Global Classical Solutions of the Boltzmann Equation with Long-Range Interactions and Soft Potentials

TL;DR

This paper establishes global stability and convergence rates for solutions to the Boltzmann equation with long-range interactions and soft potentials across all inverse power laws, providing sharp bounds and spectral gap conditions.

Contribution

It proves the first comprehensive global stability results for the Boltzmann equation with physical long-range kernels and characterizes decay rates and spectral gaps in this setting.

Findings

Solutions decay exponentially when b3 + 2s b7= 0 or greater.

Solutions decay polynomially when b3 + 2s < 0.

Spectral gap exists only when b3 + 2s b7= 0 or greater.

Abstract

In this work we prove global stability for 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 $p>2$. This completes the work which we began in (arXiv:0912.0888v1). We more generally cover collision kernels with parameters $s\in (0,1)$ and $\gamma$ satisfying $\gamma > -(n-2)-2s$ in arbitrary dimensions $\mathbb{T}^n \times \mathbb{R}^n$ with $n\ge 2$. Moreover, we prove rapid convergence as predicted by the Boltzmann H-Theorem. When $\gamma + 2s \ge 0$, we have exponential time decay to the Maxwellian equilibrium states. When $\gamma + 2s < 0$, our solutions decay polynomially fast in time with any rate. These results are 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 $\gamma + 2s \ge 0$, as conjectured in Mouhot-Strain (2007).