The nonlinear diffusion limit for generalized Carleman models: the initial-boundary value problem

摘要

Consider the initial-boundary value problem for the 2-speed Carleman model ofthe Boltzmann equation of the kinetic theory of gases set in some boundedinterval with boundary conditions prescribing the density of particles enteringthe interval. Under the usual parabolic scaling, a nonlinear diffusion limit isestablished for this problem. In fact, the techniques presented here allowtreating generalizations of the Carleman system where the collision frequencyis proportional to some power of the macroscopic density, with exponent in[-1,1].