  For second-order elliptic or parabolic equations with subcritical or critical
drifts, it is well-known that the Harnack inequality holds and their bounded
weak solutions are H\"older continuous. We construct time-independent
supercritical drifts in $L^{n-\lambda}(\mathbb{R}^n)$ with arbitrarily small
$\lambda>0$ such that the Harnack inequality and the H\"older continuity fail
in both the elliptic and the parabolic cases, thus confirming a conjecture by
Seregin, Silvestre, Sverak and Zlatos. These results are sharp, and they also
apply to a toy model of the axi-symmetric Navier-Stokes equations in space
dimension $3$.
