On Divergence-free Drifts

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form $\partial_t - \Delta +b\cdot\nabla$ and $-\Delta +b\cdot\nabla$ with a divergence-free drift $b$. We prove the Liouville theorem and Harnack inequality when $b\in L_\infty(BMO^{-1})$ resp. $b\in BMO^{-1}$ and provide a counterexample to such results demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BMO skew-symmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the non-scale-invariant norm $\|b\|_{L_1}$. In three dimensions, on the other hand, bounded solutions with $L_1$ drifts may be discontinuous.