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VARIATION ON A THEME BY KISELEV AND NAZAROV: HÖLDER ESTIMATES FOR NONLOCAL TRANSPORT-DIFFUSION, ALONG A NON-DIVERGENCE-FREE BMO FIELD

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  08 January 2021

Ioann Vasilyev  and
François Vigneron Open the ORCID record for François Vigneron [Opens in a new window]
Ioann Vasilyev
Affiliation:
Université Paris-Est, LAMA, UMR 8050, UPEC, UPEM, CNRS, 61, avenue du Général de Gaulle, CréteilF94010, France ([email protected])
François Vigneron
Affiliation:
Université de Reims Champagne-Ardenne, Laboratoire de Mathématiques de Reims, UMR 9008, Moulin de la Housse, BP 1039, Reims cedex251687 ([email protected])
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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.

Keywords

Transport-diffusionHölder regularityfractional diffusion equationnonlocal operatorBMO driftdual equationconservation lawfunctional analysisatoms

MSC classification

Primary: 35B65: Smoothness and regularity of solutions
Secondary: 35R11: Fractional partial differential equations 35Q35: PDEs in connection with fluid mechanics
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 5 , September 2022 , pp. 1651 - 1675
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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