The regularity from part (i) makes $u\omega$ and its first weak derivatives square-integrable. The product rule therefore holds in the distributional derivative sense:Since $u={\nabla }^{\perp }\Psi$, equality of mixed weak derivatives gives $\nabla \cdot u=0$. Both remaining expressions belong to $L_{\mathrm{per}}^2$, so their distributional equality is an equality in that space:

The second estimate in part (a) bounds $({\omega }_m)$ in $H_{\mathrm{per}}^2$. The periodic Poisson equation $-\Delta {\Psi }_m={\omega }_m$ and the supplied curl identity then bound $({\Psi }_m)$ in $H_{\mathrm{per}}^4$ and $(u_m)$ in $H_{\mathrm{per}}^3$. After passing to a subsequence,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 $L^2$, soPassing to the limit in the Galerkin equations givesThese identities have the claimed Sobolev regularity, and the first holds in $L_{\mathrm{per}}^2$. Finally, the weak lower semicontinuity of the Hilbert norm preserves the estimates

Use on ${\tilde{H}}_m$ 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,The Poincare inequality bounds $\Vert w{\Vert }_2\le {\mu }_1^{-1/2}\Vert \nabla w{\Vert }_2$. Thus $(F_m(w),w{)}_1<0$ on every $H^1$ sphere whose radius is larger than $\Vert g{\Vert }_2/(\nu \sqrt{{\mu }_1})$. 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 ${\tilde{H}}_m$, 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.

Take the $L^2$ inner product of the first Galerkin equation with ${\omega }_m$. The orthogonal projection may be removed against ${\omega }_m\in {\tilde{H}}_m$, and skew-symmetry of incompressible transport cancels the nonlinear term. HenceThe Cauchy-Schwarz inequality and Young inequality giveThus both quantities requested in the first estimate are bounded by, for example,

Next take the inner product with $-\Delta {\omega }_m$. Periodicity and incompressibility giveThe given curl identity and the two-dimensional Gagliardo-Nirenberg inequality implyApplying Young's inequality to this term and to $(g,-\Delta {\omega }_m)$ yieldsThe first estimate bounds the right-hand side independently of $m$. Therefore one may choose a constant $R_2=R_2(\Vert g{\Vert }_2,\gamma ,\nu ,{\mu }_1)$ such that