Vol. 293, No. 2, 2018

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ISSN: 0030-8730
On the differentiability issue of the drift-diffusion equation with nonlocal Lévy-type diffusion

Liutang Xue and Zhuan Ye

Vol. 293 (2018), No. 2, 471–510
DOI: 10.2140/pjm.2018.293.471
Abstract

We investigate the differentiability property of the drift-diffusion equation with nonlocal Lévy-type diffusion at either supercritical- or critical-type cases. Under the suitable conditions on the velocity field and the forcing term in terms of the spatial Hölder regularity, and for the initial data without regularity assumption, we show the a priori differentiability estimates for any positive time. If additionally the velocity field is divergence-free, we also prove that the vanishing viscosity weak solution is differentiable with some Hölder continuous derivatives for any positive time.

Keywords
Drift-diffusion equation, differentiability, Lévy-type operator, fractional Laplacian operator, smoothness
Mathematical Subject Classification 2010
Primary: 35B65, 35Q35, 35R11, 35K99
Milestones
Received: 1 March 2017
Revised: 30 August 2017
Accepted: 11 September 2017
Published: 23 November 2017
Authors
Liutang Xue
School of Mathematical Sciences
Beijing Normal University
Laboratory of Mathematics and Complex Systems
Ministry of Education
Beijing
China
Zhuan Ye
Department of Mathematics and Statistics
Jiangsu Normal University
Xuzhou
China