The estimates from part b make $(u_m)$ bounded in
$$
L^\infty(0,T;H)\cap L^2(0,T;V),
$$
and $(\partial_tu_m)$ bounded in $L^2(0,T;V')$. Since the periodic embedding $V\Subset H$ is compact, the <Aubin-Lions lemma> supplies a subsequence such that
$$
u_m\rightharpoonup^*u
\quad\hbox{in }L^\infty(0,T;H),
$$
$$
u_m\rightharpoonup u
\quad\hbox{in }L^2(0,T;V),
\qquad
u_m\to u
\quad\hbox{in }L^2(0,T;H).
$$
The derivatives converge weakly in $L^2(0,T;V')$ to $\partial_tu$.

Because $P_NH$ is fixed and finite dimensional, strong convergence in $H$ implies
$$
P_Nu_m\to P_Nu
\quad\hbox{strongly in }L^2(0,T;V).
$$
Combining this with the uniform $L^\infty H$ and $L^2V$ bounds in the estimate from part a identifies the weak limit
$$
B(P_Nu_m,u_m)\rightharpoonup B(P_Nu,u)
\quad\hbox{in }L^2(0,T;V').
$$
Passing to the limit in the Galerkin identity gives
$$
\boxed{
\frac{du}{dt}+\nu Au+B(P_Nu,u)=0
\quad\hbox{in }L^2(0,T;V')}.
$$
The projected initial data converge to $u_0$ in $H$, so $u(0)=u_0$.

The <weak continuity from evolution-space bounds> gives
$$
\boxed{
u\in C([0,T];H_{\rm weak})
\cap L^\infty(0,T;H)
\cap L^2(0,T;V),
\qquad
u_t\in L^2(0,T;V')}.
$$
Since $T$ was arbitrary, this is a global weak solution of the <Navier-Stokes equation with spectrally truncated advection>.

Solution

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

Replacing the advecting velocity by a fixed finite Stokes projection $P_Nu$ makes the three-dimensional nonlinearity controlled by $\lambda_N^{1/4}|u|\|v\|\|w\|$. This yields global existence and uniqueness at the weak energy level.


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 Incoming links: Navier-Stokes equation with spectrally truncated advection