On the continuity of solutions to advection-diffusion equations with slightly super-critical divergence-free drifts

We address the regularity of solutions to elliptic and parabolic equations of the form -Delta u + b . del u = 0 and u(t) - Delta u + b . del 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 [7] and [8], we prove the uniform continuity of solutions when the drift lies in a slightly supercritical logarithmic Morrey spaces.