A regularity result for elliptic equations with low-regularity divergence-free coefficients

We study the equation $-\mathrm{\Delta }u+\mathbf{b}\cdot \mathrm{\nabla }u=f$ for divergence-free vector fields $\mathbf{b}$ with low regularity. Classical theory for general (not necessarily divergence-free) $\mathbf{b}$ requires at least $\mathbf{b}\in L^n$. It has been known for some time that for divergence-free coefficients, one can get results when $\mathbf{b}\in L^{n/2+ϵ}$. We show that for $n\ge 5$ one can even go below the exponent $n/2$, and $\mathbf{b}\in L^{(n-1)/2+ϵ}$ is sufficient.