ELLIPSOIDAL BGK MODEL NEAR A GLOBAL MAXWELLIAN

The BGK model has been widely used in place of the Boltzmann equation because of the qualitatively satisfactory results it provides at relatively low computational cost. There is, however, a major drawback to the BGK model: The hydrodynamic limit at the Navier-Stokes level is not correct. One piece of evidence is that the Prandtl number computed using the BGK model does not agree with what is derived from the Boltzmann equation. To overcome this problem, Holway [Rarefied Gas Dynamics, Vol. 1, Academic, New York, 1966, pp. 193-215] introduced the ellipsoidal BGK (ES-BGK) model where the local Maxwellian is replaced by a nonisotropic Gaussian. In this paper, we prove the existence of classical solutions of the ES-BGK model when the initial data are a small perturbation of the global Maxwellian. The key observation is that, even though the linearized relaxation operator for the ES-BGK model takes the more complicated form, the degeneracy is comparable to the original BGK model or the Boltzmann equation.