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On the asymptotic behavior of a run and tumble equation for bacterial chemotaxis
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Evans, Josephine and Yoldaş, Havva (2023) On the asymptotic behavior of a run and tumble equation for bacterial chemotaxis. SIAM Journal on Mathematical Analysis, 55 (6). pp. 7635-7664. doi:10.1137/22m1539332 ISSN 1095-7154.
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Official URL: https://doi.org/10.1137/22m1539332
Abstract
We prove that linear and weakly nonlinear run and tumble equations converge to a unique steady state solution with an exponential rate in a weighted total variation distance. In the linear setting, our result extends the previous results to an arbitary dimension
while relaxing the assumptions. The main challenge is that even though the equation is a mass-preserving, Boltzmann-type kinetic-transport equation, the classical
hypocoercivity methods, e.g., by J. Dolbeault, C. Mouhot, and C. Schmeiser [Trans. Amer. Math. Soc., 367 (2015), pp. 3807–3828], are not applicable for dimension
. We overcome this difficulty by using a probabilistic technique, known as Harris’s theorem. We also introduce a weakly nonlinear model via a nonlocal coupling on the chemoattractant concentration. This toy model serves as an intermediate step between the linear model and the physically more relevant nonlinear models. We build a stationary solution for this equation and provide a hypocoercivity result.
Item Type: | Journal Article | ||||||
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Subjects: | Q Science > QA Mathematics Q Science > QR Microbiology |
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Divisions: | Faculty of Science, Engineering and Medicine > Medicine > Warwick Medical School | ||||||
SWORD Depositor: | Library Publications Router | ||||||
Library of Congress Subject Headings (LCSH): | Bacteria -- Motility -- Mathematical models, Chemotaxis, Probabilities | ||||||
Journal or Publication Title: | SIAM Journal on Mathematical Analysis | ||||||
Publisher: | Society for Industrial & Applied Mathematics (SIAM) | ||||||
ISSN: | 1095-7154 | ||||||
Official Date: | 8 November 2023 | ||||||
Dates: |
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Volume: | 55 | ||||||
Number: | 6 | ||||||
Page Range: | pp. 7635-7664 | ||||||
DOI: | 10.1137/22m1539332 | ||||||
Status: | Peer Reviewed | ||||||
Publication Status: | Published | ||||||
Re-use Statement: | First Published in SIAM Journal on Mathematical Analysis in 55(6) 2023, published by the Society for Industrial and Applied Mathematics (SIAM)” © 2023 Society for Industrial and Applied Mathematics. Unauthorized reproduction of this article is prohibited.” | ||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||
Copyright Holders: | © 2023 Society for Industrial and Applied Mathematics | ||||||
Date of first compliant deposit: | 11 December 2023 | ||||||
Date of first compliant Open Access: | 11 December 2023 |
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