On divergence-free (form-bounded type) drifts

TL;DR

This paper develops a regularity theory for elliptic Kolmogorov operators with divergence-free or singular divergence drifts, employing advanced iterative methods to handle complex drift behaviors.

Contribution

It introduces novel regularity results for elliptic Kolmogorov operators with divergence-free drifts, utilizing combined Caccioppoli, De Giorgi, and Moser iteration techniques.

Findings

Established regularity results for operators with divergence-free drifts.

Extended the theory to include drifts with singular divergence.

Applied advanced iterative methods to prove regularity.

Abstract

We develop regularity theory for elliptic Kolmogorov operator with divergence-free drift in a large class (or, more generally, drift having singular divergence). A key step in our proofs is "Caccioppoli's iterations", used in addition to the classical De Giorgi's iterations and Moser's method.