On some properties of weak solutions to elliptic equations with divergence-free drifts

TL;DR

This paper investigates the local properties of weak solutions to elliptic equations with divergence-free drift terms, especially focusing on the less-studied case where the drift is in L2, highlighting the importance of the divergence-free condition.

Contribution

It extends the theory of weak solutions to elliptic equations with divergence-free drifts from the well-understood L_n case to the less-explored L_2 case, emphasizing the role of divergence-free condition.

Findings

Divergence-free condition is crucial for L_2 drifts.

Properties of solutions differ significantly from the L_n case.

The structure of solutions is better understood under divergence-free assumption.

Abstract

We discuss the local properties of weak solutions to the equation $-\Delta u + b\cdot\nabla u=0$. The corresponding theory is well-known in the case $b\in L_n$, where $n$ is the dimension of the space. Our main interest is focused on the case $b\in L_2$. In this case the structure assumption $\operatorname{div} b=0$ turns out to be crucial.