The classical Chapman Enskog expansions for the pressure deviator P and the heat flux q provide a natural bridge between the kinetic description of gas dynamics as given by the Boltzmann equation and continuum mechanics as given by the balance laws of mass, momentum, and energy supplemented by the expansions or P and q. Truncation of these expansions beyond first (Navier-Stokes order) yields instability of the rest state and is inconsistent with thermodynamics. In the work reported here done with Shi Jin, I present a visco-elastic relaxation that eliminates the instability paradox, and provides a system of local equations allowing numerical approximation of gas dynamics.

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