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

TL;DR

This paper establishes the existence and exponential decay of global strong solutions to the Boltzmann equation without angular cut-off, covering long-range interactions and broad angular singularities, advancing understanding of grazing collisions.

Contribution

It provides the first proof of unique global solutions for the Boltzmann equation with long-range interactions and angular singularities, using new methods to analyze grazing collisions.

Findings

Proves existence of global strong solutions without angular cut-off.

Demonstrates exponential decay of solutions over time.

Includes a broad class of physical cross-sections and singularities.

Abstract

We prove the existence and exponential decay of global in time strong solutions to the Boltzmann equation without any angular cut-off, i.e., for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential $r^{-(p-1)}$ with $p>3$, and more generally, the full range of angular singularities $s=\nu/2 \in(0,1)$. These appear to be the first unique global solutions to this fundamentally important model, which 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 effects of grazing collisions in the Boltzmann theory.