Asymptotic Stability of the Relativistic Boltzmann Equation for the Soft Potentials

TL;DR

This paper proves the global existence and polynomial decay to equilibrium of solutions to the relativistic Boltzmann equation with soft potentials in a periodic domain, under near-equilibrium initial conditions.

Contribution

It establishes the first global existence result for the relativistic Boltzmann equation with soft potentials, resolving an open problem.

Findings

Solutions exist globally for all time.

Solutions decay polynomially to the relativistic Maxwellian.

Continuity of solutions is preserved if initial data is continuous.

Abstract

In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $L^\infty_\ell$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of (Dudy{\'n}ski and Ekiel-Je{\.z}ewska, Comm. Math. Phys., 1988); this resolves the open question of global existence for the soft potentials.