The evolution-space regularity permits pairing the equation with $u(t)$ and gives
$$
\frac12\frac d{dt}|u|^2
+\nu\|u\|^2
+\langle B(P_Nu,u),u\rangle=0
$$
for almost every $t$. Since $P_Nu$ is divergence free, periodic integration by parts gives
$$
\langle B(P_Nu,u),u\rangle=0.
$$
Integration in time, using $u(0)=u_0$, yields the exact energy balance
$$
\boxed{
\frac12|u(t)|^2
+\nu\int_0^t\|u(\tau)\|^2\,d\tau
=\frac12|u_0|^2}
$$
for every $t\in[0,T]$. Unlike the usual three-dimensional Leray construction, the improved $L^2(0,T;V')$ control of the truncated nonlinearity permits an equality rather than only an inequality.


Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-359/1/c/ii/solution