Kinetic and viscous boundary layers for broadwell equations

In this paper, we investigate the boundary layer behavior of solutions to the one dimensional Broadwet1 modet of the nonlinear Boltzrnann equation for small mean free path. We consider the analogue of Maxwell’s diffusive and the reflexive boundary conditions. I t is found that even for such a simple model, there are boundary layers due to purely kinetic effects which cannot be detected by the corresponding Navier-Stokes system. It is also found numerically that a compressive boundary layer i s not always stable in the sense that i t may detach from the boundary and move into the int,erior of the gas as a shock layer. 1. I N T R O D U C T I O N We investigate the boundary layer behavior of the solutions to the one dimensional Broadwell model [l] of the nonlinear Boltzinann equation with analogue of Maxwell’s diffusive and diffusive-reflexive boundary conditions at small mean free path. The general Boltzmann equation of kinetic theory gives a statistical description of a gas of interacting particles. An important property of this equation is its asymptotic equivalence to the Euler or Navier-Stokes equations of the compressible fluid dynamics, in the limit of small mean free path. One expects that away from initial, boundary, and shock layers, the Boltzmann solution should relax t,o its equilibrium state (local Maxwellian state) i n the limit of small mean free path, and the gas should be governed by the macroscopic fluid equations as suggested by Hilbert and the Chapman-Enskog expansions [a]. Both the formal and rigorous mathematical justification of the fluid-dynamic approximations of Boltzmann solutions pose challenging open problems i n most physically interesting cases. Most of the work i n the literature concentrate on the initial layer problems for some models and general Boltzmann equation [3,4,5,6] with notable exceptions [7,8,9]. The asymptotic behavior at small mean free path of solutions to the Boltzmann equation in the presence of shocks or boundaries remains far from being well-understood (not even formally), but see ( [ l O , l l ] ) . In particular, for the boundary layer problem, a qualitative theory