# The cauchy problem for radially symmetric homogeneous boltzmann equation with shubin class initial datum and gelfand-shilov smoothing effect

## TL;DR

This paper investigates the Cauchy problem for a radially symmetric homogeneous non-cutoff Boltzmann equation with initial data in Shubin space, demonstrating Gelfand-Shilov smoothing effects similar to fractional harmonic oscillator evolution.

## Contribution

It constructs weak solutions for initial data in Shubin space and proves the Gelfand-Shilov smoothing effect for this class of solutions.

## Key findings

- Existence of weak solutions with Shubin space initial data
- Gelfand-Shilov smoothing effect for the solutions
- Spectral decomposition approach using harmonic oscillators

## Abstract

In this paper, we study the Cauchy problem for radially symmetric homogeneous non-cutoff Boltzmann equation with Maxwellian molecules, the initial datum belongs to Shubin space of the negative index which can be characterized by spectral decomposition of the harmonic oscillators. The Shubin space of the negative index contains the measure functions. Based on this spectral decomposition, we construct the weak solution with Shubin class initial datum, we also prove that the Cauchy problem enjoys Gelfand-Shilov smoothing effect, meaning that the smoothing properties are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.