Confinement by biased velocity jumps: Aggregation of  escherichia coli

Vincent  Calvez; Gaël  Raoul; Christian  Schmeiser

doi:10.3934/krm.2015.8.651

Article Contents

2015, Volume 8, Issue 4: 651-666. Doi: 10.3934/krm.2015.8.651

This issue Previous Article Some a priori estimates for the homogeneous Landau equation with soft potentials Next Article A free boundary problem for a class of parabolic type chemotaxis model

Confinement by biased velocity jumps: Aggregation of escherichia coli

1.
CNRS UMR 5669 "Unité de Mathématiques Pures et Appliquées", and project-team Inria NUMED, Ecole Normale Supérieure de Lyon, Lyon
2.
CNRS UMR 7641 "Centre de Mathématiques Appliquées", Ecole Polytechnique, Palaiseau
3.
Fakultät für Mathematik, Universität Wien

Received: March 31, 2014

Revised: April 30, 2015

Published: November 2015

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Abstract

We investigate a one-dimensional linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint, it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.

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Mathematics Subject Classification: Primary: 35B40, 35Q92, 92C17.

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