Asymptotic Stability of the Boltzmann Equation with Maxwell Boundary Conditions

In a general $C^1$ domain, we study the perturbative Cauchy theory for the Boltzmann equation with Maxwell boundary conditions with an accommodation coefficient $\alpha$ in $(\sqrt{2\mathrm{/}3},1]$, and discuss this threshold. We consider polynomial or stretched exponential weights $m(v)$ and prove existence, uniqueness and exponential trend to equilibrium around a global Maxwellian in $L_{x,v}^{\mathrm{\infty }}(m)$. Of important note is the fact that the methods do not involve contradiction arguments.