On the Boltzmann equation with soft potentials: Existence, uniqueness and smoothing effect of mild solutions

TL;DR

This paper proves local existence, uniqueness, and a sharp smoothing effect for mild solutions of the spatially inhomogeneous Boltzmann equation with soft potentials, without angular cutoff, under certain initial conditions.

Contribution

It provides the first results on the smoothing effect in analytic and Gevrey spaces for soft potentials in the Boltzmann equation, along with existence and uniqueness.

Findings

Established local-in-time existence and uniqueness of mild solutions.

Proved sharp smoothing effects in analytic and Gevrey spaces.

Applied to initial data with bounded mass, energy, and entropy, away from vacuum.

Abstract

We consider the spatially inhomogeneous Boltzmann equation without angular cutoff for soft potentials. For any given initial datum such that the mass, energy and entropy densities are bounded and the mass is away from vacuum, we establish the local-in-time existence and uniqueness of mild solutions, and further provide the first result on sharp smoothing effect in analytic space or Gevrey space for soft potentials.