Global Dynamics and Photon Loss in the Kompaneets Equation

TL;DR

This paper analyzes the long-term behavior of solutions to the Kompaneets equation, demonstrating photon loss at zero energy, convergence to equilibrium, and conditions for photon accumulation similar to Bose--Einstein condensation.

Contribution

It provides new conditions for photon loss, proves convergence to equilibrium without boundary conditions, and characterizes the zero-energy photon accumulation in the Kompaneets equation.

Findings

Photon number can decrease over time due to outflux at zero energy.

Solutions converge to Bose--Einstein equilibrium as time approaches infinity.

Photon accumulation at zero energy is analogous to Bose--Einstein condensation.

Abstract

The Kompaneets equation governs dynamics of the photon energy spectrum in certain high temperature (or low density) plasmas. We prove several results concerning the long-time convergence of solutions to Bose--Einstein equilibria and the failure of photon conservation. In particular, we show the total photon number can decrease with time via an outflux of photons at the zero-energy boundary. The ensuing accumulation of photons at zero energy is analogous to Bose--Einstein condensation. We provide two conditions that guarantee that photon loss occurs, and show that once loss is initiated then it persists forever. We prove that as $t\to \infty$, solutions necessarily converge to equilibrium and we characterize the limit in terms of the total photon loss. Additionally, we provide a few results concerning the behavior of the solution near the zero-energy boundary, an Oleinik inequality, a comparison principle, and show that the solution operator is a contraction in $L^1$. None of these results impose a boundary condition at the zero-energy boundary.