= Solution
The uniform bounds and the <Banach-Alaoglu theorem> provide a subsequence, not relabelled, and a function $\theta$ such that
$$
\theta_m\rightharpoonup^\ast\theta
\quad\hbox{in }L^\infty(0,T;L^2_{\rm per}),
\qquad
\theta_m\rightharpoonup\theta
\quad\hbox{in }L^2(0,T;H^1_{\rm per}),
$$
and
$$
\partial_t\theta_m\rightharpoonup\partial_t\theta
\quad\hbox{in }L^2(0,T;H^{-1}_{\rm per}).
$$
Since $H^1_{\rm per}$ embeds compactly into $L^2_{\rm per}$, the <Aubin-Lions lemma> strengthens the first convergence to
$$
\boxed{\theta_m\longrightarrow\theta
\quad\hbox{strongly in }L^2(0,T;L^2_{\rm per})}.
$$
The <weak continuity from evolution-space bounds> gives a representative
$$
\boxed{\theta\in C([0,T];L^2_{\rm weak})
\cap L^\infty(0,T;L^2_{\rm per})
\cap L^2(0,T;H^1_{\rm per}),\qquad
\partial_t\theta\in L^2(0,T;H^{-1}_{\rm per})}.
$$
Testing against fixed spatial modes and using $\theta_m(0)=\Pi_m\theta_0$ shows that this representative satisfies $\theta(0)=\theta_0$ weakly.
Solved by gpt-5.6-sol high.
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