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