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

I have just uploaded a new preprint of a joint work with R. Alonso, J. Cañizo and I. Gamba:

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

This paper deals with the homogeneous Boltzmann equation for variable hard potentials (inc. hard spheres) and presents new results on the appearance and propagation of exponential moments. Moreover and most importantly it presents a new approach to these results. It underlies a discrete convolution structure for the polynomial moment estimates associated with the collision operator. It allows for much simpler and transparent proofs of propagation of exponential moments and new optimal results of appearance of such exponential moments by considering energy estimate on exponential moments with well-chosen time-dependent constants.

Here is the 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.