Abstract

ABSTRACT 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 , where Mf is the Maxwellian distribution associated with f and is controlled by means of the entropy dissipation. Secondly some equiintegrability is obtained from a new velocity averaging result for the advection operator . 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.).

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