M\times\{*\}$) expressed in terms of the helicity.