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Nonlinear stability of chemotactic clustering with discontinuous advection

Calvez, Vincent and Hoffmann, Franca (2020) Nonlinear stability of chemotactic clustering with discontinuous advection. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20201110-154946006

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Abstract

We perform the nonlinear stability analysis of a chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents the key challenge for the stability analysis. We follow a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion, allowing us to circumvent the discontinuity issue. Further, the homogeneity of the problem leads to two conservation laws, which express themselves in differently weighted functional spaces. This discrepancy between the weights represents another key methodological challenge. We derive an improved Poincaré inequality that allows to transfer the information encoded in the conservation laws to the appropriately weighted spaces. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. A numerical investigation illustrates our results.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2009.11048arXivDiscussion Paper
ORCID:
AuthorORCID
Calvez, Vincent0000-0002-3674-1965
Hoffmann, Franca0000-0002-1182-5521
Additional Information:This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No 639638). FH was partially supported by the von Karman postdoctoral instructorship at California Institute of Technology, and through the Engineering and Physical Sciences Research Council (UK) grant number EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis. The authors benefited from fruitful discussion with Jean Dolbeault and Ivan Gentil about Proposition 3.1. FH is grateful to Camille, Marine and Constance Bichet, and Joachim Schmitz-Justen and Rita Zimmermann for their hospitality during the SARS-CoV-2 outbreak that allowed to finish this project.
Funders:
Funding AgencyGrant Number
European Research Council (ERC)639638
Caltech Von Kármán instructorshipUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)EP/H023348/1
University of CambridgeUNSPECIFIED
Subject Keywords:bacterial chemotaxis, rate of convergence, asymptotic behaviour, Keller-Segel, stability, entropy decay
Record Number:CaltechAUTHORS:20201110-154946006
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20201110-154946006
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:106600
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Nov 2020 00:02
Last Modified:31 Mar 2021 23:04

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