Discrete time Navier-Stokes limit for the BGK Boltzmann Equation

The discrete time Navier-Stokes equations in whole space are obtained as a fluid limit for the properly scaled discrete time BGK equation by the method developed by Bardos, Golse and Levermore to study hydrodynamic limits of the Boltzmann equation (Bardos, C.; Golse, F.; Levermore, C.D. Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation. Comm. Pure Appl. Math. 1993, 46(5), 667-753.). Two new ideas lead to a completely rigorous result in the BGK case. First we get some control on large velocities by remarking that the microscopic density can be decomposed as f = (f - M-f) + M-f, where M-f is the Maxwellian distribution associated with f and (f - M-f) is controlled by means of the entropy dissipation. Secondly some equiintegrability is obtained from a new velocity averaging result for the advection operator (nu . del(x)). Both estimates allow to fulfill the program in (Bardos, C.; Golse, F.; Levermore, C.D. Fluid Dynamic Limits of Kinetic Equations II: Convergence Proofs for the Boltzmann Equation. Comm. Pure Appl. Math. 1993, 46(5), 667-753.).