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

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.

This paper is a combination, simplification, and extension of two separate preprints originally posted on the arXiv as (arXiv:0912.0888v1) and (arXiv:1002.3639v1). It was revised in July 2010 for the referee. In particular we now estimate all $\gamma >-n$. Please cite this version. 77 pages

References in corpus (4)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity
• Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential
• Smoothing effect for Boltzmann equation with full-range interactions

Cited by in corpus (5)

• Global Classical Solutions of the Boltzmann Equation without Angular Cut-off
• Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential
• Smoothing effect for Boltzmann equation with full-range interactions
• Global hypoelliptic estimates for a linear model of non-cutoff Boltzmann equation
• Hypoelliptic Estimates for a Linear Model of the Boltzmann Equation without Angular Cutoff