I know by Rellich-Kondrachov theorem, $H^1(\Omega)$ is compactly embedded $L^2(\Omega)$. Also, $L^2(0,T;H^1(\Omega))$ is continuously embedded in $L^2(0,T;L^2(\Omega))$. However, is this embedding compact? Is there a reference related to it?
Is L^2(0,T;H^1(\Omega)) compactly embedded in L^2(0,T;L^2(\Omega))
compactnessfunctional-analysissobolev-spaces
Related Solutions
Yes, such Hilbert spaces exist and they are a special case of fractional Sobolev spaces. For $\alpha\in(0,1/2)$ we have $H^\alpha(0,1)\subset L^2(0,1)$ by definition, and one can show that the step function which is $1$ on $(1/2,1)$ and $0$ else is in $H^\alpha(0,1)$. Since this function is not continuous, $H^\alpha(0,1)$ does not embed in $C^0[0,1]$.
See also Proof that the characteristic function of a bounded open set is in $H^{\alpha}$ iff $\alpha < \frac{1}{2}$ and To what fractional Sobolev spaces does the step function belong? (Sobolev-Slobodeckij norm of step function) for more details.
It is also known that $H^\alpha(0,1)$ embeds compactly into $L^2(0,1)$ for $\alpha\in (0,1/2)$. This follows from Theorem 7.1 in this pdf.
Yes, the reasoning is correct, but you need $\Omega$ to be a Lipschitz domain, otherwise there are counterexamples where we fail even the embedding. Assuming a Lipschitz boundary means you assume "some regularity" of the boundary.
Moreover, note how important is that $\Omega$ is bounded for these inclusions, otherwise all the "nested" spaces are no more nested and the reasoning fails: indeed give a look here where $\Omega$ is not bounded to see that your Sobolev embedding fails. Hence, your claim $W^{1,n}(\Omega) \hookrightarrow W^{1,q}(\Omega)$ is correct since $q \le n$, and actually note that another interesting question is if we can gain some integrability conditions by adding some "differentiability condition": see my answer here as well.
Let me add some references in cases you wanted to go deeper with some embeddings result: “Giovanni Leoni - A first course in Sobolev Spaces” “Adams - Sobolev Spaces”.
Other two referenes, if you are interested in Sobolev Spaces from the point of view of PDEs then:
Brezis’ book Functional Analysis, Sobolev Spaces and Partial Differential Equations (mainly chapter 8 of the second edition)
Evans’ book Partial Differential Equations (mainly chapter 5 of the second edition).
Best Answer
By considering the set of constant functions $\Omega \ni x \mapsto c$ (which belong to $H^1$ and $L^2$) show that both of your spaces "contain a copy of $L^2(0,T)$". Since the embedding $L^2(0,T) \hookrightarrow L^2(0,T)$ is not compact, your embedding cannot be compact either.