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

We study the equation $-\Delta u + \mathbf{b}\cdot\nabla u = f$ 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 $\mathbf{b}\in L^{n/2 +\epsilon}$. We show that for $n \geq 5$ one can even go below the exponent $n/2$, and $\mathbf{b}\in L^{(n-1)/2 +\epsilon}$ is sufficient.