Boundary-layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation

In this paper, we study the fluid-dynamic limit for the one-dimensional Broadwell model of the nonlinear Boltzmann equation in the presence of boundaries. We consider an analogue of Maxwell's diffusive and reflective boundary conditions. The boundary layers can be classified as either compressive or expansive in terms of the associated characteristic fields. We show that both expansive and compressive boundary layers (before detachment) are nonlinearly stable and that the layer effects are localized so that the fluid dynamic approximation is valid away from the boundary. We also show that the same conclusion holds for short time without the structural conditions on the boundary layers. A rigorous estimate for the distance between the kinetic solution and the fluid-dynamic solution in terms of the mean-free path in theL-norm is obtained provided that the interior fluid flow is smooth. The rate of convergence is optimal.