The aim of this talk is the analysis of the long time behaviour for the inhomogeneous nonlinear BGK model for the Boltzmann equation with an external confining potential.
As a first step we study global existence of solutions for this model.
For an
initial datum $f_0\ge0$ with bounded mass, entropy and total energy we prove
existence and strong convergence in $L^1$ to a Maxwellian equilibrium state, by
compactness arguments and multipliers techniques.
The main part of this talk is devoted to the particularly interesting case of 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 $f_0$ to identify the equilibrium state, both in $L^1$ and in Lyapunov sense. Under further assumptions on $f$, these conditions become also sufficient for the identification of the equilibrium in $L^1$. This is a joint work with Roberta Bosi.
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