Global existence for an isotropic modification of the Boltzmann equation

TL;DR

This paper introduces an isotropic modification of the Boltzmann equation and proves global existence in the space homogeneous case for certain soft potentials, addressing an open problem in kinetic theory.

Contribution

It presents a new isotropic Boltzmann model that simplifies anisotropic collision kernels and establishes global existence results in regimes previously unresolved.

Findings

Global existence for isotropic Boltzmann in soft potentials

Convergence to isotropic Landau collision operator

Application of weighted fractional Hardy inequalities

Abstract

Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann, by analogy with the isotropic Landau equation introduced by Krieger and Strain [Comm. Partial Differential Equations 37(4), 2012, 647--689]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau. Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the "very soft potentials" regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani-Guillen [J. Funct. Anal. 283(6), 2022, Paper No. 109559] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities.