1D BOLTZMANN EQUATION IN A PERIODIC BOX (Mathematical Analysis in Fluid and Gas Dynamics)

Abstract

We study the nonlinear stability of the Boltzmann equation in the 1D periodic box with size 1/ $epsilon$, where 0 < $epsilon$ ll 1 is the Knudsen number. The convergence rate is (1+t)^{-1/2}ln(1+t) for small time region and exponential for large time region. Moreover, the exponential rate depends on the size of the domain (Knudsen number). This problem is highly nonlinear and hence we need more careful analysis to control the nonlinear term.