Existence and uniqueness of $L^p$ solutions to the Boltzmann equation with an angle-potential concentrated collision kernel

TL;DR

This paper establishes the existence and uniqueness of $L^p$ solutions for the space homogeneous Boltzmann equation with a singular collision kernel that models Coulomb to hard sphere interactions, and shows convergence to the Landau operator in a grazing collisions limit.

Contribution

It provides the first rigorous proof of existence and uniqueness of $L^p$ solutions for this specific singular collision kernel and demonstrates the limit to the Landau operator.

Findings

Proved existence and uniqueness of $L^p$ solutions.

Demonstrated convergence to the Landau operator in grazing collisions limit.

Unified treatment of Coulomb and hard sphere cases.

Abstract

We solve the Cauchy problem associated to the space homogeneous Boltzmann equation with an angle-potential singular concentration modeling the collision kernel, proposed in 2013 by Bobylev and Potapenko. The potential under consideration ranges from Coulomb to hard spheres cases. However, the motivation of such a collision kernel is to treat the case of Coulomb potentials, on which this particular form of collision operator is well defined. We also show that the scaled angle-potential singular concentration in a grazing collisions limit makes the Boltzmann operator converge in the sense of distributions to the Landau operator acting on the Boltzmann solutions.