Calder\'on-Zygmund estimates for parabolic $p$-Laplacian systems with non-divergence form right-hand sides

Abstract

We establish local Calder\'on-Zygmund type estimates for weak solutions to nonlinear parabolic systems with $p$-growth and VMO coefficients. In particular, we prove that if the right-hand side belongs locally to $L^{\mu s}$, where the exponent $\mu$ depends explicitly on $p$, $N$, and a prescribed target exponent $s>p$, then the spatial gradient of the solution enjoys improved integrability $Du \in L^s_{\rm{loc}}$. The result provides a sharp transfer of integrability from the data to the gradient, consistent with the natural parabolic scaling, and recovers the optimal exponents in the linear case $p=2$. The proof combines intrinsic scaling techniques with a Calder\'on-Zygmund type iteration scheme.