Regularity properties of passive scalars with rough divergence-free drifts

We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation [ partial_t theta - Delta theta + b cdot abla theta = 0 ] to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts $b$ belonging to $L^q_t L^p_x$, where $frac{2}{q} + frac{n}{p} < 2$, or $L^p_x L^q_t$, where $frac{3}{q} + frac{n-1}{p} < 2$. For steady drifts, the condition reduces to $b in L^{frac{n-1}{2}+}$. The space $L^1_t L^infty_x$ of drifts with `bounded total speed is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain `before they can be dissipated.