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The estimates from part b make $(u_m)$ bounded inand $({\partial }_t{u}_m)$ bounded in $L^2(0,T;V^{\prime })$. Since the periodic embedding $V⋐H$ is compact, the Aubin-Lions lemma supplies a subsequence such thatThe derivatives converge weakly in $L^2(0,T;V^{\prime })$ to ${\partial }_{t}u$.

Because $P_{N}H$ is fixed and finite dimensional, strong convergence in $H$ impliesCombining this with the uniform $L^{\infty }H$ and $L^{2}V$ bounds in the estimate from part a identifies the weak limitPassing to the limit in the Galerkin identity givesThe projected initial data converge to $u_0$ in $H$, so $u(0)=u_0$.

The weak continuity from evolution-space bounds givesSince $T$ was arbitrary, this is a global weak solution of the Navier-Stokes equation with spectrally truncated advection.

The uniform bounds and the Banach-Alaoglu theorem provide a subsequence, not relabelled, and a function $\theta$ such thatandSince $H_{\mathrm{per}}^1$ embeds compactly into $L_{\mathrm{per}}^2$, the Aubin-Lions lemma strengthens the first convergence toThe weak continuity from evolution-space bounds gives a representativeTesting against fixed spatial modes and using ${\theta }_m(0)={\Pi }_m{\theta }_0$ shows that this representative satisfies $\theta (0)={\theta }_0$ weakly.