Gaussian-BGK model of Boltzmann equation with small Prandtl number

In this paper we prove the entropy inequality for the Gaussian-BGK model of Boltzmann equation. This model, also called ellipsoidal statistical model, was introduced in order to fit realistic values of the transport coefficients (Prandtl number, second viscosity) in the Navier-Stokes approximation, which cannot be achieved by the usual relaxation towards isotropic Maxwellians introduced in standard BGK models. Moreover, we introduce new entropic kinetic models for polyatomic gases which suppress the internal energy variable in the phase space by using two distribution functions (one for particles mass and one for their internal energy). This reduces the cost of their numerical solution while keeping a kinetic description well adapted to desequilibrium regions.