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Non-conservation of dimension in divergence-free solutions of passive and active scalar systems
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Fefferman, Charles, Pooley, Benjamin C. and Rodrigo, Jose L. (2021) Non-conservation of dimension in divergence-free solutions of passive and active scalar systems. Archive for Rational Mechanics and Analysis (242). pp. 1445-1478. doi:10.1007/s00205-021-01708-6 ISSN 0003-9527.
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WRAP-non-conservation-dimension-divergence-free-solutions-passive-active-scalar-systems-Rodrigo-2021.pdf - Accepted Version - Requires a PDF viewer. Download (933Kb) | Preview |
Official URL: https://doi.org/10.1007/s00205-021-01708-6
Abstract
For any h∈(1,2], we give an explicit construction of a compactly supported, uniformly continuous, and (weakly) divergence-free velocity field in R2 that weakly advects a measure whose support is initially the origin but for positive times has Hausdorff dimension h. These velocities are uniformly continuous in space-time and compactly supported, locally Lipschitz except at one point and satisfy the conditions for the existence and uniqueness of a Regular Lagrangian Flow in the sense of Di Perna and Lions theory. We then construct active scalar systems in R2 and R3 with measure-valued solutions whose initial support has co-dimension 2 but such that at positive times it only has co-dimension 1. The associated velocities are divergence free, compactly supported, continuous, and sufficiently regular to admit unique Regular Lagrangian Flows. This is in part motivated by the investigation of dimension conservation for the support of measure-valued solutions to active scalar systems. This question occurs in the study of vortex filaments in the three-dimensional Euler equations.
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Library of Congress Subject Headings (LCSH): | Hausdorff measures, Lipschitz spaces , Lagrangian functions , Lagrange equations , Scalar field theory | |||||||||
Journal or Publication Title: | Archive for Rational Mechanics and Analysis | |||||||||
Publisher: | Springer | |||||||||
ISSN: | 0003-9527 | |||||||||
Official Date: | December 2021 | |||||||||
Dates: |
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Number: | 242 | |||||||||
Page Range: | pp. 1445-1478 | |||||||||
DOI: | 10.1007/s00205-021-01708-6 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Reuse Statement (publisher, data, author rights): | This is a post-peer-review, pre-copyedit version of an article published in Archive for Rational Mechanics and Analysis. The final authenticated version is available online at: https://doi.org/10.1007/s00205-021-01708-6 | |||||||||
Access rights to Published version: | Restricted or Subscription Access | |||||||||
Date of first compliant deposit: | 22 November 2021 | |||||||||
Date of first compliant Open Access: | 27 September 2022 | |||||||||
RIOXX Funder/Project Grant: |
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Open Access Version: |
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