Gevrey regularity of spatially homogeneous Boltzmann equation without cutoff

TL;DR

This paper proves the propagation of Gevrey regularity for solutions to the spatially homogeneous Boltzmann equation without cutoff, extending previous results to a broader range of parameters and using a modified kinetic factor.

Contribution

It extends the Gevrey regularity results for the Boltzmann equation to a wider range of the parameter b3 and s, with a different kinetic factor a6(v)=|v|^b3, building upon Morimoto's framework.

Findings

Gevrey regularity propagates for solutions with Maxwellian decay.

Extended the range of b3 and s for which regularity holds.

Used a modified kinetic factor a6(v)=|v|^b3 instead of bb v b1.

Abstract

In this paper, we study the Gevrey regularity of spatially homogeneous Boltzmann equation without angular cutoff. We prove the propagation of Gevrey regularity for $C^\infty$ solutions with the Maxwellian decay to the Cauchy problem of spatially homogeneous Boltzmann equation. The idea we use here is based on the framework of Morimoto's recent paper (See Morimoto: J. Pseudo-Differ. Oper. Appl. (2010) 1: 139-159, DOI:10.1007/s11868-010-0008-z), but we extend the range of the index $\gamma$ satisfying $\gamma + 2s \in (-1,1)$, $s\in (0,1/2)$ and in this case we consider the kinetic factor in the form of $\Phi(v)=|v|^\gamma$ instead of $\la v \ra ^\gamma$ as Morimoto did before.