# The Cauchy problem for the inhomogeneous non-cutoff Kac equation in critical Besov space

## TL;DR

This paper proves well-posedness and regularizing effects for the inhomogeneous non-cutoff Kac equation in critical Besov spaces, demonstrating optimal Gelfand-Shilov and Gevrey regularity improvements.

## Contribution

It establishes the first smoothing effect results for the kinetic equation within Besov spaces and improves the Gelfand-Shilov regularity index to be optimal.

## Key findings

- Well-posedness of weak solutions in critical Besov space
- Gelfand-Shilov regularity in velocity variable
- Gevrey regularity in position variable

## Abstract

In this work, we investigate the Cauchy problem for the spatially inhomogeneous non-cutoff Kac equation. If the initial datum belongs to the spatially critical Besov space, we can prove the well-posedness of weak solution under a perturbation framework. Furthermore, it is shown that the solution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable. In comparison with the recent result in [18], the Gelfand-Shilov regularity index is improved to be optimal. To the best of our knowledge, our work is the first one that exhibits smoothing effect for the kinetic equation in Besov spaces.