  The model of inelastic Maxwell particles (IMP) allows one to derive some
exact results which show the strong influence of inelasticity on the
nonequilibrium properties of a granular gas. The aim of this work is to propose
a simple model kinetic equation that preserves the most relevant properties of
the Boltzmann equation (BE) for IMP and reduces to the BGK kinetic model in the
elastic limit. In the proposed kinetic model the collision operator is replaced
by a relaxation-time term toward a reference Maxwellian distribution plus a
term representing the action of a friction force. It contains three parameters
(the relaxation rate, the effective temperature of the reference Maxwellian,
and the friction coefficient) which are determined by imposing consistency with
basic exact properties of the BE for IMP. As a consequence, the kinetic model
reproduces the true shear viscosity and predicts accurate expressions for the
transport coefficients associated with the heat flux. The model can be exactly
solved for the homogeneous cooling state, the solution exhibiting an algebraic
high-energy tail with an exponent in fair agreement with the correct one.
