# Well-posedness and scattering for the Boltzmann equations: Soft potential with cut-off

## TL;DR

This paper establishes the global existence, uniqueness, and scattering of solutions for the cut-off Boltzmann equation with soft potential, using advanced estimates and function space analysis.

## Contribution

It proves the global well-posedness and scattering for the soft potential Boltzmann equation with small initial data in $L^N_{x,v}$, extending understanding of solution behavior.

## Key findings

- Global existence and uniqueness of mild solutions for small initial data.
- Solutions scatter with respect to the kinetic transport operator.
- Characterization of function space connections for local well-posedness with large data.

## Abstract

We prove the global existence of the unique mild solution for the Cauchy problem of the cut-off Boltzmann equation for soft potential model $\gamma=2-N$ with initial data small in $L^N_{x,v}$ where $N=2,3$ is the dimension. The proof relies on the existing inhomogeneous Strichartz estimates for the kinetic equation by Ovcharov and convolution-like estimates for the gain term of the Boltzmann collision operator by Alonso, Carneiro and Gamba. The global dynamics of the solution is also characterized by showing that the small global solution scatters with respect to the kinetic transport operator in $L^N_{x,v}$. Also the connection between function spaces and cut-off soft potential model $-N<\gamma<2-N$ is characterized in the local well-posedness result for the Cauchy problem with large initial data.