Hypocoercivity for a thermostatted Boltzmann equation

Abstract

We study the Boltzmann equation for a gas on the three-dimensional torus. Its particles collide with one another and with a thermal reservoir whose temperature varies in space. The reservoir heats the gas in some regions and cools it in others, driving a nonequilibrium energy current. For the isotropic cutoff kernel, with Maxwell molecules or hard potentials, we prove global well-posedness and exponential convergence to the same stationary state for initial data with a common Gaussian upper bound and Gaussian moment bound. These bounds determine how weak self collisions must be relative to bath collisions. The estimates are uniform over bath profiles between fixed positive temperature bounds. For any nonconstant reservoir profile on the three-dimensional torus, this stationary state cannot be a local Maxwellian. When particles collide only with the bath, relative entropy to the stationary state cannot increase. Once self collisions are added, arbitrarily small perturbations can initially increase it, even though they later return exponentially.