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

Abstract

We address the regularity of solutions to elliptic and parabolic equations of the form -Δu+b·∇u=0${- \Delta u+b\cdot \nabla u=0}$ and ut-Δu+b·∇u=0${u_t- \Delta u+b\cdot \nabla u=0}$ 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 [`The Harnack inequality for second-order elliptic equations with divergence-free drift', Commun. Math. Sci., to appear] and [`The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity', preprint (2013)], we prove the uniform continuity of solutions when the drift b lies in a slightly supercritical logarithmic Morrey spaces.

Keywords

• harnack inequality
• regularity
• drift-diffusion equations
• 35b65
• 35j15
• 35k10
• 35q35