On divergence-free drifts

Author: 

Seregin, G
Silvestre, L
Šverák, V
Zlatoš, A

Publication Date: 

1 January 2012

Journal: 

Journal of Differential Equations

Last Updated: 

2020-07-18T04:04:42.64+01:00

Issue: 

1

Volume: 

252

DOI: 

10.1016/j.jde.2011.08.039

page: 

505-540

abstract: 

We investigate the validity and failure of Liouville theorems and Harnack inequalities for parabolic and elliptic operators with low regularity coefficients. We are particularly interested in operators of the form ∂t-Δ+b-∇ resp. -Δ+b-∇ with a divergence-free drift b. We prove the Liouville theorem and Harnack inequality when b∈L∞(BMO-1) resp. b∈BMO-1 and provide a counterexample demonstrating sharpness of our conditions on the drift. Our results generalize to divergence-form operators with an elliptic symmetric part and a BMO skew-symmetric part. We also prove the existence of a modulus of continuity for solutions to the elliptic problem in two dimensions, depending on the non-scale-invariant norm ||b||L1. In three dimensions, on the other hand, bounded solutions with L1 drifts may be discontinuous. © 2011 Elsevier Inc.

Symplectic id: 

196153

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article