VARIATION ON A THEME BY KISELEV AND NAZAROV: HÖLDER ESTIMATES FOR NONLOCAL TRANSPORT-DIFFUSION, ALONG A NON-DIVERGENCE-FREE BMO FIELD. (8th September 2022)

• Record Type: Journal Article
• Authors: Vasilyev, Ioann; Vigneron, François
• Is Part Of: Journal of the Institute of Mathematics of Jussieu. Volume 21:Number 5(2022)
• Page Start: 1651
• Page End: 1675
• Publication Date: 2022-09-08
• Subjects: Transport-diffusion -- Hölder regularity -- fractional diffusion equation -- nonlocal operator -- BMO drift -- dual equation -- conservation law -- functional analysis -- atoms; 35B65 -- 35R11 -- 35Q35; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S1474748020000699
• Languages: English
• ISSNs: 1474-7480

Abstract

Abstract: We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator and a general advection field in of bounded mean oscillation, as long as the order of the diffusion dominates the transport term at small scales; our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by Silvestre, our advection field does not need to be bounded. A similar result can be obtained in the supercritical case if the advection field is Hölder continuous. Our proof is inspired by Kiselev and Nazarov and is based on the dual evolution technique. The idea is to propagate an atom property (i.e., localisation and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.