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arXiv · 2106.02408

On supercritical divergence-free drifts

Bian Wu
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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 $3$.

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Bian Wu. 2021-06-04. On supercritical divergence-free drifts. https://arxiv.org/abs/2106.02408

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