Vanishing Angular Singularity Limit to the Hard-Sphere Boltzmann Equation

In this note we study Boltzmann’s collision kernel for inverse power law interactions U_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 θ≃ 0 in the limit s→∞ . Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.