Good day!


Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.


Consider the space
$W = \{ y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V') \}$.


It is well-known that $W \subset C([0,T];H)$ where $H = L^2(\Omega)$.


My question: for what $\alpha$ and $\beta$ we may assert that $W \subset L^\alpha(0,T;L^\beta(\Omega))$?


For example, $W \subset L^\infty(0,T;L^2(\Omega))$ and $W \subset L^2(0,T;L^6(\Omega))$ because $H^1(\Omega) \subset L^6(\Omega)$.


May we assert that $W \subset L^8(0,T;L^{24/5}(\Omega))$?


Thank you!



Embedding to $L^\alpha(0,T;L^\beta(\Omega))$
This is an interpolation problem.
$$
L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;L^6(\Omega))
\hookrightarrow L^{2/\theta}(0,T;L^{1/[(1-\theta)/2+\theta/6]}(\Omega)) ,
$$  

for any $\theta\in(0,1)$. You can check that this space coincides with 
$L^2(0,T;L^6(\Omega))$ when $\theta=1$, and this space coincides with 
$L^\infty(0,T;L^2(\Omega))$ when $\theta=0$.


