Work

Stability of Equilibria of the Spatially Inhomogeneous Vicsek-BGK Equation Across a Bifurcation

Year: 2025

Type: article

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.

Source: SIAM Journal on Mathematical Analysis

DOI: 10.1137/24m1715908

Authors: Sara Merino-Aceituno, Christian Schmeiser, Raphael Winter

Institutions: University of Vienna, Cardiff University

Language: English

Topic: Gas Dynamics and Kinetic Theory

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Funders: Vienna Science and Technology Fund, Austrian Science Fund

Awards: G3878859087, G5528860119, G6044443304