Journal of Statistical Physics, Vol. 103, Nos. 5/6, 2001. p. 1009-1033.
The classical Chapman–Enskog expansions for the pressure deviator P andheat flux q provide a natural bridge between the kinetic description of gas dynamics as given by the Boltzmann equation andcontinuum mechanics as given by the balance laws of mass, momentum, energy supplementedby the expansions for P and q. Truncation of these expansions beyond first (Navier–Stokes) order yields instability of the rest state andis inconsistent with thermodynamics. In this paper we propose a visco-elastic relaxation approximation that eliminates the instability paradox. This system is weakly parabolic, has a linearly hyperbolic convection part, andis endowedwith a generalizedentropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman–Enskog expansion.