Boltzmann equations for mixtures of Maxwell gases: exact solutions and power like tails

TL;DR

This paper derives explicit self-similar solutions with power-like tails for the Boltzmann equations of Maxwell gas mixtures, revealing that such tails can arise in classical particle interactions and establishing non-existence results for certain models.

Contribution

It provides explicit integral representations of self-similar solutions and proves non-existence of positive solutions with finite moments in a broad class of Maxwell models.

Findings

Self-similar solutions with power-like tails exist for Maxwell gas mixtures.

Explicit integral forms of these solutions are constructed.

Non-existence of positive solutions with finite moments is proven for many Maxwell models.

Abstract

We consider the Boltzmann equations for mixtures ofMaxwell gases. It is shown that in certain limiting case the equations admit self-similar solutions that can be constructed in explicit form. More precisely, the solutions have simple explicit integral representations. The most interesting solutions have finite energy and power like tails. This shows that power like tails can appear not just for granular particles (Maxwell models are far from reality in this case), but also in the system of particles interacting in accordance with laws of classical mechanics. In addition, non-existence of positive self-similar solutions with finite moments of any order is proven for a wide class of Maxwell models.