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.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 1 / c / i / Solution

{c}
{wiki}

If $X_0$ embeds compactly into $X$ and $X$ embeds continuously into $X_1$, then boundedness in $L^p(0,T;X_0)$ together with a suitable time-derivative bound in $L^q(0,T;X_1)$ makes a family relatively compact in $L^p(0,T;X)$. A standard case is
$$
L^2(0,T;H^1)\cap H^1(0,T;H^{-1})Subset L^2(0,T;L^2).
$$


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 Incoming links: Aubin-Lions lemma