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

Advances in Nonlinear Analysis

article

Mihaela Ignatova¹ 

[1] Department of Mathematics, Stanford University

关键词: Harnack inequality;    regularity;    drift-diffusion equations;   

DOI  :  10.1515/anona-2013-0031

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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.

【 授权许可】

CC BY