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In proving the weak solution for Navier-Stokes equations, one can conclude from the estimates $$u_n \in L^\infty(0,T;L^2(\Omega)) \cap L^2(0,T;H^1(\Omega)) \quad \partial_t u_n \in L^p(0,T;H^{-1}(\Omega))$$ that a subsequence $u_{n_k}$ of $u_n$ converge as $$u_{n_k} \rightharpoonup^* u \quad \text{in } L^\infty(0,T;L^2(\Omega))$$ $$u_{n_k} \rightharpoonup u \quad \text{in } L^2(0,T;H^1(\Omega))$$ $$u_{n_k} \rightarrow u \quad \text{in } L^2(0,T;L^2(\Omega))$$

I have known that the last two convergence result come from the Banach-Alaoglu lemma and the Aubin-Lions theorem. But why does the first hold?

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For $p\ge2$ the functions $u_n$ are uniformly bounded in $C([0,T],L^2(\Omega))$ as well by continuous embeddings. Thus, they are bounded in $L^\infty(0,T;L^2(\Omega))$, which is the dual space of the separable space $L^1(0,T;L^2(\Omega))$. Then one can apply Banach-Alaoglu again.

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