2015 / June Volume 10 No.2
The Steady Boltzmann and Navier-Stokes Equations
| Published Date |
2015 / June
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|---|---|
| Title | The Steady Boltzmann and Navier-Stokes Equations |
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| Pagination | 205-257 |
| Abstract | 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.
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| AMS Subject Classification |
5Q30, 35Q20 (76P05, 76D05, 82C40)
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| Received |
2015-05-21
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| Accepted |
2015-05-11
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