Velocity Decay Estimates for Boltzmann equation with hard potentials

TL;DR

This paper derives pointwise polynomial decay estimates in velocity space for the inhomogeneous Boltzmann equation with hard potentials, showing these decay properties are self-generated without initial decay assumptions.

Contribution

It extends previous results on decay estimates from soft to hard potentials, providing new self-generating decay bounds for the Boltzmann equation without cutoff.

Findings

Established polynomial decay estimates for hard potentials

Extended decay results to a broader class of potentials

Provided self-generating decay bounds independent of initial data

Abstract

We establish pointwise polynomial decay estimates in velocity space for the spatially inhomogeneous Boltzmann equation without cutoff, in the case of hard potentials ($\gamma +2s > 2$), under the assumption that the mass, energy, and entropy densities are bounded above, and the mass density is bounded below. These estimates are self-generating, i.e. they do not require corresponding decay assumptions on the initial data. Our results extend the recent work of Imbert-Mouhot-Silvestre (arXiv:1804.06135), which addressed the case of moderately soft potentials ($\gamma + 2s \in [0,2]$).