Global regularity and decay estimates for the relativistic Landau equation

TL;DR

This paper proves that solutions to the relativistic Landau equation remain smooth and decay over time under certain conditions, providing new regularity and decay estimates in a complex kinetic setting.

Contribution

It establishes the first comprehensive regularity and decay estimates for the relativistic Landau equation in an inhomogeneous, far-from-equilibrium regime.

Findings

Solutions stay smooth if certain bounds hold.

Polynomial and exponential decay in momentum are preserved over time.

A Schauder estimate for linear relativistic kinetic equations is developed.

Abstract

We consider the relativistic Landau equation in the spatially inhomogeneous, far-from-equilibrium regime. We establish regularity estimates of all orders, implying that solutions remain smooth for as long as some zeroth-order conditional bounds hold. We also prove that polynomial and exponential decay in the momentum variable is propagated forward in time. As part of our proof, we establish a Schauder estimate for linear relativistic kinetic equations, that may be of independent interest.