Gevrey smoothing effect for the spatially inhomogeneous Boltzmann equations without cut-off

TL;DR

This paper proves that solutions to the spatially inhomogeneous Boltzmann equation without angular cutoff become smooth in Gevrey class over time, revealing a regularization effect influenced by angular singularity.

Contribution

It establishes the Gevrey regularization effect for the nonlinear Boltzmann equation without cutoff, using symbolic calculus and subelliptic estimates.

Findings

Solutions gain Gevrey regularity at positive times

Regularity depends on the angular singularity

Method involves symbolic calculus and subelliptic estimates

Abstract

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in Gevrey class at positive time, with Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of linearized Boltzmann operator.