On divergence-free drifts - Seregin, G; Silvestre, L; Sverak, V; Zlatos, A (2012)
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.