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 $|v-v_{\ast }{|}^{\beta }b(\cos (\theta ))$ for $\beta \in (0,2]$ with $\cos (\theta )=|v-v_{\ast }{|}^{-1}(v-v_{\ast })\cdot \sigma$ 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.

14 pages. Many typos corrected in this revised version

References in corpus (1)

• On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates

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