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

Abstract

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 $L_{x,v}^{\infty }\left({\left(1+|v|\right)}^{\beta }{e}^{|v{|}^2/4}\right)$, 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.

47 pages

References in corpus (3)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• The Boltzmann equation for a multi-species mixture close to global equilibrium
• Stability of global equilibrium for the multi-species Boltzmann equation in $L^{\infty }$ settings

Cited by in corpus (2)

• Decay of the Boltzmann equation with the specular boundary condition in non-convex cylindrical domains
• From Vlasov-Maxwell-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Maxwell system with Ohm's law: convergence for classical solutions