Abstract

The Kompaneets equation models the energy spectrum of photons interacting with a gas of electrons by Compton scattering. It is fundamental in modern cosmology for explaining the Sunyaev-Zeldovich effect which involves deformation of the cosmic microwave background. We establish L1 convergence to Bose-Einstein equilibria in large time and prove several results on the existence and behavior of a `condensate' of photons at the zero-energy boundary.

The Kompaneets equation is a scalar conservation law with a degenerate parabolic nature that permits a loss of photons in finite time due to shock formation at the zero-energy boundary. Solutions satisfy entropy decay, an Oleinik one-sided slope bound, and L1 contraction and comparison principles.

This is joint work with Gautam Iyer, Hailiang Liu, Josh Ballew, and Dave Levermore.