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Relativistic BGK model of Marle for polyatomic gases near equilibrium

Byung-Hoon Hwang

TL;DR

This work analyzes the Marle-type relativistic BGK equation for polyatomic gases near equilibrium, employing a generalized Jüttner distribution to model polyatomic internal states. By linearizing around the global equilibrium and using a micro-macro decomposition with a conserved, orthonormal projection, the authors construct a coercive linear operator $\\mathcal{L}$ and a nonlinear remainder $\\Gamma(f)$, establishing local and then global well-posedness for small perturbations. The key result is an exponential decay of the perturbation energy $\\mathcal{E}(f)(t)$, i.e. $\\mathcal{E}(f)(t)\\le e^{-\\lambda_0 t}\\mathcal{E}(f_0)$, showing stability of the global equilibrium under polyatomic dynamics. This work extends the relativistic BGK theory to polyatomic gases by connecting relativistic extended thermodynamics with kinetic theory, providing rigorous near-equilibrium results and informing future studies on relativistic polyatomic gas dynamics.

Abstract

In this paper, we consider the direct application of the relativistic extended thermodynamics theory of polyatomic gases developed in [Ann. Phys. 377 (2017) 414--445] to the relativistic BGK model proposed by Marle. We present the perturbed Marle model around the generalized Jüttner distribution and investigate the properties of the linear operator. Then we prove the global existence and large-time behavior of classical solutions when the initial data is sufficiently close to a global equilibrium.

Relativistic BGK model of Marle for polyatomic gases near equilibrium

TL;DR

This work analyzes the Marle-type relativistic BGK equation for polyatomic gases near equilibrium, employing a generalized Jüttner distribution to model polyatomic internal states. By linearizing around the global equilibrium and using a micro-macro decomposition with a conserved, orthonormal projection, the authors construct a coercive linear operator and a nonlinear remainder , establishing local and then global well-posedness for small perturbations. The key result is an exponential decay of the perturbation energy , i.e. , showing stability of the global equilibrium under polyatomic dynamics. This work extends the relativistic BGK theory to polyatomic gases by connecting relativistic extended thermodynamics with kinetic theory, providing rigorous near-equilibrium results and informing future studies on relativistic polyatomic gas dynamics.

Abstract

In this paper, we consider the direct application of the relativistic extended thermodynamics theory of polyatomic gases developed in [Ann. Phys. 377 (2017) 414--445] to the relativistic BGK model proposed by Marle. We present the perturbed Marle model around the generalized Jüttner distribution and investigate the properties of the linear operator. Then we prove the global existence and large-time behavior of classical solutions when the initial data is sufficiently close to a global equilibrium.

Paper Structure

This paper contains 9 sections, 11 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Relativistic BGK model for polyatomic gases
  3. Main result
  4. Brief history
  5. Linearization around the global equilibrium
  6. Technical lemmas
  7. Proof of theorem \ref{['main']}
  8. Local existence
  9. Global existence and asymptotic behavior of solutions

Key Result

Theorem 1.1

Let $N\geq 3$. Assume that the initial data $F_0$ is non-negative and it shares the same total mass, momentum, and energy as the global equilibrium $\mathcal{F}_0$, i.e. If $\mathcal{E}(f_0)(t)$ is sufficiently small, then there exists a global-in-time unique solution $F(t,x,p,\mathcal{I})\ge 0$ to the relativistic BGK model PMarle satisfying for some positive constant $\lambda_0$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Remark 3.1
  • ...and 14 more