Vanishing angular singularity limit to the hard-sphere Boltzmann equation

TL;DR

This paper investigates the limit of the Boltzmann collision kernel for inverse power law interactions as it approaches the hard-sphere case, providing asymptotic formulas and convergence results.

Contribution

It establishes the limit of the non-cutoff kernel to the hard-sphere kernel and derives precise asymptotics near zero scattering angles as s approaches infinity.

Findings

The non-cutoff kernel converges to the hard-sphere kernel as s→∞.

Asymptotic formulas describe the singular layer near θ≈0 in the limit s→∞.

Solutions to the homogeneous Boltzmann equation converge in this limit.

Abstract

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 $\theta\simeq 0$ in the limit $ s\to \infty $. Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions.