Gevrey regularizing effect of the Cauchy problem for non-cutoff homogeneous Kac's equation

TL;DR

This paper proves that solutions to the non-cutoff homogeneous Kac's equation become Gevrey regular over time, demonstrating a smoothing effect even from non-smooth initial data, using Fourier analysis and mollifiers.

Contribution

It establishes the Gevrey regularizing effect for the Cauchy problem of the non-cutoff homogeneous Kac's equation, a novel result in kinetic theory.

Findings

Solutions become Gevrey regular for positive time

The regularizing effect applies to non-smooth initial data

Fourier analysis and mollifiers are key tools in the proof

Abstract

In this work, we consider a spatially homogeneous Kac's equation with a non cutoff cross section. We prove that the weak solution of the Cauchy problem is in the Gevrey class for positive time. This is a Gevrey regularizing effect for non smooth initial datum. The proof relies on the Fourier analysis of Kac's operators and on an exponential type mollifier.