Cutoff Boltzmann equation with polynomial perturbation near Maxwellian

TL;DR

This paper establishes the global existence, uniqueness, and decay rates of solutions to the cutoff Boltzmann equation near Maxwellian, using polynomial perturbations and advanced inequalities.

Contribution

It introduces new inequalities and semigroup techniques to prove global solutions with large oscillations and decay properties for the cutoff Boltzmann equation.

Findings

Proved global existence and uniqueness in polynomial weighted spaces for all b3 a0(-3, 1]

Established polynomial decay from polynomial initial decay in velocity

Showed exponential decay for exponential initial decay

Abstract

In this paper, we consider the cutoff Boltzmann equation near Maxwellian, we proved the global existence and uniqueness for the cutoff Boltzmann equation in polynomial weighted space for all $\gamma \in (-3, 1]$. We also proved initially polynomial decay for the large velocity in $L^2$ space will induce polynomial decay rate, while initially exponential decay will induce exponential rate for the convergence. Our proof is based on newly established inequalities for the cutoff Boltzmann equation and semigroup techniques. Moreover, by generalizing the $L_x^\infty L^1_v \cap L^\infty_{x, v}$ approach, we prove the global existence and uniqueness of a mild solution to the Boltzmann equation with bounded polynomial weighted $L^\infty_{x, v}$ norm under some small condition on the initial $L^1_x L^\infty_v$ norm and entropy so that this initial data allows large amplitude oscillations.