Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 359 / 1 / c

The uniform bounds and the Banach-Alaoglu theorem provide a subsequence, not relabelled, and a function $\theta$ such thatandSince $H_{\mathrm{per}}^1$ embeds compactly into $L_{\mathrm{per}}^2$, the Aubin-Lions lemma strengthens the first convergence toThe weak continuity from evolution-space bounds gives a representativeTesting against fixed spatial modes and using ${\theta }_m(0)={\Pi }_m{\theta }_0$ shows that this representative satisfies $\theta (0)={\theta }_0$ weakly.

DefineThe first equation and the bounded inverse of the Stokes operator show that $u\in L^{\infty }(0,T;D(A))$ and thatin $L^{\infty }(0,T;H)$. In fact, the strong convergence from part (i) and convergence of the spectral projections implyIn three dimensions $D(A)\subset H^2\hookrightarrow L^{\infty }$, so $u_m{\theta }_m\to u\theta$ in $L^1(0,T;L^2)$. Because $\nabla \cdot u_m=0$,and the nonlinear term consequently converges in distributions and in the required weak $L^2(0,T;H_{\mathrm{per}}^{-1})$ sense. The linear terms pass by weak convergence, while ${\Pi }_m$ tends strongly to the identity. Hencein $L^2(0,T;H_{\mathrm{per}}^{-1})$. Together with part (i), this proves existence of a global weak solution of the Rayleigh-Bénard convection system on every finite interval.