Well/Ill-posedness of the Boltzmann Equation with Soft Potential

TL;DR

This paper investigates the well-posedness and ill-posedness of the Boltzmann equation with soft potential, establishing sharp regularity thresholds in Sobolev spaces using dispersive PDE techniques.

Contribution

It identifies the precise regularity threshold for well- and ill-posedness of the Boltzmann equation with soft potential, surpassing previous scaling-invariant expectations.

Findings

Well-posedness established for regularity s > (d-1)/2

Ill-posedness shown for regularity s < (d-1)/2

Separation point at s = (d-1)/2, higher than the scaling-invariant index

Abstract

We consider the Boltzmann equation with the soft potential and angular cutoff. Inspired by the methods from dispersive PDEs, we establish its sharp local well-posedness and ill-posedness in $H^{s}$ Sobolev space. We find the well/ill-posedness separation at regularity $s=\frac{d-1}{2}$, strictly $\frac{1}{2}$-derivative higher than the scaling-invariant index $s=\frac{d-2}{2}$, the usually expected separation point.