Regularization of the Burnett equations via relaxation. (English) Zbl 1019.82018

Jin, Shi; Slemrod, Marshall

J. Stat. Phys. 103, No. 5-6, 1009-1033 (2001).

Summary: The classical Chapman-Enskog expansions for the pressure deviator \({\mathbf P}\) and heat flux \({\mathbf q}\) 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, energy supplemented by the expansions for \({\mathbf P}\) and \({\mathbf q}\). Truncation of these expansions beyond first (Navier-Stokes) order yields instability of the rest state and is 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, and is endowed with a generalized entropy inequality. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman-Enskog expansion.

MSC:

82C40	Kinetic theory of gases in time-dependent statistical mechanics
76P05	Rarefied gas flows, Boltzmann equation in fluid mechanics
74A99	Generalities, axiomatics, foundations of continuum mechanics of solids

Keywords:

Burnett equations; Chapman-Enskog expansions; Boltzmann equation; visco-elastic relaxation approximation; weakly parabolic; generalized entropy inequality