Considering a weak solution $u\in L^2(0,1;H^1(B_1))$ with $\partial_t u \in L^2(0,1;H^{-1}(B_1))$ to $$\partial_t u-\operatorname{div}(A(x,t)\nabla u)=f+\operatorname{div}(F) \hspace0.5cm \text{in} \hspace0.5cm B_1\times(0,1);$$ where $B_1\subset\mathbb{R}^n$, $A$ is a uniformly elliptic matrix in space variable $x$ (i.e. $\lambda\lvert\xi\rvert^2\le A(x,t)\xi\cdot\xi\le\Lambda\lvert\xi\rvert^2$ for a.e. $t$, where $0<\lambda\le\Lambda$), $F$ is a field and $f$ a function. Namely, $u$ satisfies $$\int_0^1\langle\partial_t u,\phi\rangle dt+\int_{B_1\times(0,1)}\nabla u\cdot\nabla\phi =\int_{B_1\times(0,1)}f\phi+\int_{B_1\times(0,1)}F\cdot\nabla\phi,$$ for all $\phi\in C_c^\infty(B_1\times[0,1))$. Remarking that $\langle,\rangle$ denotes the pairing between $H^{-1}(B_1)$ and $H_0^1(B_1)$.

I want an $L^p$ estimate for the gradient of $u$, with $p>2$. I wonder what the minimum assumptions will be on $f$, $F$ and $A$ to obtain $$\lVert\nabla u\rVert_{L_\text{loc}^2(0,1;L_\text{loc}^p(B_1))}\le C,$$ where $C$ will depends on $A$, $F$, $f$. There is an abuse of notation in the last inequalities! It means for every compact subset of $B_1\times(0,1)$.

Any references or suggestions to prove this are welcome!

`\langle\rangle`

is preferable to`<>`

for pairings (note $1 + \langle u, \phi\rangle$ versus $1 + <u, \phi>$); and`\rm`

runs to the end of its group, so you must use`{\rm …}`

, not`\rm{…}`

, to avoid`\rm`

ing the rest of the line (note $\rm{div}(F)$`\rm{div}(F)`

versus ${\rm div}(F)$`{\rm div}(F)`

). But really it should be`\mathrm`

, or, even better in this case,`\operatorname`

; note the manual spacing needed in ${\rm div\ }F$`{\rm div\ }F`

versus $\operatorname{div} F$`\operatorname{div} F`

. I have edited accordingly. $\endgroup$