Abstract
We study the stationary solution of the Boltzmann equation in a slab with a constant external force parallel to the boundary and complete accommodation condition on the walls at a specified temperature. We prove that when the force is sufficiently small there exists a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip boundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.
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Hilbert, D.: Grundzüge einer Allgemeinen Theorie der Linearen Intergral Gleichungen. New York: Chelsea 1953
Spohn, H.: Large Scale Dynamics of Interacting Particles. Berlin, Heidelberg, New York: Springer 1991
Nishida, T.: Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Commun. Math. Phys.61, 119–148 (1978)
Caflisch, R.E.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math.33, 651–666 (1980)
Ukai, S.: The Euler limit and initial layer of the nonlinear Boltzmann equation. Hokkaido Math. J.12, 303–324 (1983)
Cercignani, C.: The Boltzmann Equation and Its Applications. New York: Springer-Verlag 1988
De Masi, A., Esposito, R., Lebowitz, J.L.: Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Commun. Pure Appl. Math.42, 1189–1214 (1989)
Bardos, C., Ukai, S.: The classical incompressible Navier-Stokes limit of the Boltzmann equation. Preprint, 1991
Asano, K.: The fluid Dynamical Limit of the Boltzmann Equation. Preprint, 1991
Bardos, C., Golse, F., Levermore, D.: Fluid dynamical limits of kinetic equations I. Formal derivations. J. Stat. Phys.63, 323–344 (1991)
Bardos, C., Golse, F., Levermore, D.: Fluid dynamical limits of kinetic equations II. Convergence proof for the Boltzmann equation. Preprint, 1991
Esposito, R., Lebowitz, J.L., Marra, R.: Navier-Stokes limit of the Boltzmann equation. Preprint, 1991
Hannon, L., Lie, G.C., Clementi, E.: Micro-Hydrodynamics. J. Stat. Phys.51, 965–979 (1988)
Bardos, C., Caflisch, R.E., Nicolaenko, B.: Thermal layer solutions of the Boltzmann Equation. In: Fritz, J., Jaffe, A. and Szasz, D. (eds.) Random Fields: Rigorous results in Statistical Physics. Proceedings, Koszeg 1984, Boston: Birkhauser 1985
Maslova, N.B.: Kramers problem in the kinetic theory of gases. USSR Comput. Math. Phys.22, 208–219 (1982)
Bardos, C., Caflisch, R.E., Nicolaenko, B.: The Milne and Kramers problems for the Boltzmann Equation of a hard sphere gas. Commun. Pure and Appl. Math.XXXIX, 323–352 (1986)
Cercignani, C.: Half-space problems in the kinetic theory of gases. Lect. Notes in Phys.249, Berlin, Heidelberg, New York: Springer 1986, pp. 35–50.
Grad, A.: Asymptotic Theory of the Boltzmann equation. Phys. Fluids6, 147–181 (1963)
Grad, A.: Asymptotic Theory of the Boltzmann equation II. In: Rarefied Gas Dynamics II, Paris, 1962, pp. 26–59
Grad, A.: Asymptotic equivalence of the Navier-Stokes and nonlinear Boltzmann operator. Proc. Symp. Appl. Math.XVII, 154–183 (1965)
Costantini, C., Marra, R.: Hydrodynamic Limits for the Boltzmann proc. J. Stat. Phys.67, 229–249 (1992)
Golse, F., Perthame, B., Sulem, C.: On a boundary layer problem for the nonlinear Boltzmann equation. Arch. Rat. Mechanics104, 81–96 (1988)
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Communicated by Ya.G. Sinai
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Esposito, R., Lebowitz, J.L. & Marra, R. Hydrodynamic limit of the stationary Boltzmann equation in a slab. Commun.Math. Phys. 160, 49–80 (1994). https://doi.org/10.1007/BF02099789
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DOI: https://doi.org/10.1007/BF02099789