Quantitative stability of constant equilibria in a non-linear alignment model of self-propelled particles

TL;DR

This paper rigorously analyzes the long-time behavior and stability of equilibria in a kinetic model of self-propelled particles, using hypocoercivity methods adapted to the sphere velocity space.

Contribution

It develops an adapted algebraic framework and nonlinear stability analysis for the kinetic Vicsek equation near uniform equilibria, including decay estimates and regularity results.

Findings

Finite time explosion does not occur near equilibria below the critical threshold.

Quantitative decay estimates are established for the whole space case.

Gains in regularity enable well-posedness and stability in lower Sobolev spaces.

Abstract

We are interested in the long-time behaviour of the kinetic Vicsek equation, rigorously derived as the mean-field limit~\cite{bolley2012meanfield} of a coupled system of~$N$ stochastic differential equations describing particles moving at unit velocity and aligning with their neighbours. We focus on the local-in-space version (that may for instance appear as a moderate interaction limit instead of mean-field), which is not a priori globally well-posed and could explode in finite time. Despite its simple expression, little is rigorously established about the behaviour of its solutions. We use hypocoercivity methods to show that finite time explosion does not occur in the vicinity of uniform and homogeneous equilibria in space below the critical threshold. We recast the now-classic~\cite{villani2009hypocoercivity} approach of modifying Sobolev-type norms by adding cross-terms, linked to commutators between the different operators appearing in the kinetic equation. However, the fact that the velocity space is the sphere adds significant subtleties and requires to develop an adapted algebraic framework of operators. Taking advantage of this new framework, we manage to perform an approach \textit{\`a la} H\'erau~\cite{herau2007short} to show the nonlinear stability. Our main results are a quantitative decay estimate in the case of the whole space, despite the absence of control of the $L^1$ norm of the perturbation, and a gain in regularity at the nonlinear level which allows to have well-posedness and stability in the space~$H^{s,0}(\mathbb{R}^d\times\S)$ for some~$s<\frac{d}2$ (that is to say without a priori uniform bound in space on the~$L^2$ norm in velocity, that would come with Sobolev injection in the case~$s>\frac{d}2$).