We address the regularity of solutions to elliptic and parabolic equations of the form -δu + b - ∇u = 0 and ut - Δu + b - ∇u = 0 with divergence-free drifts b. We are particularly interested in the case when the drift velocity b is assumed to be at the supercritical regularity level with respect to the natural scaling of the equations. Using Harnack-type inequalities obtained in our previous works  and , we prove the uniform continuity of solutions when the drift b lies in a slightly supercritical logarithmic Morrey spaces.
All Science Journal Classification (ASJC) codes
- Drift-diffusion equations
- Harnack inequality