Recently, a new class of semi-Lagrangian methods for the BGK model of the Boltzmann equation has been introduced [8,17,18]. These methods work in a satisfactory way either in rarefied or fluid regime. Moreover, because of the semi-Lagrangian feature, the stability property is not restricted by the CFL condition. These aspects make them very attractive for practical applications. In this paper, we investigate the convergence properties of the method and prove that the discrete solution of the scheme converges in a weighted L 1 norm to the unique smooth solution by deriving an explicit error estimate.2010 Mathematics Subject Classification. 35Q20, 76P05, 65M12, 65M25.

Abstract. In this paper, we are interested in the Cauchy problem for the Boltzmann-BGK model for a general class of collision frequencies. We prove that the Boltzmann-BGK model linearized around a global Maxwellian admits a unique global smooth solution if the initial perturbation is sufficiently small in a high order energy norm. We also establish an asymptotic decay estimate and uniform L 2 -stability for nonlinear perturbations.

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.

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 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 [21] introduced the ellipsoidal BGK model where the local Maxwellian is replaced by a non-isotropic Gaussian. In this paper, we prove the existence of classical solutions of the ES-BGK model when the initial data is a small perturbation of the global Maxwellian. The key observation is that the degeneracy of the ellipsoidal BGK model is comparable to that of the original BGK model or the Boltzmann equation in the range −1/2 < ν < 1.

The ellipsoidal BGK model is a generalized version of the original BGK model designed to reproduce the physical Prandtl number in the Navier-Stokes limit. In this paper, we propose a new implicit semi-Lagrangian scheme for the ellipsoidal BGK model, which, by exploiting special structures of the ellipsoidal Gaussian, can be transformed into a semi-explicit form, guaranteeing the stability of the implicit methods and the efficiency of the explicit methods at the same time. We then derive an error estimate of this scheme in a weighted L ∞ norm. Our convergence estimate holds uniformly in the whole range of relaxation parameter ν including ν = 0, which corresponds to the original BGK model.where the macroscopic density, velocity, temperature and the stress tensor are defined byThe temperature tensor T ν is given by a convex combination of T and Θ:where Id is the d 2 × d 2 identity matrix. The ellipsoidal relaxation operator satisfies the following cancellation property:which leads to the conservation of mass, momentum and energy:When Holway first suggested this model, H-theorem was not verified, which was the main reason why the ES-BGK model has been neglected in the literature until very recently. It was resolved in [2] (See also [10,53]):and ignited the interest on this model [1,10,22,25,31,32,44,51,52,53,54]. It can be verified via the Chapman-Enskog expansion that the Prandtl number computed using the ES-BGK model is 1/(1 − ν). Therefore, the correct physical Prandtl number can be recovered by choosing appropriate ν, namely, ν = 1 − 1/P r ≈ −1/2, where P r denotes the correct Prandtl number. When ν = 0, the ES-BGK model reduces to the original BGK model. Hence, any results for the ES-BGK model automatically hold for the original BGK model either. We also mention that, in the range −1/2 < ν < 1, the only possible equilibrium state of the ellipsoidal relaxation operator is the usual Maxwellian, not the ellipsoidal Gaussian. That is, the only solution satisfying the relation M ν (f ) = f is the local Maxwellian (See [51,53] for the proof.):