Vanishing Angular Singularity Limit to the Hard-Sphere Boltzmann Equation

In this notewe study Boltzmann's collision kernel for inverse power lawinteractionsU(s)(r) = 1/r(s-1) for s > 2 in dimension d = 3. We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near. theta similar or equal to 0 in the limit s -> infinity. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.