Weak continuity from evolution-space bounds

If $u\in L^{\infty }(0,T;H)$ and $u_t\in L^q(0,T;V^{\prime })$ for a dense continuous embedding $V\hookrightarrow H$ and some $q>1$, then $u$ has a representative in $C([0,T];H_{\mathrm{weak}})$. Indeed, its pairing with each element of a dense subset of $V$ is absolutely continuous, and the uniform $H$ bound extends continuity to every element of $H$.

If $u:[0,T]\to H$ is weakly continuous and an energy equality makes $t\mapsto \Vert u(t){\Vert }_H$ continuous, then $u$ is strongly continuous. This follows from the Radon-Riesz theorem applied whenever $t_n\to t$.