Confinement by biased velocity jumps: aggregation of Escherichia coli

Vincent Calvez (1, 2) , Gaël Raoul (3) , Christian Schmeiser (4)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 NUMED - Numerical Medicine
3 CEFE - Centre d’Ecologie Fonctionnelle et Evolutive
4 Fakultät für Mathematik [Wien]

We investigate a linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in the 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.

Mots clés: Kinetic equation; Velocity jump process; Chemotaxis; Hypocoercivity; Milne problem

Domaines: Equations aux dérivées partielles [math.AP]

Vincent Calvez, Gaël Raoul, Christian Schmeiser. Confinement by biased velocity jumps: aggregation of Escherichia coli. Kinetic and Related Models , 2015, ⟨10.3934/krm.2015.8.651⟩. ⟨hal-00971302⟩