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

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.