On the Loss of Continuity for Super-Critical Drift-Diffusion Equations

We show that there exist solutions of drift-diffusion equations in two dimensions with divergence-free super-critical drifts that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and time-independent drifts, we prove that solutions satisfy a modulus of continuity depending only on the local L¹ norm of the drift, which is a super-critical quantity.