Abstract
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).
Highlights
The half-space problem for the steady Boltzmann equation is to find solutions F ≡ F(x, v) to the Boltzmann equation in the half-space with slab symmetry— meaning that F depends on one space variable only, denoted by x > 0, and on three velocity variables v = (v1, v2, v3)—converging to some Maxwellian equilibrium as x → +∞
We prove the existence of the curve C corresponding to solutions of (1)–(12) in some neighborhood of the point (1, 0, 1) converging as x → +∞ with exponential speed uniformly in u
We achieve much less: for instance we do not know whether the solution M(1 + fu) of the steady Boltzmann equation obtained in Theorem 1 satisfies M(1 + fu) ≥ 0
Summary
The quadratic collision integral above is polarized so as to define a symmetric bilinear operator as follows: B(F, G. |v|2 for all rapidly decaying, continuous functions F, G defined on R3—see §3.1 in [9]. |ξ |2 for all rapidly decaying, continuous functions f defined on R3. Has a unique solution f in some class of functions that is invariant under the action of O3(R) on the velocity variable ξ (such as, for instance, the Lebesgue space L∞(R+ × R3; Mdξ dx)). The linear integral operator K is compact on L2(R3; Mdξ ) and satisfies the identity K(φ ◦ R) = (Kφ) ◦ R, where R is defined in (8).
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