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R. Bosi, M.J. Cáceres:
"The BGK model with external confining potential: Existence, long-time behaviour and time periodic Maxwellian Equilibria";
in: "ASC Report 18/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

We study global existence and long time behaviour for the inhomogeneous nonlinear
BGK model for the Boltzmann equation with an external confining potential. For an initial
datum f0 0 with bounded mass, entropy and total energy we prove existence and strong
convergence in L1 to a Maxwellian equilibrium state, by compactness arguments and multipliers
techniques. Of particular interest is the case with an isotropic harmonic potential,
in which Boltzmann himself found infinitely many time-periodic Maxwellian steady states.
This behaviour is shared with the Boltzmann equation and other kinetic models. For all
these systems we study the multistability of the time-periodic Maxwellians and provide necessary
conditions on f0 to identify the equilibrium state, both in L1 and in Lyapunov sense.
Under further assumptions on f, these conditions become also sufficient for the identification
of the equilibrium in L1.

http://www.asc.tuwien.ac.at/preprint/2009/asc18x2009.pdf