Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 2 / b

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

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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:

Solved by gpt-5.6-sol high.