# Helicity integral in differential forms

1. Oct 30, 2016

### spaghetti3451

1. The problem statement, all variables and given/known data

Let $V^{3}(t)$ be a compact region moving with the fluid.

Assume that at $t=0$ the vorticity $2$-form $\omega^{2}$ vanishes when restricted to the boundary $\partial V^{3}(0)$; that is, $i^{*}\omega^{2}=0$, where $i$ is the inclusion of $\partial V$ in $\mathbb{R}^{3}$.

(This does $\textit{not}$ say that $\omega^{2}$ itself vanishes, rather only that $\omega({\vec{u}},{\vec{w}})=0$ for $\vec{u}$,$\vec{w}$ tangent to $\partial V^{3}(0)$.)

Then the $\textbf{helicity}$ integral $\displaystyle{\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx\wedge dy\wedge dz}$ can be constant in time.

Prove that the helicity integral is constant in time.

2. Relevant equations

3. The attempt at a solution

$$\frac{d}{dt}\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx\wedge dy\wedge dz$$
$$= \frac{d}{dt}\int_{V(t)}{\vec{v}}\cdot{\vec{\omega}}\ dx^{1}\wedge dx^{2}\wedge dx^{3}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} {\vec{v}}\cdot{\vec{\omega}}\ \text{vol}^{3}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu^{1}\wedge \omega^{2}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu^{1}\wedge {\rm d}\nu^{1}$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{V(t)} \nu\wedge {\rm d}\nu$$
$$= \frac{1}{o(x)\sqrt{g}(x)}\frac{1}{2} \frac{d}{dt}\int_{W(t)} (\nu\wedge {\rm d}\nu)+(\nu\wedge {\rm d}\nu)$$
$$= \frac{1}{o(x)\sqrt{g}(x)}\frac{1}{2} \frac{d}{dt}\int_{W(t)} (\nu\wedge {\rm d}\nu)-({\rm d}\nu\wedge \nu)$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\int_{W(t)} {\rm d}(\nu\wedge \nu)$$
$$= \frac{1}{o(x)\sqrt{g}(x)} \frac{d}{dt}\oint_{\partial W(t)} \nu\wedge \nu$$
$$= 0.$$

I think this is wrong because I haven't used the assumption that at $t=0$ the vorticity $2$-form $\omega^{2}$ vanishes when restricted to the boundary $\partial V^{3}(0)$; that is, $i^{*}\omega^{2}=0$, where $i$ is the inclusion of $\partial V$ in $\mathbb{R}^{3}$.

Last edited: Oct 31, 2016
2. Nov 5, 2016