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

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 $j_{\partial Q}^{\ast }B=0$, 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 $\mathcal{D}\times \mathcal{B}\to \mathcal{B}$, $(h,B)\mapsto h^{\ast }B$. 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 $M\times \{\ast \}$) expressed in terms of the helicity.

5 pages

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