Local well-posedness for the periodic Boltzmann equation with constant collision kernel

TL;DR

This paper proves local well-posedness for the periodic Boltzmann equation with a constant collision kernel in higher dimensions, using dispersive PDE techniques and establishing new Strichartz estimates.

Contribution

It introduces a novel approach applying dispersive PDE methods to the Boltzmann equation, establishing local well-posedness in specific Sobolev spaces.

Findings

Proves local well-posedness in L^{2,r}_vH^s_x for s > d/2 - 1/4 and r > d/2.

Establishes L^4 Strichartz estimate for the linear Boltzmann equation.

Extends techniques from dispersive PDEs to kinetic theory.

Abstract

We study the Boltzmann equation with the constant collision kernel in the case of spatially periodic domain $\mathbb{T}^d$, $d\geq 2$. Using the existing techniques from nonlinear dispersive PDEs, we prove the local well-posedness result in $L^{2,r}_vH^s_x$ for $s>\frac{d}{2}-\frac{1}{4}$ and $r>\frac{d}{2}$. To reach the result, the main tool we establish is the $L^4$ Strichartz estimate for solutions to the corresponding linear equation.