The Ellipsoidal Statistical BGK model (also Gaussian BGK model) is an improvment of the classical relaxation towards a local Maxwellian (BGK model). It has been devised in order to provide the right transport coefficients (viscosity and heat transfert) in the Navier-Stokes asymptotics. It was proposed in the 70's by Holway and independently by Cercignani. The question of the entropy property (H-Theorem) was left open because the model involves non convex combinations of the second order velocity moments and of the usual isotropic matrix built on the temperature, and this discarded the model.

In fact, and unexpectedly, the entropy property for the ES-BGK model holds true. In this talk we will give a proof (relying on Brunn-Minkowski inequality) and we will recall the motivation and the construction of the model.

We will also give sharp numerical comparisons between the BGK, the ES-BGK and the Boltzmann models on realistic two dimensional computations.

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