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 $$\partial_t \theta - \Delta \theta + b \cdot \nabla \theta = 0$$ 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.