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We study the equilibrating effects of the boundary and intermolecularcollision in the kinetic theory for rarefied gases. We consider theMaxwell-type boundary condition, which has weaker equilibrating effect than thecommonly studied diffuse reflection boundary condition. The gas region is thespherical domain in $\mathbb{R}^d$, $d=1,2$. First, without the equilibratingeffect of collision, we obtain the algebraic convergence rates to the steadystate of free molecular flow with variable boundary temperature. Theconvergence behavior has intricate dependence on the accommodation coefficientof the Maxwell-type boundary condition. Then we couple the boundary effect withthe intermolecular collision and study their interaction. We are able toconstruct the steady state solutions of the full Boltzmann equation for largeKnudsen numbers and small boundary temperature variation. We also establish thenonlinear stability with exponential rate of the stationary Boltzmannsolutions. Our analysis is based on the explicit formulations of the boundarycondition for symmetric domains.

Equilibrating effect of Maxwell-type boundary condition in highly rarefied gas