The BGK Model with External Confining Potential: Existence, Long-Time Behaviour and Time-Periodic Maxwellian Equilibria
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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 f 0≥0 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. 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 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.
KeywordsBGK model Boltzmann equation External force Long time behaviour Maxwellian steady states
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