Global existence proof for the spatially homogeneous relativistic Boltzmann equation with soft potentials

We study the spatially homogeneous solutions for the relativistic kinetic equations. It is shown that the Cauchy problem for the relativistic Boltzmann and Landau equation with soft potentials admits a global weak solution if the mass, energy and entropy of the initial data are finite. Besides the asymptotic behavior of grazing collisions of the relativistic Boltzmann equation is concerned. We prove that the subsequences of solutions to the relativistic Boltzmann equation weakly converge to the solutions of the relativistic Landau equation when almost all the collisions are grazing. These results are extensions of the work of Villani for the spatially homogeneous Boltzmann and Landau equations in the classical case.

Automated summary

We study the spatially homogeneous solutions for the relativistic kinetic equations and prove the existence of global weak solutions for the Cauchy problem of the relativistic Boltzmann and Landau equation with finite mass, energy, and entropy in the initial data. We also investigate the asymptotic behavior of grazing collisions in the relativistic Boltzmann equation and prove that solutions weakly converge to the solutions of the relativistic Landau equation when almost all collisions are grazing. These results extend Villani's work on the spatially homogeneous Boltzmann and Landau equations in the classical case.