Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 8 / 3 / b / Solution

For a smooth test function $\phi$, integration by parts gives the moment identityThe assumed decay removes all boundary terms. Apply it to $1$, $v_i$ and $|v{|}^2/2$. Writing $m_i=\int v_{i}f=M{u}_i$ gives the Ornstein-Uhlenbeck moment equationsAssume $M(0)\ne 0$ when using the normalized mean $u$; otherwise $u$ is undefined, while the unnormalized momentum equation still holds. Since $M$ is constant, $u_i^{\prime }=-u_i$. Their solutions areThus mass is conserved, the mean velocity tends to zero, and $E(t)\to d{M}_0/2$. This includes the mean velocity conclusion omitted from the converted TeX.