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Consider the steady Boltzmann equation with slab symmetry for a monatomic, hard sphere gas in a half space. At the boundary of the half space, it is assumed that the gas is in contact with its condensed phase. The present paper discusses the existence and uniqueness of a uniformly decaying boundary layer type solution of the Boltzmann equation in this situation, in the vicinity of the Maxwellian equilibrium with zero bulk velocity, with the same temperature as that of the condensed phase, and whose pressure is the saturating vapor pressure at the temperature of the interface. This problem has been extensively studied, first by Sone, Aoki and their collaborators, by means of careful numerical simulations. See section 2 of (Bardos et al. in J Stat Phys 124:275–300, 2006) for a very detailed presentation of these works. More recently, Liu and Yu (Arch Ration Mech Anal 209:869–997, 2013) proposed an extensive mathematical strategy to handle the problems studied numerically by Sone, Aoki and their group. The present paper offers an alternative, possibly simpler proof of one of the results discussed in Liu and Yu (2013). [ABSTRACT FROM AUTHOR]

This paper proposes two different approaches for deriving the Vlasov-Navier–Stokes (or Vlasov-Stokes) system used in the theory of 'thin' aerosol flows from two kinds of microscopic models. One approach is based on homogenization techniques for elliptic equations in domains with holes, whereas the other approach is based on the various mathematical tools used to study the fluid dynamic limits of the Boltzmann equation. [ABSTRACT FROM AUTHOR]

Abstract: The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius r centered at the integer points of the Euclidian plane, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order as . To cite this article: E. Caglioti, F. Golse, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: The half-space problem of evaporation and condensation of a binary mixture of vapors is investigated on the basis of the kinetic theory of gases. Assuming the Mach number of the normal component of the flow is small, a solution of the Boltzmann equation that varies slowly in the scale of the molecular mean-free-path (slowly varying solution) is introduced. Then a fluid-dynamic system that describes the behavior of the slowly varying solution is derived by a systematic asymptotic analysis. The analytical expression of the conditions allowing steady evaporation or condensation is derived from that system. We analyze the qualitative difference between the conditions in the evaporation and condensation cases: four conditions are needed in the former case while only one condition is required in the latter case. The present paper extends a earlier contribution of the first author for the BGK-type model equation [S. Takata, Half-space problem of weak evaporation and condensation of a binary mixture of vapors, in: Capitelli M. (Ed.), Rarefied Gas Dynamics, AIP, New York, 2005, pp. 503–508] to the Boltzmann equation. The extension is achieved by considering the linear stability of the far field in the case of evaporation and the H theorem, the monotonic decrease of the flux of Boltzmann''s H function, in the case of condensation. [Copyright &y& Elsevier]

For r ∈ (0, 1), let Zr = {x ∈; R2 | dist(x,Z2) > r/2} and define τr (x, v) = inf{t > 0 | x + tv € ∂Zr }. Let Φr (t) be the probability that τr (x, v) ≥ t for x and v uniformly distributed in Zr and S1 respectively. We prove in this paper that [This symbol cannot be presented in ASCII format] as t → +∞. This result improves upon the bounds on Φr in Bourgain­Golse­Wennberg [Commun. Math. Phys. 190, 491–508 (1998)]. We also discuss the applications of this result in the context of kinetic theory. [ABSTRACT FROM AUTHOR]

In the kinetic theory of gases, half‐space problems are important because they govern the behavior of correctors to the Hilbert expansion near the boundaries. This paper takes up the discussion in Coron et al., Comm. Pure Appl. Math. 41 (1988) by a different method, based on the index formula for Wiener–Hopf operators on the half‐line. This method leads naturally to a new result on analogues of Chandrasekhar’s H function, which does not seem easy to obtain by the energy method of Coron et al. (1988). [ABSTRACT FROM AUTHOR]

A Correction to this paper has been published: https://doi.org/10.1007/s00205-021-01608-9 [ABSTRACT FROM AUTHOR]