On divergence-free drifts
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 a...
| Auteurs principaux: | , , , |
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| Format: | Journal article |
| Langue: | English |
| Publié: |
2012
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| _version_ | 1826291973202378752 |
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| author | Seregin, G Silvestre, L Sverak, V Zlatos, A |
| author_facet | Seregin, G Silvestre, L Sverak, V Zlatos, A |
| author_sort | Seregin, G |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-07T03:07:33Z |
| format | Journal article |
| id | oxford-uuid:b3108ee2-36e4-4167-b16f-3aa3a09bd1a8 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:07:33Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:b3108ee2-36e4-4167-b16f-3aa3a09bd1a82022-03-27T04:16:22ZOn divergence-free driftsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b3108ee2-36e4-4167-b16f-3aa3a09bd1a8EnglishSymplectic Elements at Oxford2012Seregin, GSilvestre, LSverak, VZlatos, AWe 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. |
| spellingShingle | Seregin, G Silvestre, L Sverak, V Zlatos, A On divergence-free drifts |
| title | On divergence-free drifts |
| title_full | On divergence-free drifts |
| title_fullStr | On divergence-free drifts |
| title_full_unstemmed | On divergence-free drifts |
| title_short | On divergence-free drifts |
| title_sort | on divergence free drifts |
| work_keys_str_mv | AT sereging ondivergencefreedrifts AT silvestrel ondivergencefreedrifts AT sverakv ondivergencefreedrifts AT zlatosa ondivergencefreedrifts |