Compact embedding of fractional space

Question

Is the space $H^\lambda((0,T); H^1(K))$ for $0 <\lambda <1$ where $K$ is compact subset of $\mathbb{R}^n$ compactly embedded in $L^2( (0,T) \times K)$?

$H^\lambda((0,T); H^1(K))\hookrightarrow \hookrightarrow L^2((0,T) ; H^1(K))\hookrightarrow \hookrightarrow L^2((0,T); L^2(K))$. Where can I find these type of results.

Answer 1

There are (at least) two ways to prove $H^\gamma(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;L^2).$

1. $H^\gamma(0,T;H^1) \hookrightarrow H^\gamma(0,T;(H^1)') \cap L^2(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;L^2).$
2. $H^\gamma(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;H^1) \hookrightarrow L^2(0,T;L^2).$

Note that you don't get that $L^2(0,T;H^1) \hookrightarrow \hookrightarrow L^2(0,T;L^2)$ as stated in the question. This is not true. For example take the sequence $f_n(t,x)=\sin(nt)x$.

First: Ben Schweizer "Partielle Differentialgleichungen", Theorem 24.2

Let $V,X,V_0$ be Hilbert spaces such that $V \hookrightarrow \hookrightarrow X \hookrightarrow V_0$. Then for $T<\infty$ and $\gamma>0$ we have $$H^\gamma(0,T;V_0) \cap L^2(0,T;V) \hookrightarrow \hookrightarrow L^2(0,T;X).$$

Second: You get $H^\gamma(0,T) \hookrightarrow \hookrightarrow L^2(0,T)$ from the fractional Sobolev embedding.