The paper discusses the similarities and the differences in the mathematical
theories of the steady Boltzmann and incompressible Navier-Stokes equations
posed in a bounded domain. First we discuss two different scaling limits in
which solutions of the steady Boltzmann equation have an asymptotic behavior
described by the steady Navier-Stokes Fourier system. Whether this system
includes the viscous heating term depends on the ratio of the Froude number to
the Mach number of the gas flow. While the steady Navier-Stokes equations with
smooth divergence-free external force always have at least one smooth
solutions, the Boltzmann equation with the same external force set in the
torus, or in a bounded domain with specular reflection of gas molecules at the
boundary may fail to have any solution, unless the force field is identically
zero. Viscous heating seems to be of key importance in this situation. The
nonexistence of any steady solution of the Boltzmann equation in this context
seems related to the increase of temperature for the evolution problem, a
phenomenon that we have established with the help of numerical simulations on
the Boltzmann equation and the BGK model.Comment: 55 pages, 4 multiple figure
