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arXiv · 2404.00347

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

Sara Merino-Aceituno·Christian Schmeiser·Raphael Winter
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Abstract

The Vicsek-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.

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Sara Merino-Aceituno, Christian Schmeiser, Raphael Winter. 2024-03-30. Stability of equilibria of the spatially inhomogeneous Vicsek-BGK equation across a bifurcation. https://arxiv.org/abs/2404.00347

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