Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 1 / c / ii

The evolution-space regularity permits pairing the equation with $u(t)$ and givesfor almost every $t$. Since $P_{N}u$ is divergence free, periodic integration by parts givesIntegration in time, using $u(0)=u_0$, yields the exact energy balancefor every $t\in [0,T]$. Unlike the usual three-dimensional Leray construction, the improved $L^2(0,T;V^{\prime })$ control of the truncated nonlinearity permits an equality rather than only an inequality.