A note on local $W^{1,p}$-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts

TL;DR

This paper establishes local weighted Sobolev regularity estimates for weak solutions of parabolic equations with divergence-free drifts, extending classical results to borderline cases using BMO space assumptions.

Contribution

It improves existing regularity results by replacing boundedness assumptions with BMO space conditions and generalizes to equations with small oscillation coefficients.

Findings

Established local weighted $L^p$-estimates for gradients of solutions.

Extended regularity results to borderline cases with BMO solutions.

Generalized results to equations with small oscillation elliptic coefficients.

Abstract

We investigate weighted Sobolev regularity of weak solutions of non-homogeneous parabolic equations with singular divergence-free drifts. Assuming that the drifts satisfy some mild regularity conditions, we establish local weighted $L^p$-estimates for the gradients of weak solutions. Our results improve the classical one to the borderline case by replacing the $L^\infty$-assumption on solutions by solutions in the John-Nirenberg \textup{BMO} space. The results are also generalized to parabolic equations in divergence form with small oscillation elliptic symmetric coefficients and therefore improve many known results.