We consider the kinetics of a three-dimensional fluid of weakly interacting bosons with supercritical densities. More precisely, we consider the postulated nonlinear BoltzmannNordheim equations for this system, in a spatially homogeneous state which has an isotropic momentum distribution. The resulting evolution equations have a surprisingly rich mathematical structure, where proper definitions play an important role. Elaborating on previous results, we propose a definition of the coupled equations for which the thermal equilibrium states are stationary. To test the validity of the equations, we study the global existence and uniqueness of solutions, as a problem about return to equilibrium from a perturbation of a thermal state with a condensate.