The uniform estimate gives a subsequence
$$
u_m\rightharpoonup u
\quad\hbox{weakly in }\widetilde V,
\qquad
\|u\|\leq R.
$$
The compact periodic embedding $\widetilde V\Subset\widetilde H$ improves this to
$$
u_m\to u
\quad\hbox{strongly in }\widetilde H.
$$
One may also take strong convergence in $L^3$ by the <Rellich-Kondrachov compactness theorem>. Combining this with weak convergence of the gradients and using the skew form when derivatives must be moved to a test function gives
$$
\widetilde B(u_m,u_m)
\rightharpoonup\widetilde B(u,u)
\quad\hbox{in }\widetilde V'.
$$
For any test function in a fixed finite-dimensional subspace, the Galerkin equation therefore passes to the limit. Density then gives
$$
\boxed{
\nu\widetilde Au+\widetilde B(u,u)=f
\quad\hbox{in }\widetilde V',
\qquad
\|u\|\leq R}.
$$
This is a weak solution of the <steady skew-symmetrized Navier-Stokes equation>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 359 / 2 / c / i / Solution

The steady equation $\nu Au+\widetilde B(u,u)=f$ retains the energy cancellation without imposing incompressibility. Galerkin bounds, compactness, and elliptic bootstrapping produce an $H^2$ periodic solution for square-integrable forcing.


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