Free molecular flow with boundary effect (Q1942288)

The purpose of this paper is to focus on the dissipative effect of the boundary condition and establish the convergence rate to equilibrium of free molecular gas flow in spatial dimensions \(d.\) The authors consider the free molecular equation: \[ \frac{\partial{\bar{f}}}{\partial t}+\sum_{i=1}^d\zeta_i\frac{\partial{\bar{f}}}{\partial x_i} =0,\;\;\; \bar{f}=\bar{f}(\mathbf{x},\zeta,t), \] \[ \bar{f}(\mathbf{x},\zeta,0)=\bar{f_{in}}(\mathbf{x},\zeta)\text{ given, } \mathbf{x}\in D\subset\mathbb R^d,\zeta\in\mathbb R^3, t\in\mathbb R_{+}, \] where \(\mathbf{x}\) is the spatial coordinate, \(\zeta\) is the microscopic velocity, \(t\) is the temporal coordinate, and \(\bar{f}=\bar{f_{in}}(\mathbf{x},\zeta)\) is the density function in \((\mathbf{x},\zeta)\) of a gas molecule at time \(t.\) The authors obtain the optimal convergence rates of \((t+1)^{-d}\).