On the Cauchy problem for Boltzmann equation modelling a polyatomic gas

Abstract

In the present manuscript we consider the Boltzmann equation that models a polyatomic gas by introducing one additional continuous variable, referred to as microscopic internal energy. We establish existence and uniqueness theory in the space homogeneous setting for the full non-linear case, under an extended Grad assumption on transition probability rate, that comprises hard potentials for both the relative speed and internal energy with the rate in the interval $(0,2]$, which is multiplied by an integrable angular part and integrable partition functions. The Cauchy problem is resolved by means of an abstract ODE theory in Banach spaces, for an initial data with finite and strictly positive gas mass and energy, finite momentum, and additionally finite $k_{\ast }$ polynomial moment, with $k_{\ast }$ depending on the rate of the transition probability and the structure of a polyatomic molecule or its internal degrees of freedom. Moreover, we prove that polynomially and exponentially weighted Banach space norms associated to the solution are both generated and propagated uniformly in time.

arXiv comment68 pages. 2 figures

References in corpus (5)

• Rate of convergence to equilibrium for the spatially homogeneous Boltzmann equation with hard potentials
• Moment inequalities and high-energy tails for the Boltzmann equations with inelastic interactions
• On Measure Solutions of the Boltzmann Equation, part I: Moment Production and Stability Estimates
• Non-Linear Maximum Entropy Principle for a Polyatomic Gas subject to the Dynamic Pressure
• The Cauchy problem and BEC stability for the quantum Boltzmann-Condensation system for bosons at very low temperature

Cited by in corpus (1)

• Linearized Boltzmann Collision Operator: II. Polyatomic Molecules Modeled by a Continuous Internal Energy Variable