Exponential Convergence to the Maxwell Distribution for Some Class of Boltzmann Equations

Abstract

We consider a class of nonlinear Boltzmann equations describing return to thermal equilibrium in a gas of colliding particles suspended in a thermal medium. We study solutions in the space \({L^{1}(\Gamma^{(1)},d\lambda),}\) where \({\Gamma^{(1)}=\mathbb{R}^{3} \times \mathbb{T}^3}\) is the one-particle phase space and \({d\lambda= d^3 v d^3 x}\) is the Liouville measure on Γ(1). Special solutions of these equations, called “Maxwellians,” are spatially homogenous static Maxwell velocity distributions at the temperature of the medium. We prove that, for dilute gases, the solutions corresponding to smooth initial conditions in a weighted L1-space converge to a Maxwellian in \({L^{1}(\Gamma^{(1)},d\lambda)}\) , exponentially fast in time.