= Solution
The second estimate in part (a) bounds $(\omega_m)$ in $H^2_{\rm per}$. The periodic <Poisson equation> $-\Delta\Psi_m=\omega_m$ and the supplied curl identity then bound $(\Psi_m)$ in $H^4_{\rm per}$ and $(u_m)$ in $H^3_{\rm per}$. After passing to a subsequence,
$$
\omega_m\rightharpoonup\omega\ \hbox{in }H^2,
\qquad
\Psi_m\rightharpoonup\Psi\ \hbox{in }H^4,
\qquad
u_m\rightharpoonup u\ \hbox{in }H^3.
$$
The <Rellich-Kondrachov compactness theorem> also gives strong convergence in the corresponding spaces with one fewer derivative. In particular, $u_m\to u$ in $L^\infty$ and $\nabla\omega_m\to\nabla\omega$ in $L^2$, so
$$
(u_m\mathbin\cdot\nabla)\omega_m
\longrightarrow(u\mathbin\cdot\nabla)\omega
\quad\hbox{in }L^2.
$$
Passing to the limit in the Galerkin equations gives
$$
-\nu\Delta\omega+\gamma\omega+(u\mathbin\cdot\nabla)\omega=g,
\qquad
u=\nabla^\perp\Psi,
\qquad
-\Delta\Psi=\omega.
$$
These identities have the claimed <Sobolev space>[Sobolev regularity], and the first holds in $L^2_{\rm per}$. Finally, the <weak lower semicontinuity of the Hilbert norm> preserves the estimates
$$
\boxed{\nu\|\Delta\omega\|_2^2\leq R_2^2,
\qquad
\gamma\|\nabla\omega\|_2^2\leq R_2^2}.
$$
Solved by gpt-5.6-sol high.
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