Ce travail est consacré à l’analyse de stabilité non-linéaire d’un modèle de chimiotactisme pour l’auto-organisation de colonies de bactéries. Une hypothèse centrale de ce modèle est que les bactéries répondent de manière très abrupte aux variations du signal chimique. Le champ d’advection discontinue qui en résulte est la principale difficulté dans l’analyse de stabilité. Nous adoptons une approche perturbative, dans laquelle le profil de densité cellulaire est détaché du mouvement global de la population, ce qui nous permet de contourner cette difficulté. De l’homogénéité du problème résultent deux lois de conservation qui font intervenir des poids différents. Cette disparité représente une autre difficulté à surmonter. Nous étalissons une inégalité de Poincaré améliorée qui permet de transférer l’information d’un poids à l’autre. En conclusion, nous déduisons de cette analyse la convergence exponentielle vers l’équilibre avec un taux explicite. Cette étude est complétée par des exemples numériques.
We perform the nonlinear stability analysis of a one-dimensional 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.
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@article{AFST_2023_6_32_2_287_0,
author = {Vincent Calvez and Franca Hoffmann},
title = {Nonlinear stability of chemotactic clustering with discontinuous advection},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {287--318},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {2},
year = {2023},
doi = {10.5802/afst.1737},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1737/}
}
TY - JOUR AU - Vincent Calvez AU - Franca Hoffmann TI - Nonlinear stability of chemotactic clustering with discontinuous advection JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2023 SP - 287 EP - 318 VL - 32 IS - 2 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1737/ DO - 10.5802/afst.1737 LA - en ID - AFST_2023_6_32_2_287_0 ER -
%0 Journal Article %A Vincent Calvez %A Franca Hoffmann %T Nonlinear stability of chemotactic clustering with discontinuous advection %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2023 %P 287-318 %V 32 %N 2 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1737/ %R 10.5802/afst.1737 %G en %F AFST_2023_6_32_2_287_0
Vincent Calvez; Franca Hoffmann. Nonlinear stability of chemotactic clustering with discontinuous advection. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 32 (2023) no. 2, pp. 287-318. doi : 10.5802/afst.1737. https://afst.centre-mersenne.org/articles/10.5802/afst.1737/
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