On the Cauchy problem for the cutoff Boltzmann equation with small initial data

TL;DR

This paper establishes the global existence and scattering of solutions for the cutoff Boltzmann equation with soft potentials in three dimensions, addressing previous gaps for specific potential cases and utilizing symmetric properties of the gain term.

Contribution

It provides the first proof of global solutions for the cutoff Boltzmann equation with < in 3D, fixing prior gaps and introducing symmetric gain term estimates.

Findings

Proves global existence for  in 3D with small initial data.

Shows solutions scatter with respect to the kinetic transport operator.

Fixes previous gaps in the mathematical analysis of the Boltzmann equation.

Abstract

We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model $-1\leq \gamma< 0$ with the small initial data in three dimensional space. Thus our result fixes the gap for the case $\gamma=-1$ in three dimensional space in the authors' previous work where the estimate for the loss term was improperly used. The other gap there for the case $\gamma=0$ in two dimensional space is recently fixed by Chen, Denlinger and Pavlovi\'{c}. The initial data $f_{0}$ is non-negative, small in weighted $L^{3}_{x,v}$ and finite in weighted $L^{15/8}_{x,v}$. We also show that the solution scatters with respect to the kinetic transport operator. The novel contribution of this work lies in the exploration of the symmetric property of the gain term in terms of weighted estimate. It is the key ingredient for solving the model $-1<\gamma<0$ when applying the Strichartz estimates.