Some remarks on the homogeneous Boltzmann equation with the fractional Laplacian term

TL;DR

This paper investigates the homogeneous Boltzmann equation with a fractional Laplacian, providing existence, uniqueness, and stability results for solutions with initial data in a natural Fourier space.

Contribution

It introduces a new initial data space $\

Findings

Existence of a unique global solution

Improved understanding of initial data in $\\mathcal{M}^\alpha$ space

Derived stability and growth estimates

Abstract

We study the homogeneous Boltzmann equation with the fractional Laplacian term. Working on the Fourier side we solve the resulting integral equation, and improve a previous result by Y.-K. Cho. We replace the initial data space with a certain space $\mathcal{M}^\alpha$ introduced by Morimoto, Wang, and Yang. This space precisely captures the Fourier image of probability measures with bounded fractional moments, providing a more natural initial condition. We show existence of a unique global solution, in addition to the expected maximal growth estimates and stability estimates. As a consequence we obtain a continuous density solution of the original equation.