A Question on the Use of Aubin-Lions Lemma I am now studying a paper about the construction of weak solutions, and I am not able to fully understand how the paper argues the convergence using the Aubin-Lions lemma:
For any $T > 0$, we have a sequence of approximate solutions $u_\epsilon \in L^2([0,T], H^1(\mathbb{R}^n))$ and $\partial_tu_\epsilon \in L^2([0,T], W^{-1,\frac{n}{n-1}}(\mathbb{R}^n))$. The paper argues that one can extract a subsequence (here WLOG we assume the original sequence does the job) such that $u_\epsilon \to u$ in $L^2([0,T],L^2_{loc}(\mathbb{R}^n))$.
To apply the Aubin-Lions lemma, we have to identify a chain of 3 spaces $X_0, X, X_1$ such that $X_0 \subset X$ compactly and $X \subset X_1$ continuously. Here I have identified those spaces as: $X_0 = H^1, X = L^2_{loc}, X_1 = W^{-1,\frac{n}{n-1}}$. It is clear that this holds true when $n=2$, and in the general case when $n \ge 2$, the compact embedding part follows from the Sobolev compact embedding theorem. Maybe I missed something pretty trivial, but it is not quite clear to me why $L^2_{loc}$ embeds in $W^{-1,\frac{n}{n-1}}$ continuously. Could someone give me a hint how to make sense of this? Thanks a lot in advance!
 A: 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.
