Global Stability of Boltzmann Equation with Large External Potential for a Class of Large Oscillation Data

In this paper, we investigate the stability of Boltzmann equation with large external potential in ${\mathbb{T}}^3$. For a class of initial data with large oscillations in $L_{x,v}^{\mathrm{\infty }}$ around the local Maxwellian, we prove the existence of a global solution to the Boltzmann equation provided the initial perturbation is suitably small in $L^2$-norm. The large time behavior of the Boltzmann solution with exponential decay rate is also obtained. This seems to be the first result on the perturbation theory of large-amplitude non-constant equilibriums for large-amplitude initial data.