On Blowup for Gain-Term-Only classical and relativistic Boltzmann equations

TL;DR

Removing the loss term from the Boltzmann equation, including relativistic versions, causes solutions to blow up in finite time, highlighting the importance of the loss term for global existence.

Contribution

This paper proves that deleting the loss term in all relevant Boltzmann equations leads to finite-time blowup, correcting previous errors and providing a more general proof.

Findings

Loss term removal causes finite-time blowup

Results apply to classical and relativistic Boltzmann equations

Corrects previous proof errors in the literature

Abstract

We show that deletion of the loss part of the collision term in all physically relevant versions of the Boltzmann equation, including the relativistic case, will in general lead to blowup in finite time of a solution and hence prevent global existence. Our result corrects an error in the proof given in Ref. [12], where the result was announced for the classical hard sphere case; here we give a simpler proof which applies much more generally.