Propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation

Abstract

In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These $L^{\infty }$ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator: first, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counter-part of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: first we give another proof of the Boltzmann $H$-theorem, and second we prove the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium.

arXiv comment33 pages, revised based upon recommendations of the referees

References in corpus (4)

• On Strong Convergence to Equilibrium for the Boltzmann Equation with Soft Potentials
• Uniqueness of Bounded Solutions for the Homogeneous Relativistic Landau Equation with Coulomb Interactions
• Global Classical Solutions to the Relativistic Boltzmann Equation without Angular Cut-off
• Stationary solutions to the boundary value problem for relativistic BGK model in a slab

Cited by in corpus (1)

• On the Determinant Problem for the Relativistic Boltzmann Equation