How to prove that this sequence has a convergent subsequence in some $L^p(\Omega)$

Question

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with $C^{\infty}$boundary, $T$ is a positive number, sequence $\{u_n(t,x)\}_{n=1}^{n=\infty}$ is the bounded sequence in Abstract Sobolev Space $L^{\infty}(0,T;L^2(\Omega))$, at the same time $\{u_{nt}(t,x)\}_{n=1}^{n=\infty}$ is the bounded sequence in $L^{\infty}(0,T;H^{-1}(\Omega))$, is there any theorem that warrants the validity of :
$$\exists p\in[1,\infty),\exists u\in L^{\infty}(0,T,L^p(\Omega))\quad s.t.\quad u_{n_k}\rightarrow u\quad in\ L^{\infty}(0,T,L^p(\Omega)) $$ where $\{u_{n_k}\}_{k=1}^{k=\infty}$ is a subsequence of $\{u_n\}_{n=1}^{n=\infty}$

Answer 1

No. Take a sequence $v_k \in L^2(\Omega)$ that does not converge in $L^1$. Then define $u_k(t,x)=v_k(x)$. It has all the properties as your sequence, but does not converge in $L^\infty(0,T;L^1(\Omega))$.

In order to get the desired strong convergence you need in addition that $u_n$ is bounded in some $L^2(0,T;H^1(\Omega))$. Search for Aubin-Lions theorem.