# Reference and hint for L^p estimates of the gradient of solutions to parabolic equation in divergence form

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!

• TeX notes: \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 \rming 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. Commented Jan 17, 2023 at 16:25

The first is a perturbative theorem assuming that the coefficient matrix $$A$$ is "flat," i.e. it has small oscillation over the domain. The conclusion is that the gradient is in $$L^p$$ for any given $$p$$ provided $$A$$ is flat enough (with the flatness depending on $$p$$). The flatness is always satisfied if $$A$$ is continuous in both variables (at small scales), and this is the most practical way to check the hypothesis. For the elliptic case, this theorem is due to Caffarelli and Peral. I'm less familiar with the parabolic literature but you can look at this paper of Peral and Soria.
If you do not know that $$A$$ is continuous, the above theorem cannot be applied (there are counterexamples in the elliptic case). However, it is still true for $$p - 2$$ sufficiently small (with the smallness depending on the ellipticity ratio). This is known as Gehring's Lemma, and involves combining local energy inequalities with some generic harmonic analysis principles. Again I am not very familiar with the parabolic literature, but there appear to be a number of papers by Arkhipova from the 90s on this topic.
Finally, under more strict assumptions on the coefficients, you may be able to use more standard regularity results. For example, if it is known that $$A \in C^{0, \alpha}$$, one obtains that the spacial derivatives are locally in $$C^{0, \alpha}$$ as well by Schauder theory, which is much better than the above.