= Solution
Use on $\widetilde H_m$ the inner product $(w,z)_1=(\nabla w,\nabla z)$. The map $F_m$ in the hint is continuous because it is a finite-dimensional polynomial map. Since the velocity $v=\nabla^\perp\Phi$ is divergence-free,
$$
\begin{aligned}
\nu(F_m(w),w)_1
&=-\nu\|\nabla w\|_2^2-\gamma\|w\|_2^2
-((v\mathbin\cdot\nabla)w,w)+(g,w)\\
&=-\nu\|\nabla w\|_2^2-\gamma\|w\|_2^2+(g,w).
\end{aligned}
$$
The <Poincare inequality> bounds $\|w\|_2\leq\mu_1^{-1/2}\|\nabla w\|_2$. Thus $(F_m(w),w)_1<0$ on every $H^1$ sphere whose radius is larger than $\|g\|_2/(\nu\sqrt{\mu_1})$. The <Brouwer inward-pointing zero lemma> supplies $w_\ast$ inside that sphere with $F_m(w_\ast)=0$.
Set $\omega_m=w_\ast$, solve $-\Delta\Psi_m=\omega_m$ in $\widetilde H_m$, and put $u_m=\nabla^\perp\Psi_m$. Expanding $F_m(w_\ast)=0$ gives exactly the first equation of the Galerkin system, so $(\omega_m,\Psi_m,u_m)$ is a solution.
Solved by gpt-5.6-sol high.
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