Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equation for soft and Coulomb potentials

TL;DR

This paper provides explicit bounds on the Wasserstein distance between solutions of Boltzmann and Landau equations, establishing convergence rates for grazing collisions in soft and Coulomb potentials.

Contribution

It introduces explicit bounds and convergence rates for the grazing collisions limit in the Boltzmann and Landau equations for soft and Coulomb potentials.

Findings

Explicit Wasserstein bounds for solutions

Convergence rates for grazing collisions limit

Local and global in time results depending on potential softness

Abstract

We give an explicit bound for the Wasserstein distance with quadratic cost between the solutions of Boltzmann's and Landau's equations in the case of soft and Coulomb potentials. This gives an explicit rate of convergence for the grazing collisions limit. Our result is local in time for very soft and Coulomb potentials and global in time for moderately soft potentials.