Stability analysis for a kinetic bacterial chemotaxis model

TL;DR

This paper establishes exponential stability for a kinetic bacterial chemotaxis model with sharp chemical response, using novel hypocoercivity techniques to handle discontinuous tumbling kernels and nonlinearities.

Contribution

It introduces new hypocoercivity methods, including $H^1$ norm use and nonlinear term treatment, to analyze stability of a kinetic chemotaxis model with discontinuous tumbling kernels.

Findings

Proves nonlinear stability in a perturbative setting without chemical degradation.

Establishes linearized stability with chemical degradation.

Demonstrates exponential relaxation to equilibrium with explicit rates.

Abstract

We perform stability analysis of a kinetic bacterial chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous tumbling kernel represents the key challenge for the stability analysis as it rules out a direct linearization of the nonlinear terms. To address this challenge we fruitfully separate the evolution of the shape of the cellular profile from its global motion. We provide a full nonlinear stability theorem in a perturbative setting when chemical degradation can be neglected. With chemical degradation we prove stability of the linearized operator. In both cases we obtain exponential relaxation to equilibrium with an explicit rate using hypocoercivity techniques. To apply a hypocoercivity approach in this setting, we develop two novel and specific approaches: i) the use of the $H^1$ norm instead of the $L^2$ norm, and ii) the treatment of nonlinear terms. This work represents an important step forward in bacterial chemotaxis modeling from a kinetic perspective as most results are currently only available for the macroscopic descriptions, which are usually parabolic in nature. Significant difficulty arises due to the lack of regularization of the kinetic transport operator as compared to the parabolic operator in the macroscopic scaling limit.