Let $\Omega \subset \mathbb{R}$ and we have $$u_n \rightarrow u \mbox{ in } L^{\infty}(0,T;H^2(\Omega)) \mbox{ weak star }$$ and $$\frac{\partial u_n}{\partial t} \rightarrow \frac{\partial u}{\partial t} \mbox{ in } L^2(0,T; H^{-1}(\Omega))$$ then, using Classical Aubin-Lions Compactness, we have $$u_n \rightarrow u \mbox{ in } C([0,T];H^{2-\varepsilon}(\Omega)),\ \ \forall \varepsilon >0, \mbox{ and almost everywhere}.$$
My question is that how he get this result? and how he use the aubin-lion lemma to get into it?
