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

TL;DR

This paper proves local Calderón-Zygmund estimates for nonlinear parabolic systems with p-growth, showing that the gradient's integrability improves based on the data's integrability, extending classical results to a nonlinear setting.

Contribution

It establishes sharp Calderón-Zygmund estimates for nonlinear parabolic systems with VMO coefficients, including explicit conditions for gradient integrability enhancement.

Findings

Gradient of solutions belongs to L^s_loc if data is in L^{μs}

Recovers optimal exponents in the linear case p=2

Uses intrinsic scaling and Calderón-Zygmund iteration techniques

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.