I am trying to show that $u$ is a weak solution to the following Navier-Stokes equations with partial dissipation \begin{equation} \begin{cases} \partial_tu+(u\cdot\nabla)u=-\nabla p+\nu\partial_{22} u, \\ \nabla\cdot u=0, \end{cases} \end{equation} So I have a sequence $u^n$ satisfying the following two condition:
$u^n$ is uniformly bounded in $X$ and it converges weakly to a limit $u$ in $X$,
$\partial_t u^n$ is uniformly bounded in $Y$ and it also converges weakly to $\partial_t u$ in $Y$, where $X \subset Y$ are two complete spaces and $T>0$.
My question is: why do we need to extract another subsequence of the sequence $u^n$ that converges strongly to u in order to show that $u$ is a weak solution of the above system?
Thank you very much.