Equilibrating Effect of Maxwell-Type Boundary Condition in Highly Rarefied Gas

Abstract

We study the equilibrating effects of the boundary and intermolecular collision in the kinetic theory for rarefied gases. We consider the Maxwell-type boundary condition, which has weaker equilibrating effect than the commonly studied diffuse reflection boundary condition. The gas region is the spherical domain in R d , d = 1, 2. First, without the equilibrating effect of the collision, we obtain the algebraic convergence rates to the steady state of free molecular flow with variable boundary temperature. The convergence behavior has intricate dependence on the accommodation coefficient of the Maxwell-type boundary condition. Then we couple the boundary effect with the intermolecular collision and study their interaction. We are able to construct the steady state solutions of the full Boltzmann equation for large Knudsen numbers and small boundary temperature variation. We also establish the nonlinear stability with exponential rate of the stationary Boltzmann solutions. Our analysis is based on the explicit formulations of the boundary condition for symmetric domains.

A semigroup approach to the convergence rate of a collisionless gas

“…The key idea is that the symmetry of the domain allows one to consider the intervals in time between two rebounds of a particle as independent and identically distributed random variables, and to deduce a law of large numbers from which one can control the flux of the solution at the boundary in the L ∞ norm. Kuo [19] later extended this result with similar tools to the case of the Maxwell boundary condition, in dimension 2. Finally let us mention that Mokhtar-Kharroubi and Seifert [22] recently obtained an explicit polynomial rate in slab geometry (dimension 1).…”

“…n for some constant C > 0. This is a slight drawback of the method, which prevents us from obtaining the optimal rate 1 (t+1) n from Kuo, Liu and Tsai [17,18,19]. However, the rate obtained here is almost optimal in the sense that it is better than 1 (1+t) n− for all > 0.…”

“…In Section 3 we prove the Lyapunov inequality (18) for several couples of weights. In Section 4 we prove the Doeblin-Harris condition satisfied by the semigroup (S t ) t≥0 , (19). In Section 5 we recall some interpolation results for L 1 -weighted space and give very slight extensions in the case of spaces defined through a projection.…”

“…Hypothesis and main result. While the methods used in [1], [17], [18] and [19] are difficult to adapt to a nonsymmetric setting, it seems intuitive to expect that the rate of convergence will be of the same order without this assumption. In this work, we give, using a slightly modified version of the subgeometric Doeblin-Harris theory of Cañizo and Mischler [7], an answer to questions i) and ii) and a partial answer to question iii) in the larger context of C 2 domains.…”

Effect of abrupt change of the wall temperature in the kinetic theory

“…Kinetic theory is the theory describing the behavior of a rarefied (or non-equilibrium) gas, and there are many interesting phenomena, peculiar to a rarefied gas and caused by the effect of the boundary, that are to be handled by kinetic theory. If we restrict ourselves to time-evolution problems, the following are some examples: The equilibrating effect of the boundary and the slow approach to the equilibrium in a highly rarefied gas [1,13,14,15,21,23], the slip flow induced by the impulsive motion of the boundary (the so-called kinetic Rayleigh problem) [12,16,18], and the temperature jump caused by the sudden change of the temperature of the boundary [19]. It should also be mentioned that some rigorous mathematical studies of boundary-value problems of the Boltzmann equation have been carried out for other specific phenomena [2,3,4,6,9,17] as well as for general settings [7,8,22].…”