# Global dynamics of the Kompaneets equation

## Location

The Kompaneets equation describes energy transport in low-density (or

high temperature) plasmas where the dominant energy exchange mechanism

is Compton scattering. The equation itself is a one dimensional

non-linear parabolic equation with a diffusion coefficient that vanishes

at the boundary. This degeneracy, combined with the nonlinearity causes

an out-flux of photons with zero energy, often interpreted as a

Bose--Einstein condensate. This talk will describe several results about

the long time behavior of two models of these these equations including

convergence to equilibrium, persistence of the condensate, sufficient

conditions under which it forms, sufficient conditions under which it

doesn't form and a loss formula for the mass of the condensate. I will

also talk about some ongoing work that obtains most of these results for

the full Kompaneets equation.