Kac's Process with Hard Potentials and a Moderate Angular Singularity

TL;DR

This paper studies Kac's stochastic gas model with hard potentials and angular singularities, proving the convergence of cutoff systems to the noncutoff system and establishing well-posedness and particle limit results.

Contribution

It demonstrates that the noncutoff particle system can be derived as a limit of cutoff systems with a rate independent of particle number, and proves well-posedness of the Boltzmann equation.

Findings

Convergence of cutoff systems to the noncutoff system with rate independent of N

Well-posedness of the Boltzmann equation for hard potentials with angular singularity

Validation of the particle limit as N approaches infinity

Abstract

We investigate Kac's many-particle stochastic model of gas dynamics in the case of hard potentials with a moderate angular singularity, and show that the noncutoff particle system can be obtained as the limit of cutoff systems, with a rate independent of the number of particles $N$. As consequences, we obtain a wellposedness result for the corresponding Boltzmann equation, and convergence of the particle system in the limit $N\rightarrow \infty$.