Regularity Properties of Degenerate Diffusion Equations with Drifts

TL;DR

This paper investigates the regularity of solutions to nonlinear degenerate drift-diffusion equations, establishing conditions on the drift's integrability for uniform Hölder regularity and demonstrating the necessity of certain bounds through examples.

Contribution

It provides new regularity estimates for degenerate diffusion equations with drifts, extending known thresholds and illustrating the sharpness of these bounds with examples.

Findings

Hölder regularity holds for drifts with $L^p$ bounds when $p > d + 4/(d+2)$

Examples show $p > d$ is necessary for regularity, even with divergence-free drifts

Scaling arguments identify the critical threshold at $p = d$

Abstract

This paper considers a class of nonlinear, degenerate drift- diffusion equations. We study well-posedness and regularity properties of the solutions, with the goal to achieve uniform H\"{o}lder regularity in terms of $L^p$-bound on the drift vector field. A formal scaling argument yields that the threshold for such estimates is $p=d$, while our estimates are for $p>d+\frac{4}{d+2}$. On the other hand we are able to show by a series of examples that one needs $p>d$ for such estimates, even for divergence free drift.