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Global renormalized solutions to Boltzmann systems modeling mixture gases of monatomic and polyatomic species

Yi-Long Luo, Jing-Xin Nie

TL;DR

This work proves global existence of renormalized solutions for a Boltzmann system modeling mixtures of monatomic and polyatomic gases with continuous internal energy, under angular-cutoff kernels. The authors construct smooth approximations of the initial data, collision kernels, and collision operator, obtaining uniform bounds that reflect mass, momentum, energy, and entropy conservation. A key contribution is the averaged velocity(-internal energy) lemma, which, together with renormalized variables, yields strong $L^1$-type convergence of the sequence and the identification of the limit as a renormalized solution that satisfies an entropy inequality. The results advance the rigorous theory of kinetic models with internal degrees of freedom, enabling large-data global existence and entropy control for complex gas mixtures. The methods combine DiPerna–Lions renormalization with precise compactness tools to handle nonlinear collision operators in the presence of internal-energy variables.

Abstract

Inspired by DiPerna-Lions' work \cite{Diperna-Lions}, we study the renormalized solutions to the large-data Cauchy problem of the Boltzmann systems modeling mixture gases of monatomic and polyatomic species, in which the distribution functions $f_α$ characterized the polyatomic species contain the continuous internal energy variable $I \in \mathbb{R}_+$. We first construct the smooth approximated problem and establish the corresponding uniform and physically natural bounds. Then, by employing the averaged velocity (-internal energy) lemma, we can show that the weak $L^1$ limit of the approximated solution is exactly a renormalized solution what we required. Moreover, we also justify that the constructed renormalized solution subjects to the entropy inequality.

Global renormalized solutions to Boltzmann systems modeling mixture gases of monatomic and polyatomic species

TL;DR

This work proves global existence of renormalized solutions for a Boltzmann system modeling mixtures of monatomic and polyatomic gases with continuous internal energy, under angular-cutoff kernels. The authors construct smooth approximations of the initial data, collision kernels, and collision operator, obtaining uniform bounds that reflect mass, momentum, energy, and entropy conservation. A key contribution is the averaged velocity(-internal energy) lemma, which, together with renormalized variables, yields strong -type convergence of the sequence and the identification of the limit as a renormalized solution that satisfies an entropy inequality. The results advance the rigorous theory of kinetic models with internal degrees of freedom, enabling large-data global existence and entropy control for complex gas mixtures. The methods combine DiPerna–Lions renormalization with precise compactness tools to handle nonlinear collision operators in the presence of internal-energy variables.

Abstract

Inspired by DiPerna-Lions' work \cite{Diperna-Lions}, we study the renormalized solutions to the large-data Cauchy problem of the Boltzmann systems modeling mixture gases of monatomic and polyatomic species, in which the distribution functions characterized the polyatomic species contain the continuous internal energy variable . We first construct the smooth approximated problem and establish the corresponding uniform and physically natural bounds. Then, by employing the averaged velocity (-internal energy) lemma, we can show that the weak limit of the approximated solution is exactly a renormalized solution what we required. Moreover, we also justify that the constructed renormalized solution subjects to the entropy inequality.
Paper Structure (33 sections, 50 theorems, 438 equations, 2 figures)

This paper contains 33 sections, 50 theorems, 438 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Model for mixture gases of monatomic and polyatomic species
  3. Collision operator and its equivalent transformation
  4. Definition of collision operator $Q_\alpha$
  5. Equivalent transformation of the collision operator
  6. Notations and main results
  7. Notations
  8. Initial data
  9. Hypotheses on collision kernels
  10. Definition of renormalized solutions
  11. Sketch of proofs
  12. Historical remarks
  13. Organization of current paper
  14. Preliminaries
  15. Properties of weakly relatively compactness in $L^1$
  16. ...and 18 more sections

Key Result

Theorem 1.1

Assume that the collision kernels satisfy Asum-angular-cutoff and collision. Let the initial data $f_0 = (f_{1,0}, \cdots, f_{s,0})$ obeys f0-p, fs0 and fs. Then the Cauchy problem BE-MP-f0 admits a renormalized solution $f=(f_1,\cdots,f_s)$ enjoying the following properties: For any given $T<\infty

Figures (2)

  • Figure 1: Transformation of velocities for monatomic molecules with same mass.
  • Figure 2: Transformation of velocities for polyatomic molecules with same mass.

Theorems & Definitions (92)

  • Definition 1.1
  • Remark 1.1
  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.1: Dunford-Pettis
  • Lemma 2.2: De La Vallée-Poussin Criterion
  • Lemma 2.3
  • proof
  • Corollary 2.1
  • Definition 2.2: DiPerna-Lions' work Diperna-Lions
  • ...and 82 more