Effects of Soft Interaction and Non-isothermal Boundary Upon Long-Time Dynamics of Rarefied Gas

Abstract

In the paper, assuming that the motion of rarefied gases in a bounded domain is governed by the angular cutoff Boltzmann equation with diffuse reflection boundary, we study the effects of both soft intermolecular interaction and non-isothermal wall temperature upon the long-time dynamics of solutions to the corresponding initial boundary value problem. Specifically, we are devoted to proving the existence and dynamical stability of stationary solutions whenever the boundary temperature has suitably small variations around a positive constant. For the proof of existence, we introduce a new mild formulation of solutions to the steady boundary-value problem along the speeded backward bicharacteristic, and develop the uniform estimates on approximate solutions in both \(L^2\) and \(L^\infty \). Such mild formulation proves to be useful for treating the steady problem with soft potentials even over unbounded domains. In showing the dynamical stability, a new point is that we can obtain the sub-exponential time-decay rate in \(L^\infty \) without losing any velocity weight, which is actually quite different from the classical results, such as those in Caflisch (Commun Math Phys 74:97–109, 1980) and Strain and Guo (Arch Ration Mech Anal 187:287–339, 2008), for the torus domain and essentially due to the diffuse reflection boundary and the boundedness of the domain.