Borderline gradient regularity estimates for quasilinear parabolic systems with data independent of time

TL;DR

This paper investigates the regularity of gradients in quasilinear parabolic systems with time-independent data, establishing optimal conditions for higher integrability and Lipschitz continuity, and linking elliptic and parabolic estimates for improved potential bounds.

Contribution

It introduces new borderline regularity criteria for gradients in parabolic systems with data independent of time, bridging elliptic and parabolic theories for enhanced estimates.

Findings

Established optimal conditions for higher gradient integrability.

Derived Lipschitz estimates for weak solutions.

Connected elliptic and parabolic potential estimates.

Abstract

In this paper, we study some regularity issues concerning the gradient of weak solutions of $u_t - {\rm div} \mathcal{A}(x,t,\nabla u) = g$, where $\mathcal{A}(x,t,\nabla u)$ is modeled after the $p$-Laplace operator. The main results we are interested in is to obtain optimal conditions on the datum $g$ (independent of time) such that borderline higher integrability of the gradient and Lipschitz estimates for the weak solution holds. Moreover, we develop a theory where we can obtain elliptic type estimates using parabolic theory, which gives improved potential estimates for the elliptic systems.