Asymptotic Stability of the Relativistic Boltzmann Equation without Angular Cut-off

TL;DR

This paper proves the global stability and uniqueness of solutions to the relativistic Boltzmann equation without angular cutoff, showing the collision operator acts like a fractional diffusion operator and establishing new asymptotic estimates.

Contribution

It introduces the relativistic analogue of Carleman's dual representation and establishes global existence and stability without Grad's angular cutoff assumption.

Findings

Collision operator behaves like a fractional diffusion operator.

Established sharp asymptotics for the frequency multiplier.

Proved global existence and uniqueness of solutions near Maxwellian.

Abstract

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness, and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudy\'nski and Ekiel-Je$\dot{\text{z}}$ewska (Comm. Math. Phys. \textbf{115}(4):607--629, 1985), and our assumptions include the case of Israel particles (J. Math. Phys. \textbf{4}:1163--1181, 1963). In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. The coercivity estimates that are needed rely crucially on the sharp asymptotics for the frequency multiplier that has not been previously established. We further derive the relativistic analogue of the Carleman dual representation for the Boltzmann collision operator. This resolves the open question of perturbative global existence and uniqueness without the Grad's angular cut-off assumption.