On the loss of continuity for super-critical drift-diffusion equations

TL;DR

This paper demonstrates that solutions to two-dimensional super-critical drift-diffusion equations can become discontinuous in finite time, but under certain conditions, solutions maintain a specific modulus of continuity.

Contribution

It establishes the existence of finite-time discontinuities for solutions with divergence-free super-critical drifts and identifies conditions under which solutions remain continuous.

Findings

Solutions can become discontinuous in finite time.

Solutions satisfy a modulus of continuity depending on local $L^1$ norm of the drift.

Classical diffusion with time-independent drifts preserves certain continuity properties.

Abstract

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 $L^1$ norm of the drift, which is a super-critical quantity.