= Solution
Take the zeroth <velocity moment> of <1/a>. The right side is a <velocity space> divergence with vanishing boundary contribution, giving the <continuity equation>
$$
\boxed{\partial_t\rho+\nabla\cdot(\rho\mathbf u)=0,\qquad \rho=\int f\,d^3v,\qquad \rho\mathbf u=\int\mathbf v f\,d^3v.}
$$
For the first <velocity moment>, use <integration by parts> componentwise:
$$
\int d^3v\,v_i\partial_{v_j}f_2=-\delta_{ij}\int d^3v\,f_2.
$$
Write $\mathbf c=\mathbf v-\mathbf u$; then $\int\mathbf c f\,d^3v=0$, and the <momentum flux> splits into bulk transport and the <pressure tensor>:
$$
\int v_iv_jf\,d^3v=\rho u_iu_j+P_{ij},\qquad \boxed{P_{ij}=\int (v_i-u_i)(v_j-u_j)f\,d^3v.}
$$
Therefore
$$
\boxed{\partial_t(\rho u_i)+\partial_{x_j}(\rho u_iu_j+P_{ij})=-\int d^3x_\star\,n_2(\mathbf x,\mathbf x_\star,t)\partial_{x_i}\phi(|\mathbf x-\mathbf x_\star|),}
$$
where repeated Cartesian indices are summed and $n_2=\int f_2\,d^3v\,d^3v_\star$. The negative sign is the attractive or repulsive <force> $-\nabla\phi$, rather than an additional sign convention for the <pressure tensor>.
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