A Question on the Use of Aubin-Lions Lemma

Recall the Aubin-Lion lemma:

Theorem (Aubin-Lion). Let $X_0$, $X$, and $X_1$ be three Banach spaces with $X_0\subset X\subset X_1$. Suppose that $X_0$ is compactly embedded in $X$ and that $X$ is continuously embedded in $X_1$. Suppose that $1<p,q<\infty$ and $$ W =\{ u \in L^p ([0,T];X_0) : \partial_t u \in L^q ([0,T];X_1) \}. $$ Then $W$ is compactly embedded into $L^p (0,T;X)$.

To apply this theorem into your setting, note that the sequence $\{u_\varepsilon\}$ consists of elements in $$ W =\{ u \in L^2([0,T];H^1(\mathbb{R}^n)) : \partial_t u_\varepsilon \in L^2([0,T];W^{-1,n/(n-1)}(\mathbb{R}^n)) \}.$$

Hence for any open ball $B$ in $\mathbb{R}^n$, it is easy to see that $\{u_\varepsilon\}$ also belong to $$ W(B) =\{ u \in L^2([0,T];H^1(B)) : \partial_t u_\varepsilon \in L^2([0,T];W^{-1,n/(n-1)}(B)) \}.$$ By Rellich-Kondrachov's theorem, $H^1(B)$ is compactly embedded into $L^2(B)$. Also, by Holder's inequality and Sobolev embedding theorem, one can see that $L^2(B)$ is continuously embedded into $W^{-1,n/(n-1)}(B)$. Hence we can apply the Aubin-Lion lemma that there exists a subsequence $\{u_\varepsilon \}$ (still denote it by same notation) such that $u_{\varepsilon}\rightarrow u$ strongly in $L^2(0,T;L^2(B))$. Since $B$ was arbitrary chosen, one can show that there exists a subsequence of $\{u_\varepsilon\}$ that converges $L^2(0,T;L^2_{loc}(\mathbb{R}^n))$.

One may find this argument in establishing weak solution of the Navier-Stokes equation in whole space. If you need a further elaboration, let me know.