= Solution
Consider an unforced <incompressible flow> with localized <vorticity> and sufficient smoothness for <integration by parts>. Define the <hydrodynamic impulse> per unit density by $\mathbf I=\tfrac12\int\mathbf x\times\boldsymbol\omega\,dV$, denoting the vector <impulse> by $\mathbf I$ to distinguish it from the later scalar $L$. The <vorticity equation> can be written
$$
\partial_t\boldsymbol\omega=\nabla\times\mathbf F,\qquad
\mathbf F=\mathbf u\times\boldsymbol\omega-\nu\nabla\times\boldsymbol\omega.
$$
The supplied identity, integrated over all space, gives $\tfrac12\int\mathbf x\times\nabla\times\mathbf F\,dV=\int\mathbf F\,dV$, because $\mathbf F$ is localized and its surface terms vanish. Therefore
$$
\frac{d\mathbf I}{dt}=\int\mathbf u\times\boldsymbol\omega\,dV-
\nu\int\nabla\times\boldsymbol\omega\,dV.
$$
The second integral is a vanishing surface integral. The vector identity $\mathbf u\times\boldsymbol\omega=\nabla(|\mathbf u|^2/2)-(\mathbf u\cdot\nabla)\mathbf u$ and incompressibility reduce the first to surface terms proportional to the quadratic velocity. Those vanish for $\mathbf u=O(R^{-3})$. Thus
$$
\boxed{d\mathbf I/dt=0}.
$$
For a large spherical control volume define $\mathbf P_R=\int_{V_R}\mathbf u\,dV$. The <impulse> relation is $\mathbf I=(3/2)\lim_{R\to\infty}\mathbf P_R$ for this localized field. The factor is important: <impulse> is not simply the integral of velocity, since an $R^{-3}$ velocity tail has a nonvanishing contribution in the <impulse> <integration by parts> involving a factor of $R$.
Using <pressure> per unit density $\Pi$ and the Navier-Stokes equation gives
$$
\frac{d\mathbf P_R}{dt}=-\int_{\partial V_R}[\mathbf u(\mathbf u\cdot\mathbf n)+\Pi\mathbf n-\nu\partial_n\mathbf u],dA.
$$
The advective flux is $O(R^{-4})$, the <pressure> force is $O(R^{-1})$, and the viscous term is $O(\nu R^{-2})$ under the regular far-field derivative decay. All vanish as $R\to\infty$. \b[The localized eddy cannot change its total <impulse> without a net external <momentum> input.] <Pressure> redistributes <momentum> internally but its resultant force on an increasingly distant control surface tends to zero. At finite $R$, boundary forces need not vanish exactly; the all-space limit is the invariant statement.
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