I am trying to study an example on how to apply Aubin-lions Lemma, but i dont get why some things happen, so first, i state the lemma
(Aubin-Lions) Let $E_{0} \subset E \subset E_{1}$ be three Banach spaces, with compact embedding $E_{0} \subset E$ (bounded sets of $E_{0}$ are relatively compact in $E$ ) and continuous embedding $E \subset E_{1} .$ Let $p, q \geq 1 .$ Then $$ L^{p}\left(0, T ; E_{0}\right) \cap W^{1, q}\left(0, T ; E_{1}\right) $$ is compactly embedded into $$ L^{p}(0, T ; E) $$
And the PDE where i want to apply the lemma is
\begin{aligned} \frac{\partial u}{\partial t} &=\Delta u+b(u) \cdot \nabla u+c(u) \\ \left.u\right|_{t=0} &=u_{0},\left.\quad u\right|_{\partial D}=0 \end{aligned}
where D is an open Domain in the reals. So i can find that for a possible weak solution $u_n$, $u_n$ is bounded in $L^2(0,T:H^1(D))$ and $\frac{du_{n}}{dt}$ is bounded in $L^2(0,T:H^{-1}(D))$. So after this the notes that i'm studying conclude the following
Using Aubin lemma we have that $$ E_{0}=W^{1,2}(D), E=L^{2}(D), E_{1}=W^{-1,2} $$ and use in essential way the compactness of the embedding $W^{1,2}(D) \subset L^{2}(D)$ (Rellich theorem). We deduce that $\left\{u_{n}\right\}$ is relatively compact in $L^{2}\left(0, T ; L^{2}(D)\right)$ in the strong topology (this is the important information in order to pass to the limit in nonlinear terms). The sequence is also weakly relatively compact in $L^{2}\left(0, T ; W^{1,2}(D)\right)$ and weak star relatively compact in $C\left([0, T] ; L^{2}(D)\right) .$ Then there exists a subsequence which converges in all these three topologies to a function $u$ which is then of class $L^{2}\left(0, T ; W^{1,2}(D)\right)$ and $C\left([0, T] ; L^{2}(D)\right) .$ Thanks to the strong convergence in $L^{2}\left(0, T ; L^{2}(D)\right)$ one can pass to the limit and prove that $u$ is a solution.
So, first of all i would like to know what weakly relatively compact means.
Second, how do we know that $u_n$ is weak star relatively compact in $C\left([0, T] ; L^{2}(D)\right) .$
And, how can we conlude that there exist a subsequence which converges in all these three topologies (what topologies does the author mean) to a function $u$ which is then of class $L^{2}\left(0, T ; W^{1,2}(D)\right)$ and $C\left([0, T] ; L^{2}(D)\right) .$
Thank you in advance and i really hope u can help me.