On Divergence-free Drifts

2010/10/28 by G. Serëgin, Seregin, Gregory, Luís Silvestre +5 · 4 citations
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1010.6025

openalex publication_date 2010/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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⋅∇ and -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 to such results 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.

Citations

• A new proof of Moser's parabolic harnack inequality using the old ideas of Nash

Cited by

• Existence and regularity for perturbed Stokes system with critical drift in 2D
• Pointwise bounds on Dirichlet Green's functions for a singular drift term
• The parabolic Dirichlet problem with continuous and Hölder boundary data, and rough coefficients
• Conditional Liouville theorems for the Navier-Stokes equations

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