Relativistic BGK model for massless particles in the FLRW spacetime

In this paper, we address the Cauchy problem for the relativistic BGK model proposed by Anderson and Witting for massless particles in the Friedmann-Lemaȋtre-Robertson-Walker (FLRW) spacetime. We first derive the explicit form of the Jüttner distribution in the FLRW spacetime, together with a set of nonlinear relations that leads to the conservation laws of particle number, momentum, and energy for both Maxwell-Boltzmann particles and Bose-Einstein particles. Then, we find sufficient conditions that guarantee the existence of equilibrium coefficients satisfying the nonlinear relations and we show that the condition is satisfied through all the induction steps once it is verified for the initial step. Using this observation, we construct explicit solutions of the relativistic BGK model of Anderson-Witting type for massless particles in the FLRW spacetime.