Classical solutions for the ellipsoidal BGK model with fixed collision frequency

We establish the existence of global in time smooth solutions for the ellipsoidal BGK model, which is a variant of the BGK model for the Boltzmann equation designed to yield the correct Prandtl number in the hydrodynamic approximation at the Navier-Stokes level. For this, we carefully design a function space which captures the growth of the solution in a weighted Sobolev norm, and show that the ellipsoidal relaxation operator is Lipschitz continuous in the induced metric. This approach is restricted to the case when the collision frequency does not depend on the macroscopic field, but no smallness on the initial data is required.