Remarks on the relaxation approximation of the Burnett equations
Methods and Applications of Analysis
0. Introduction. The purpose of this note is to present a short discussion of the relaxation regularization of the Burnett equations, a second order approximation via the Chapman-Enskog expansion on the Boltzmann equation, proposed in . 1. A model problem. The classical Chapman-Enskog expansion in term of the Knudsen number on the Boltzmann equation yields the compressible Euler, Navier-Stokes and Burnett equations in the zeroth, first and second order truncation . However, truncation at
... the Burnett order introduces instability of the rest state [3,4], violating the second law of thermodynamics. In  we introduced a viscoelastic relaxation to regularize the Burnett equations. As an illustrative example for this relaxation procedure, consider the following model problem (1.1) ut + uu x = eq x , (1-2) qt = -^, where e > 0. System (1.1)-(1.2) of course can be rewritten as (1-3) ut + uu x -eq x = 0 (1.4) eqt-u x = -q and is hyperbolic with damping, and the two characteristics are (u ± y/u 2 + 4)/2. It is easy to argue that the rest state u = 0, q = 0 is linearly stable. Now consider the Chapman-Enskog expansion for (1.1), (1.2): q-u x = -eqt (1.6) q -u x = -eu x t + higher order terms in e. If we substitute (1.6) into the balance law (1.3) we find "Navier-Stokes" order e (1.7) Ut + UU x = €U xx and u = 0 is stable. However, if we expand to "Burnett" order e 2 we see (1.8) u t + uu x = eu xx -e 2 u xxt .