# $L^2(S;W^{1,p})$-regularity for solution of parabolic pde in Hilbert space setting

I have a question regarding the regularity theory for parabolic pdes. I am considering the following "minimal working example":

Let $\Omega\subset\mathbb{R}^n$, $n\in\mathbb{N}$, be and bounded domain and $S=(0,T)$, $T>0$, a time interval. Suppose that $A:H_0^1(\Omega)\to H^{-1}(\Omega)$ is a linear second-order elliptic operator, $f\in L^2(S;H^{-1}(\Omega))$ and $u_0\in L^2(\Omega)$ and consider the following Cauchy problem

$$u^\prime(t) +Au(t)=f(t)\qquad\text{in}\quad L^2(S;H^{-1}(\Omega)),\\ u(0)=u_0 \qquad\text{in}\quad L^2(\Omega).$$

This problem is well-posed so we have a unique solution $u\in L^2(S;H_0^1(\Omega))$ such that $u^\prime\in L^2(S;H^{-1}(\Omega))$.

My question(s): Are there any results that place $u$ in the smaller space $L^2(S;W^{1,p}(\Omega))$ for $p>2$ if, for example, $f\in L^2(S;L^{p}(\Omega))$ or $f\in L^2(S;W^{1,p}(\Omega))$? And if so, does anybody has a good reference for me? There are many regularity results concerning higher smoothness of $u$ if $f$, $u_0$ and $\partial\Omega$ are well-behaved. Are there similar regularity results concerning the summability of $u$ (and $\nabla u$)?

Update: What I have come up with so far:

1. There are many regularity results concerning higher smoothness of $u$ if $f$, $u_0$ and $\partial\Omega$ are well-behaved. Using Sobolev embeddings, e.g. the Kondrachov embedding theorem, one then could show $u\in L^2(S;W^{1,p}(\Omega))$ for any $p>2$ if $u$ is sufficient smooth. But I fear that is like breaking a butterfly on a wheel since actually we have $u$ in a larger space (hence the embedding)!

2. Maximal $L^p$-regularity: If $A$ has maximal $L^p$-regularity, we would have (by definition) $u\in L^p(S;H_0^1(\Omega))$ for all $f\in L^p(S;L^2(\Omega))$ and all $p\in(1,\infty)$. However, it gives no "extra" spatial summability and since I'm interested in exactly that I don't think that this concept is of help here!?