Uniform convergence to the equilibrium of the homogeneous Boltzmann-Fermi-Dirac Equation with moderately soft potential

We concern the long-time behavior of mild solutions to the spatially homogeneous Boltzmann–Fermi–Dirac equation with moderately soft potential. Based on the well-posedness results in [X-G. Lu, J. Stat. Phys., 105, (2001), 353-388], we prove that the mild solution decays algebraically to the Fermi–Dirac statistics with an explicit rate. Under the framework of the level set analysis by De Giorgi, we derive an $L^{\mathrm{\infty }}$ estimate which is uniform with respect to the quantum parameter $\epsilon$. All quantitative estimates are independent of $\epsilon$, which implies that they also hold in the classical limit, i.e., the Boltzmann equation.