Green's Function and Pointwise Space-time Behaviors of the Three-Dimensional Relativistic Boltzmann Equation

TL;DR

This paper investigates the detailed space-time behavior of the Green's function for the 3D relativistic Boltzmann equation, revealing wave decompositions and decay properties, and provides estimates for solutions with non-smooth initial data.

Contribution

It offers a novel analysis of the Green's function's structure and decay in the relativistic Boltzmann equation, including pointwise estimates for solutions with rough initial data.

Findings

Green's function decomposes into diffusive and Huygens waves

Green's function exhibits exponential decay in space and time

Established pointwise estimates for nonlinear solutions with non-smooth initial data

Abstract

The pointwise space-time behavior of the Green's function of the three-dimensional relativistic Boltzmann equation is studied in this paper. It is shown that the Green's function has a decomposition of the macroscopic diffusive waves and Huygens waves with the speed $\sqrt{a^2+b^2}$ at low-frequency, the singular kinetic wave and the remainder term decaying exponentially in space and time. In addition, we establish the pointwise space-time estimate of the global solution to the nonlinear relativistic Boltzmann equation with non-smooth initial data based on the Green's function.