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We present sharp conditions on divergence-free drifts in Lebesgue spaces forthe passive scalar advection-diffusion equation \[  \partial_t \theta - \Delta \theta + b \cdot \nabla \theta = 0 \] to satisfylocal boundedness, a single-scale Harnack inequality, and upper bounds onfundamental solutions. We demonstrate these properties for drifts $b$ belongingto $L^q_t L^p_x$, 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 conditionreduces to $b \in L^{\frac{n-1}{2}+}$. The space $L^1_t L^\infty_x$ of driftswith `bounded total speed' is a borderline case and plays a special role in thetheory. To demonstrate sharpness, we construct counterexamples whose goal is totransport anomalous singularities into the domain `before' they can bedissipated.

Regularity properties of passive scalars with rough divergence-free drifts