Stationary Boltzmann Equation for Polyatomic Gases in a slab

TL;DR

This work analyzes the stationary polyatomic Boltzmann equation in a slab with evaporation and condensation at the boundaries, reformulating the problem on a unit interval with Knudsen number $1/\varepsilon$. The authors establish a unique mild stationary solution for sufficiently small $\varepsilon$ under boundary data in carefully designed weighted norms, by proving invariance and contraction of a solution map $\Psi$ in a Banach space. A key novelty is the polyatomic regularization arising from integrating over the internal-energy distribution parameter, which strengthens the gain-term estimates in norms $\|\cdot\|_0$, $\|\cdot\|_{1-\gamma}$, and $\|\cdot\|_P$, enabling a fixed-point argument in the hard-potential regime for $\tfrac{1}{2} \le \gamma \le 1$. The results extend Maslova’s monatomic slab theory to polyatomic molecules with a total-energy collision model, providing rigorous existence, uniqueness, and a framework for modeling steady, out-of-equilibrium polyatomic gas flows across phase interfaces.

Abstract

We consider the existence of steady rarefied flows of polyatomic gas between two parallel condensed phases, where evaporation and condensation processes occur. To this end, we study the existence problem of stationary solutions in a one-dimensional slab for the polyatomic Boltzmann equation, which takes into account the effect of internal energy in the collision process of the gas molecules. We show that, under suitable norm bound assumptions on the boundary condition functions, there exists a unique mild solution to the stationary polyatomic Boltzmann equation when the slab is sufficiently small. This is based on various norm estimates - singular estimates, hyperplane estimates - of the collision operator, for which genuinely polyatomic techniques must be employed. For example, in the weighted and singular estimates of the collision operator, we carry out integration with respect to the parameter describing the internaltranslational energy distribution, which provides a regularizing effect in the estimate.

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Lemma 2.2: Symmetry property of the Collision Operator
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof