Gradient estimates for parabolic nonlinear nonlocal equations

TL;DR

This paper derives pointwise gradient estimates for solutions to a broad class of parabolic nonlinear nonlocal equations, revealing their regularity properties resemble those of the fractional heat equation, with implications for gradient continuity.

Contribution

It introduces new pointwise gradient estimates for nonlinear nonlocal parabolic equations using caloric Riesz potentials, advancing understanding of their regularity.

Findings

Solutions have gradient estimates in terms of caloric Riesz potentials.

First-order regularity of solutions resembles fractional heat equation.

Gradients are H"older continuous under optimal tail conditions.

Abstract

The primary objective of this work is to establish pointwise gradient estimates for solutions to a class of parabolic nonlinear nonlocal measure data problems, expressed in terms of caloric Riesz potentials of the data. As a consequence of our pointwise estimates, we obtain that the first-order regularity properties of solutions to such general parabolic nonlinear nonlocal equations, both in terms of size and oscillations of the spatial gradient, closely resemble the ones of the fractional heat equation even at highly refined scales. Along the way, we show that solutions to homogeneous parabolic nonlinear nonlocal equations have H\"older continuous spatial gradients under optimal assumptions on the nonlocal tails.