Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation

Merino-Aceituno, Sara, Schmeiser, Christian and Winter, Raphael 2025. Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation. SIAM Journal on Mathematical Analysis 57 (6) , pp. 6017-6038. 10.1137/24m1715908

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Abstract

The Vicsek-Bhatnagar–Gross–Krook (BGK) equation is a kinetic model for alignment of particles moving with constant speed between stochastic reorientation events with sampling from a von Mises distribution. The spatially homogeneous model shows a steady state bifurcation with exchange of stability. The main result of this work is an extension of the bifurcation result to the spatially inhomogeneous problem under the additional assumption of a sufficiently large Knudsen number. The mathematical core is the proof of linearized stability, which employs a new hypocoercivity approach based on Laplace–Fourier transformation. The bifurcation result includes global existence of smooth solutions for close-to-equilibrium initial data. For large data, smooth solutions might blow up in finite time, whereas weak solutions with bounded Boltzmann entropy are shown to exist globally.

Item Type:	Article
Date Type:	Published Online
Status:	Published
Schools:	Schools > Mathematics
Publisher:	Society for Industrial and Applied Mathematics
ISSN:	0036-1410
Date of First Compliant Deposit:	18 November 2025
Date of Acceptance:	28 July 2025
Last Modified:	18 Nov 2025 10:00
URI:	https://orca.cardiff.ac.uk/id/eprint/182475

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