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

Abstract

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.

Keywords:

• Relativistic Boltzmann equation
• relativistic Landau equation
• soft potentials
• grazing collision

2010 Mathematics Subject Classification:

• 35Q20

Acknowledgements

The authors would like to thank the referees of this paper for their helpful suggestions on this work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by NSFC 11171356.