Perturbative theory for the Boltzmann equation in bounded domains with different boundary conditions

We study the Boltzmann equation near a global Maxwellian in the case of bounded domains. We consider the boundary conditions to be either specular reflections or Maxwellian diffusion. Starting from the reference work of Guo in $L_{x,v}^{\mathrm{\infty }}\left({(1+\mathrm{\mid }v\mathrm{\mid })}^{\beta }{e}^{\mathrm{\mid }v{\mathrm{\mid }}^2\mathrm{/}4}\right)$, we prove existence, uniqueness, continuity and positivity of solutions for less restrictive weights in the velocity variable; namely, polynomials and stretch exponentials. The methods developed here are constructive.