Regularity of the Gain Term and Strong $L^1 $ Convergence to Equilibrium for the Relativistic Boltzmann Equation

Abstract

The main purpose of the paper is to show that the gain term of the relativistic collision operator is regularizing. This is a generalization of P. L. Lions’ analogous result in the nonrelativistic situation. The regularizing theorem has many applications in kinetic theory, and a few are discussed in this paper. In particular, the asymptotic behaviour of periodic solutions to the relativistic Boltzmann equation is studied. We show that such solutions converge strongly in $L^1 $ to a global Jüttner equilibrium solution (sometimes called a relativistic Maxwellian) provided that the initial data satisfy the physically natural bounds of finite energy and entropy.