= Solution
For a smooth test function $\varphi$, <integration by parts> gives the moment identity
$$
\frac d{dt}\int\varphi(v)f(t,v)\,dv
=\int\bigl(\Delta\varphi-v\cdot\nabla\varphi\bigr)f(t,v)\,dv.
$$
The assumed decay removes all boundary terms. Apply it to $1$, $v_i$ and $|v|^2/2$. Writing $m_i=\int v_if=M u_i$ gives the <Ornstein-Uhlenbeck moment equations>
$$
\boxed{M'=0,\qquad m_i'=-m_i,\qquad E'=dM-2E.}
$$
Assume $M(0)\ne0$ when using the normalized mean $u$; otherwise $u$ is undefined, while the unnormalized <momentum> equation still holds. Since $M$ is constant, $u_i'=-u_i$. \b[Their solutions are]
$$
\boxed{M(t)=M_0,\qquad
u_i(t)=e^{-t}u_i(0),\qquad
E(t)=\frac{dM_0}{2}+
\left(E(0)-\frac{dM_0}{2}\right)e^{-2t}.}
$$
Thus <mass> is conserved, the <mean velocity> tends to zero, and $E(t)\to dM_0/2$. This includes the <mean velocity> conclusion omitted from the converted TeX.
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