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Article: On a generalized maximum principle for a transport-diffusion model with log-modulated fractional dissipation
Title | On a generalized maximum principle for a transport-diffusion model with log-modulated fractional dissipation |
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Authors | |
Keywords | Fractional dissipation Generalized maximum principle Nonlocal decomposition Nonlocal operators Transport-diffusion equations |
Issue Date | 2014 |
Citation | Discrete and Continuous Dynamical Systems- Series A, 2014, v. 34, n. 9, p. 3437-3454 How to Cite? |
Abstract | We consider a transport-diffusion equation of the form ∂tΦ + v·δ Φ + vAΦ = 0, where v is a given time-dependent vector field on Rd. The operator A represents log-modulated fractional dissipation: A = /δ/γ/logβ(λ /δ/) and the parameters v ≥ 0, β 0, 0 ≤ γ 2, λ > 1. We introduce a novel nonlocal decomposition of the operator A in terms of a weighted integral of the usual fractional operators /delta;/s, 0 ≤ s ≤ γ plus a smooth remainder term which corresponds to an L1 kernel. For a general vector field v (possibly non-divergence-free) we prove a generalized L∞ maximum principle of the form ∥ 0(t) ∥ ∞ ≤ eCt∥ Φ0∥ ∞ where the constant C = C(v, β, γ) > Φ. In the case div(u) = 0 the same inequality holds for ∥0 (t) ∥p with 1≤ p≤ ∞. Under the additional assumption that Φ0 2 L2, we show that ∥ Φ (t) ∥p is uniformly bounded for 2 ≤ p ≤ ∞. At the cost of a possible exponential factor, this extends a recent result of Hmidi [7] to the full regime d ≥ 1, 0 ≤ γ ≤ 2 and removes the incompressibility assumption in the L∞ case. |
Persistent Identifier | http://hdl.handle.net/10722/326990 |
ISSN | 2023 Impact Factor: 1.1 2023 SCImago Journal Rankings: 1.104 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Dong, Hongjie | - |
dc.contributor.author | Li, Dong | - |
dc.date.accessioned | 2023-03-31T05:28:00Z | - |
dc.date.available | 2023-03-31T05:28:00Z | - |
dc.date.issued | 2014 | - |
dc.identifier.citation | Discrete and Continuous Dynamical Systems- Series A, 2014, v. 34, n. 9, p. 3437-3454 | - |
dc.identifier.issn | 1078-0947 | - |
dc.identifier.uri | http://hdl.handle.net/10722/326990 | - |
dc.description.abstract | We consider a transport-diffusion equation of the form ∂tΦ + v·δ Φ + vAΦ = 0, where v is a given time-dependent vector field on Rd. The operator A represents log-modulated fractional dissipation: A = /δ/γ/logβ(λ /δ/) and the parameters v ≥ 0, β 0, 0 ≤ γ 2, λ > 1. We introduce a novel nonlocal decomposition of the operator A in terms of a weighted integral of the usual fractional operators /delta;/s, 0 ≤ s ≤ γ plus a smooth remainder term which corresponds to an L1 kernel. For a general vector field v (possibly non-divergence-free) we prove a generalized L∞ maximum principle of the form ∥ 0(t) ∥ ∞ ≤ eCt∥ Φ0∥ ∞ where the constant C = C(v, β, γ) > Φ. In the case div(u) = 0 the same inequality holds for ∥0 (t) ∥p with 1≤ p≤ ∞. Under the additional assumption that Φ0 2 L2, we show that ∥ Φ (t) ∥p is uniformly bounded for 2 ≤ p ≤ ∞. At the cost of a possible exponential factor, this extends a recent result of Hmidi [7] to the full regime d ≥ 1, 0 ≤ γ ≤ 2 and removes the incompressibility assumption in the L∞ case. | - |
dc.language | eng | - |
dc.relation.ispartof | Discrete and Continuous Dynamical Systems- Series A | - |
dc.subject | Fractional dissipation | - |
dc.subject | Generalized maximum principle | - |
dc.subject | Nonlocal decomposition | - |
dc.subject | Nonlocal operators | - |
dc.subject | Transport-diffusion equations | - |
dc.title | On a generalized maximum principle for a transport-diffusion model with log-modulated fractional dissipation | - |
dc.type | Article | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.3934/dcds.2014.34.3437 | - |
dc.identifier.scopus | eid_2-s2.0-84898866056 | - |
dc.identifier.volume | 34 | - |
dc.identifier.issue | 9 | - |
dc.identifier.spage | 3437 | - |
dc.identifier.epage | 3454 | - |
dc.identifier.eissn | 1553-5231 | - |
dc.identifier.isi | WOS:000333556300008 | - |