  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, which
is shown to be necessary. A counterexample is derived in the form of an
exponentially increasing travelling wave solution of the linearized equation,
indicating a possible explanation of traveling bands, which have been observed
in the literature in simulations with an individual based model. 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.
