Quantitative global well-posedness of Boltzmann-Bose-Einstein equation and incompressible Navier-Stokes-Fourier limit

In the diffusive scaling and in the whole space, we prove the global well-posedness of the scaled Boltzmann-Bose-Einstein (briefly, BBE) equation with high temperature in the low regularity space $H^2_xL^2$. In particular, we quantify the fluctuation around the Bose-Einstein equilibrium $\mathcal{M}_{\lambda,T}(v)$ with respect to the parameters $\lambda$ and temperature $T$. Furthermore, the estimate for the diffusively scaled BBE equation is uniform to the Knudsen number $\epsilon$. As a consequence, we rigorously justify the hydrodynamic limit to the incompressible Navier-Stokes-Fourier equations. This is the first rigorous fluid limit result for BBE.