On divergence-free drifts

By G A Seregin, Luis Silvestre, Vladimír Sverák, Andrej Zlatoš. 2010

137 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/94df524acab2c41b07e08c880cc33c5095156f0a

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

On a transport equation with nonlocal drift (2014, result): “The above estimate is in direct analogy with the fact that in 2D, H 1 functions that obey the maximum principle on every ball have a logarithmic modulus of continuity [39]. so that all terms in the distributional deﬁnition of the weak solution are well-deﬁned.”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, methodology): “1)), if the drift u is assumed to be divergence-free and in the critical space L∞(BMO−1), then one can obtain a Hölder estimate on the solution θ by extending the methods of De Giorgi, Nash, and Moser (see [23] or [31]).”