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Acoustic limit of Boltzmann equations for gas mixture

Gaofeng Wang, Tianfang Wu, Linjie Xiong

Abstract

In this paper, we study the hydrodynamic and acoustic limit from Boltzmann equations for two species gas mixture with potential $γ\in \left(-3, 1\right]$. % in the whole space $(x \in \mathbb{R}^3)$.Here the particle masses are different which derives to the loss of symmetry to the linearized collision operator. %This paper resolves it precisely by using a framework based on vector-valued functions. We construct the hydrodynamic limit for two species based on the Hilbert expansion method when the Knudsen number is small. The key observation is the precise properties of the linearized collision operators, including the extra operators due to the different particle masses $(m^A \neq m^B)$. In additional, the acoustic limit of the Boltzmann equations for gas mixtures is rigorously justified by assuming the strength of the initial data depends on the Knudsen number.

Acoustic limit of Boltzmann equations for gas mixture

Abstract

In this paper, we study the hydrodynamic and acoustic limit from Boltzmann equations for two species gas mixture with potential . % in the whole space .Here the particle masses are different which derives to the loss of symmetry to the linearized collision operator. %This paper resolves it precisely by using a framework based on vector-valued functions. We construct the hydrodynamic limit for two species based on the Hilbert expansion method when the Knudsen number is small. The key observation is the precise properties of the linearized collision operators, including the extra operators due to the different particle masses . In additional, the acoustic limit of the Boltzmann equations for gas mixtures is rigorously justified by assuming the strength of the initial data depends on the Knudsen number.
Paper Structure (12 sections, 10 theorems, 260 equations, 2 figures)

This paper contains 12 sections, 10 theorems, 260 equations, 2 figures.

Table of Contents

  1. Introduction and main result
  2. Introduction
  3. The properties of the collision operator.
  4. The main results
  5. The compressible Euler limit
  6. The determination of $\mathbf{F}_k~~(k=0,1,\cdots,5)$.
  7. The estimates of the remainder
  8. The hydrodynamic limit
  9. Acoustic Limit
  10. The Acoustic System for gas mixture
  11. The second-order perturbation in $\delta$ of Euler solutions
  12. Second-order expansion approximation of the local Maxwellian $\mu^{\alpha}_{\delta}$.

Key Result

Theorem 1.1

Let $(n_{\delta}^{A, \rm in}, n_{\delta}^{B, \rm in}, \mathbf{u}^{\rm in}_{\delta}, \theta^{\rm in}_{\delta})$ is given as nonlindt. Suppose that there exists a series of integers $s_0 > s_1 > \cdots > s_4 > s_5 > 3,$ such that the initial data satisfy and Then, there exists some constant $\delta_0>0$ independent of $\varepsilon$ such that, for any $0<\delta<\delta_0$, ORDERZONG admit $\mathb

Figures (2)

  • Figure 1: The collision between nitrogen and oxygen molecules
  • Figure 2: The illustration of variable substitution

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 10 more