Existence and regularity for perturbed Stokes system with critical drift

TL;DR

This paper investigates the existence and regularity of solutions to perturbed Stokes systems with divergence-free critical drift, providing new conditions under which gradient estimates hold in bounded Lipschitz domains.

Contribution

It introduces novel existence and regularity results for perturbed Stokes systems with critical divergence-free drift, including cases with small and non-small drift in weak $L^n$ spaces.

Findings

Existence of solutions under $L^n$ and small weak $L^n$ drift conditions.

Gradient estimates for $q$ close to 2 with non-small weak $L^n$ drift.

Results applicable to bounded Lipschitz domains in $ abla$-space.

Abstract

We consider the existence and $L^q$ gradient estimates for perturbed Stokes systems with divergence-free critical drift in a bounded Lipschitz domain in $\mathbb{R}^n$, $n \ge 3$. The first two results assume the drift is either in $L^n$ or sufficiently small in weak $L^n$. The third result assumes the drift is in weak $L^n$ without smallness, and obtain results for $q$ close to 2.