Interior pointwise $C^{\alpha}$ regularity for elliptic and parabolic equations with divergence-free drifts

Yuanyuan Lian

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arXiv · 2402.18161

Interior pointwise $C^{\alpha}$ regularity for elliptic and parabolic equations with divergence-free drifts

Yuanyuan Lian

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Abstract

We investigate the interior pointwise $C^{\alpha}$ regularity for weak solutions of elliptic and parabolic equations with divergence-free drifts. For such equations, the integrability condition on the drift can be relaxed and the interior $C^{\alpha}$ regularity for some $0<\alpha<1$ has been obtained previously with the aid of Harnack inequality. In this paper, we prove the interior pointwise $C^{\alpha}$ regularity for any $0<\alpha<1$ provided that the drift is small. We obtain the regularity under three different types conditions on the drift. The proof is based on the energy inequality and the perturbation technique.

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Yuanyuan Lian. 2024-02-28. Interior pointwise $C^{\alpha}$ regularity for elliptic and parabolic equations with divergence-free drifts. https://arxiv.org/abs/2402.18161

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