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Formal derivations from Boltzmann equation to three stationary equations

Zhendong Fang

Abstract

In this paper, we concentrate on the connection between Boltzmann equation and stationary equations. To our knowledge, the stationary Navier-Stokes-Fourier system, the stationary Euler equations and the stationary Stokes equations are formally derived by moment estimate in the first time and extend the results of Bardos, Golse, and Levermore in J. Statist. Phys. 63(1-2), 323-344, 1991.

Formal derivations from Boltzmann equation to three stationary equations

Abstract

In this paper, we concentrate on the connection between Boltzmann equation and stationary equations. To our knowledge, the stationary Navier-Stokes-Fourier system, the stationary Euler equations and the stationary Stokes equations are formally derived by moment estimate in the first time and extend the results of Bardos, Golse, and Levermore in J. Statist. Phys. 63(1-2), 323-344, 1991.
Paper Structure (8 sections, 3 theorems, 50 equations)

This paper contains 8 sections, 3 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. The Boltzmann equation
  3. Well-posedness and hydrodynamic limits
  4. Notations, difficulty and ideas
  5. Notations
  6. Difficulty and ideas
  7. Main results and the proof
  8. Main result

Key Result

Proposition 2.1

There exist $\hat{A}=(\hat{A}_i)$ and $\hat{B}=(\hat{B}_{ij})$ uniquely in $Ker^{\perp}(L)$ such that Moreover, there exist two scalar positive functions $\alpha$ and $\beta$ such that Furthermore, we have where positive constants $\nu$ and $\kappa$ are given by

Theorems & Definitions (5)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Remark 3.1
  • proof