Global dynamics of Bose-Einstein condensation for a model of the Kompaneets equation

TL;DR

This paper investigates a model of the Kompaneets equation to understand Bose-Einstein condensation of photons, proving existence, uniqueness, and long-term behavior of solutions, including condensate formation.

Contribution

It introduces a new mathematical analysis of a model of the Kompaneets equation, establishing global solutions and condensation phenomena with novel proof techniques.

Findings

Existence and uniqueness of global weak solutions.

Finite-time formation of Bose-Einstein condensates.

Solutions tend to stationary states over time.

Abstract

The Kompaneets equation describes a field of photons exchanging energy by Compton scattering with the free electrons of a homogeneous, isotropic, non-relativistic, thermal plasma. This paper strives to advance our understanding of how this equation captures the phenomenon of Bose-Einstein condensation through the study of a model equation. For this model we prove existence and uniqueness theorems for global weak solutions. In some cases a Bose-Einstein condensate will form in finite time, and we show that it will continue to gain photons forever afterwards. Moreover we show that every solution approaches a stationary solution for large time. Key tools include a universal super solution, a one-sided Oleinik type inequality, and an $L^1$ contraction.