= Solution
For each $\nu>0$, choose the solution from parts (a)--(c). The maximum estimate gives a $\nu$-independent bound
$$
\|\omega_\nu\|_\infty\leq\gamma^{-1}\|g\|_\infty.
$$
Thus a sequence $\nu_j\downarrow0$ has $\omega_{\nu_j}\rightharpoonup^\ast\omega$ in $L^\infty$. The periodic elliptic estimates for
$$
-\Delta\Psi_{\nu_j}=\omega_{\nu_j},
\qquad
u_{\nu_j}=\nabla^\perp\Psi_{\nu_j}
$$
bound $\Psi_{\nu_j}$ in $W^{2,p}$ and $u_{\nu_j}$ in $W^{1,p}$ for every finite $p$. Taking $p>2$ and using compact <sobolev embedding> gives, after a further subsequence, uniform convergence $u_{\nu_j}\to u$. It follows that $u_{\nu_j}\omega_{\nu_j}\rightharpoonup u\omega$ distributionally.
The first energy estimate gives
$$
\|\nu_j\nabla\omega_{\nu_j}\|_2
\leq\sqrt{\nu_j}\,R_1\longrightarrow0.
$$
Therefore the viscous term vanishes in $H^{-1}$, and the weak formulation passes to the limit as
$$
\boxed{\gamma\omega+\nabla\mathbin\cdot(u\omega)=g}.
$$
The elliptic relations pass to the limit as well and give
$$
u=\nabla^\perp\Psi,\qquad-\Delta\Psi=\omega.
$$
Since $\omega\in L^\infty\subset L^2$, periodic elliptic regularity gives $\Psi\in H^2_{\rm per}$ and $u\in H^1_{\rm per}\cap H$. Moreover $u\omega\in L^2$, so its divergence lies in $H^{-1}_{\rm per}$. This constructs the required weak solution of the damped-driven Euler system by a <vanishing-viscosity limit>.
Solved by gpt-5.6-sol high.
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