The Steady Boltzmann and Navier-Stokes Equations

Kazuo Aoki (1) , François Golse (2) , Shingo Kosuge (1)

1 Department of Mechanical Engineering and Science, Graduate School of Engineering
2 CMLS - Centre de Mathématiques Laurent Schwartz

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.

Mots clés: Heat diffusion; Viscous heating; Periodic solutions; Steady Boltzmann equation; Steady Navier-Stokes equation

Domaines: Equations aux dérivées partielles [math.AP]

Kazuo Aoki, François Golse, Shingo Kosuge. The Steady Boltzmann and Navier-Stokes Equations. Bulletin of the Institute of Mathematics, Academia Sinica (New Series), 2015, 10 (2), pp.205-257. ⟨hal-01141244⟩