Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel

TL;DR

This paper studies the well-posedness of the 3D Boltzmann equation with constant collision kernel, identifying a sharp threshold at Sobolev regularity s=1, using methods from nonlinear dispersive PDEs.

Contribution

It establishes the precise Sobolev regularity threshold for well/ill-posedness of the Boltzmann equation with constant collision kernel, revealing a bifurcation at s=1.

Findings

Well-posedness holds for s>1

Ill-posedness occurs for s<1

Threshold at s=1 is sharp and exact

Abstract

We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near equilibrium nor self-similar, to the equation, and prove that the well/ill-posedness threshold in $H^{s}$ Sobolev space is exactly at regularity $s=1$, despite the fact that the equation is scale invariant at $s=\frac{1}{2}$.