On the Boltzmann equation with the symmetric stable Levy process

TL;DR

This paper studies a kinetic model combining the Boltzmann equation with symmetric stable Levy processes, proving global existence, uniqueness, and stability of solutions under certain conditions.

Contribution

It introduces a new analysis of the Fourier-transformed Boltzmann equation with Levy processes, establishing key mathematical properties for this model.

Findings

Proved global existence of solutions.

Established uniqueness and stability.

Provided maximum growth estimates.

Abstract

As for the spatially homogeneous Boltzmann equation of Maxwellian molecules with the fractional Fokker-Planck diffusion term, we consider the Cauchy problem for its Fourier-transformed version, which can be viewed as a kinetic model for the stochastic time-evolution of characteristic functions associated with the symmetric stable Levy process and the Maxwellian collision dynamics. Under a non-cutoff assumption on the kernel, we establish a global existence theorem with maximum growth estimate, uniqueness and stability of solutions.