A new approach to the creation and propagation of exponential moments in the Boltzmann equation

TL;DR

This paper introduces a new method to analyze the creation and propagation of exponential moments in solutions to the spatially homogeneous Boltzmann equation with specific collision kernels, establishing conditions for finiteness of these moments over time.

Contribution

It presents a novel proof technique using a single differential inequality with time-dependent coefficients for exponential moments in the Boltzmann equation.

Findings

Existence of a positive constant a ensuring finite exponential moments for solutions.

The method applies under classical cut-off conditions for the collision kernel.

Finite exponential moments are propagated for all positive times.

Abstract

We study the creation and propagation of exponential moments of solutions to the spatially homogeneous $d$-dimensional Boltzmann equation. In particular, when the collision kernel is of the form $|v-v_*|^\beta b(\cos(\theta))$ for $\beta \in (0,2]$ with $\cos(\theta)= |v-v_*|^{-1}(v-v_*)\cdot \sigma$ and $\sigma \in \mathbb{S}^{d-1}$, and assuming the classical cut-off condition $ b(\cos(\theta))$ integrable in $\mathbb{S}^{d-1}$, we prove that there exists $a > 0$ such that moments with weight $\exp(a \min{t,1} |v|^\beta)$ are finite for $t>0$, where $a$ only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.