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

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 $|v-v_*|^βb(\cos(θ))$ for $β\in (0,2]$ with $\cos(θ)= |v-v_*|^{-1}(v-v_*)\cdot σ$ and $σ\in \mathbb{S}^{d-1}$, and assuming the classical cut-off condition $ b(\cos(θ))$ integrable in $\mathbb{S}^{d-1}$, we prove that there exists $a > 0$ such that moments with weight $\exp(a \min{t,1} |v|^β)$ 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.