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

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.