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Interior pointwise $C^α$ regularity for elliptic and parabolic equations with divergence-free drifts (2402.18161v1)
Published 28 Feb 2024 in math.AP
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.