On supercritical divergence-free drifts

TL;DR

This paper constructs specific supercritical divergence-free drifts that demonstrate the failure of Harnack inequality and Hölder continuity in elliptic and parabolic equations, confirming a conjecture and highlighting sharpness of previous results.

Contribution

It provides explicit examples of supercritical drifts where classical regularity results fail, confirming a conjecture and extending to models related to Navier-Stokes equations.

Findings

Harnack inequality fails for certain supercritical drifts

Hölder continuity fails under these conditions

Results are sharp and applicable to Navier-Stokes models

Abstract

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 $3$.