 Solution

Part (c)(i) already gives weak continuity of $t\mapsto \theta (t)$ in $L^2$. In the assumed energy equality, the dissipation integralis absolutely continuous. The forcing integrand is in $L^1(0,T)$ because $u\in L^{\infty }(0,T;L^2)$ and $\theta \in L^{\infty }(0,T;L^2)$. The equality therefore makes $t\mapsto \Vert \theta (t){\Vert }_2^2$ continuous.

Whenever $t_n\to t$, weak continuity gives $\theta (t_n)\rightharpoonup \theta (t)$ and the energy equality gives convergence of their norms. The Radon-Riesz theorem, or directly the strong continuity from weak continuity and an energy equality, now gives $\theta (t_n)\to \theta (t)$ in $L^2$. Thus

Solved by gpt-5.6-sol high.