Moment estimates for polyatomic Boltzmann equation with frozen collisions

TL;DR

This work analyzes a polyatomic Boltzmann equation with continuous internal energy in the space-homogeneous setting and introduces frozen collisions that conserve kinetic energy while keeping internal energy fixed. It establishes a priori moment estimates for the frozen operator, including exact propagation of internal-energy moments and Povzner-type generation/propagation for velocity moments, yielding explicit decay-inspired generation rates $m_k[f](t)\le E_k + C t^{-(k-2)/\zeta}$ for $k>2$. The authors then study a convex combination of frozen and pure polyatomic collisions, $Q^{\omega}$, and prove unified generation and propagation bounds for moments, with constants depending on $\omega$ and the respective collision rates $\zeta$ and $\zeta_f$. Overall, the results provide rigorous control of high-order moments for polyatomic gases under mixed collision dynamics, informing transport properties and long-time kinetic behavior.

Abstract

In this paper, a polyatomic gas with continuous internal energy is considered, allowing for frozen collisions, in which the kinetic energy of the colliding particle pair is conserved, and the internal energy of each particle remains unchanged. A priori moment estimates are derived for solutions of the space-homogeneous Boltzmann equation with a collision kernel of the hard potentials type with cut-off. The model with frozen collisions is first analyzed, followed by a review of general collisions--referred to as pure polyatomic--which preserve the total kinetic and internal energy. By combining existing results for pure polyatomic collisions with the newly derived estimates for frozen collisions, moment estimates are established for the Boltzmann equation with a collision operator that convexly combines both types of collisions. In particular, the moment generation property is shown to be driven by the rate of the pure polyatomic operator, and the moment propagation property holds.

Theorems & Definitions (10)

• proposition 1: Polynomial $I-$moment propagation estimate
• lemma 1: Povzner $\sigma$-averaging
• proposition 2: Polynomial moment estimates on the frozen collision operator
• proof
• theorem 1: Polynomial $v-$moment generation and propagation estimate
• lemma 2: Povzner $( \sigma, r, R)$-averaging, Lemma 4.3 from Gamba-Colic-poly
• proposition 3: Lemmas 5.6 and 5.8 from Alonso-Gamba-Colic
• proposition 4: Polynomial moments of the $\omega-$convex collision operator
• proof
• theorem 2: Polynomial moment generation and propagation estimate