Global Classical Solutions of the Boltzmann Equation without Angular Cut-off

TL;DR

This paper establishes the global stability and convergence rates of solutions to the Boltzmann equation without angular cutoff for a broad class of collision kernels, providing new spectral gap insights and constructive bounds.

Contribution

It proves global existence, stability, and convergence rates for the Boltzmann equation with physical kernels, including sharp bounds and spectral gap conditions, extending previous results.

Findings

Exponential decay when b3 a0a0 -2s

Polynomial decay for b3 < -2s

Existence of spectral gap only when b3 a0a0 -2s

Abstract

This work proves the global stability of the Boltzmann equation (1872) with the physical collision kernels derived by Maxwell in 1866 for the full range of inverse-power intermolecular potentials, $r^{-(p-1)}$ with $p>2$, for initial perturbations of the Maxwellian equilibrium states, as announced in \cite{gsNonCutA}. We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -n$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the celebrated Boltzmann $H$-theorem. When $\gamma \ge -2s$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma <-2s$, our solutions decay polynomially fast in time with any rate. These results are completely constructive. Additionally, we prove sharp constructive upper and lower bounds for the linearized collision operator in terms of a geometric fractional Sobolev norm; we thus observe that a spectral gap exists only when $\gamma \ge -2s$, as conjectured in Mouhot-Strain \cite{MR2322149}. It will be observed that this fundamental equation, derived by both Boltzmann and Maxwell, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the grazing collisions in the Boltzmann theory.