Local well-posedness for Boltzmann's equation and the Boltzmann hierarchy via Wigner transform

TL;DR

This paper establishes local well-posedness results for the Boltzmann equation and hierarchy in   using dispersive analysis and the Wigner transform, covering Maxwell molecules and hard spheres.

Contribution

It introduces a novel approach employing the Wigner transform and spacetime Fourier analysis to prove local well-posedness for Boltzmann models in  , including cases without initial data factorization.

Findings

Proves local well-posedness for Boltzmann equation with Maxwell molecules and hard spheres.

Establishes local well-posedness for the Boltzmann hierarchy for Maxwell molecules.

Uses dispersive properties of Schrd6dinger equation in kinetic theory context.

Abstract

We use the dispersive properties of the linear Schr\"{o}dinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain $\mathbb{R}^d$ for $d\geq 2$. The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms for the initial data $f_0$ are weighted versions of the Sobolev spaces $L^2_v H^\alpha_x$ with $\alpha \in \left( \frac{d-1}{2},\infty\right)$. Our main results are local well-posedness for the Boltzmann equation for cutoff Maxwell molecules and hard spheres, as well as local well-posedness for the Boltzmann hierarchy for cutoff Maxwell molecules (but not hard spheres); the latter result holds without any factorization assumption for the initial data.