Hilbert-class or ‘‘normal’’ solutions for stationary heat flow

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Abstract
The concept of a Hilbert-class or ‘‘normal’’ solution in kinetic theory refers to one whose space and time dependence is determined entirely through the hydrodynamic variables. Such solutions are expected to apply sufficiently far from the boundaries. We investigate this concept quantitatively for the nonlinear Bhatnagar-Gross-Krook equation to describe a gas between two infinite parallel planes at different uniform temperatures. For sufficiently small average Knudsen numbers, spatial domains are identified for which a normal solution applies. It is shown that these conditions include states far from equilibrium, in the sense that deviations of the normal solution from the first Chapman-Enskog approximation can be large. The special exact solution of the preceding paper is found to be the normal solution in the limit of constant temperature gradient.