@article{7fde47ab3cf74ce59a21a9e7ce3d6142,
title = "On the Asymptotic Behavior of a Run and Tumble Equation for Bacterial Chemotaxis",
abstract = "We prove that linear and weakly nonlinear 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 arbitary dimension d≥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 J. Dolbeault, C. Mouhot, and C. Schmeiser [Trans. Amer. Math. Soc., 367 (2015), pp. 3807–3828], are not applicable for dimension d≥1. We overcome this difficulty by using a probabilistic technique, known as Harris{\textquoteright}s theorem. We also introduce a weakly nonlinear model via a nonlocal coupling on the chemoattractant concentration. This toy model serves as an intermediate step between the linear model and the physically more relevant nonlinear models. We build a stationary solution for this equation and provide a hypocoercivity result.",
keywords = "run and tumble, chemotaxis, hypocoercivity, Harris theorem",
author = "Josephine Evans and Havva Yoldas",
note = "Green Open Access added to TU Delft Institutional Repository 'You share, we take care!' - Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.",
year = "2023",
doi = "10.1137/22M153933",
language = "English",
volume = "55",
pages = "7635--764",
journal = "SIAM Journal on Mathematical Analysis",
issn = "0036-1410",
publisher = "Society for Industrial and Applied Mathematics",
number = "6",
}