Fluid-Dynamic Limit for the Centered Rarefaction Wave of the Broadwell Equation

We study the asymptotic equivalence of the 1-d Broadwell model of the nonlinear Boltzmann equation to its corresponding Euler equation of compressible gas dynamics in the limit of small mean free path. We consider the case where the initial data are allowed to have jump discontinuities such that the corresponding solutions to the Euler equation contain centered rarefaction waves. In particular, Riemann data connected by rarefaction curves are included. We show that, as long as the initial data are a small perturbation of a non-vacuum constant state, the solution for the Broadwell equation exists globally in time and converges, in the small mean free path limit, to the solution of the corresponding Euler equation uniformly except for an initial layer whose width is essentially the order of the mean free path.