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.