On the asymptotic behaviour of a run and tumble equation for bacterial chemotaxis

TL;DR

This paper proves exponential convergence to a steady state for linear and weakly non-linear run and tumble equations modeling bacterial chemotaxis, extending previous results to higher dimensions and relaxing assumptions using probabilistic methods.

Contribution

It extends convergence results to arbitrary dimensions and introduces a probabilistic approach to hypocoercivity for these equations.

Findings

Linear model converges exponentially to a steady state in any dimension.

Weakly non-linear model admits a stationary solution with hypocoercivity.

Probabilistic techniques overcome limitations of classical methods in higher dimensions.

Abstract

We prove that linear and weakly non-linear 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 arbirtary dimension $d \geq 1$ while relaxing the assumptions. The main challenge is that even though the equation is a mass-preserving, Boltzmann-type kinetic-transport equation, the classical $L^2$ hypocoercivity methods, e.g., by Dolbeault, Mouhot, Schmeiser (Tans. Amer. Math. Soc., 367(6):3807-3828, 2015) are not applicable for dimension $d >1$. We overcome this difficulty by using a probabilistic technique, known as Harris's theorem. We also introduce a weakly non-linear model via a non-local coupling on the chemoattractant concentration. This toy model serves as an intermediate step between the linear model and the physically more relevant non-linear models. We build a stationary solution for this equation and provide a hypocoercivity result.