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Kinetic-fluid boundary layers and acoustic limit for the Boltzmann equation with general Maxwell reflection boundary condition

Ning Jiang, Yulong Wu

TL;DR

This work proves the acoustic (low-Mach) limit for the Boltzmann equation in a half-space with a general Maxwell boundary condition and accommodation $\alpha\in(0,1]$, by constructing an expansion that couples the interior fluid part with viscous and Knudsen boundary layers. The authors implement a Hilbert expansion truncated at order $6$ and rigorously justify convergence to the linear acoustic system with slip-type boundary data, achieving a rate of $\mathcal{O}(\varepsilon^{1/4})$ due to the viscous layer. A key novelty is the rigorous treatment of the $\alpha=O(1)$ regime, where the Knudsen layer provides the necessary boundary information for the fluid equations, in line with Sone’s formal analysis. The results extend prior works that covered $\alpha=0$ or $\alpha=O(\sqrt{\varepsilon})$ and establish a comprehensive link between kinetic boundary effects and compressible acoustics in a half-space setting, supported by uniform bounds for expansion terms and robust $L^2$–$L^\infty$ remainder estimates.

Abstract

We prove the acoustic limit from the Boltzmann equation with hard sphere collisions and the Maxwell reflection boundary condition. Our construction of solutions include the interior fluid part and Knudsen-viscous coupled boundary layers. The main novelty is that the accommodation coefficient is in the full range $0<α\leq 1$. The previous works in the context of classical solutions only considered the simplest specular reflection boundary condition, i.e. $α=0$. The mechanism of the derivation of fluid boundary conditions in the case $α=O(1)$ is quite different with the cases $α=0$ or $α=o(1)$. This rigorously justifies the corresponding formal analysis in Sone's books \cite{sone2002kinetic,sone2007molecular}. In particular, this is a smooth solution analogue of \cite{jiang2010remarks}, in which the renormalized solution was considered and the boundary layers were not visible.

Kinetic-fluid boundary layers and acoustic limit for the Boltzmann equation with general Maxwell reflection boundary condition

TL;DR

This work proves the acoustic (low-Mach) limit for the Boltzmann equation in a half-space with a general Maxwell boundary condition and accommodation , by constructing an expansion that couples the interior fluid part with viscous and Knudsen boundary layers. The authors implement a Hilbert expansion truncated at order and rigorously justify convergence to the linear acoustic system with slip-type boundary data, achieving a rate of due to the viscous layer. A key novelty is the rigorous treatment of the regime, where the Knudsen layer provides the necessary boundary information for the fluid equations, in line with Sone’s formal analysis. The results extend prior works that covered or and establish a comprehensive link between kinetic boundary effects and compressible acoustics in a half-space setting, supported by uniform bounds for expansion terms and robust – remainder estimates.

Abstract

We prove the acoustic limit from the Boltzmann equation with hard sphere collisions and the Maxwell reflection boundary condition. Our construction of solutions include the interior fluid part and Knudsen-viscous coupled boundary layers. The main novelty is that the accommodation coefficient is in the full range . The previous works in the context of classical solutions only considered the simplest specular reflection boundary condition, i.e. . The mechanism of the derivation of fluid boundary conditions in the case is quite different with the cases or . This rigorously justifies the corresponding formal analysis in Sone's books \cite{sone2002kinetic,sone2007molecular}. In particular, this is a smooth solution analogue of \cite{jiang2010remarks}, in which the renormalized solution was considered and the boundary layers were not visible.
Paper Structure (18 sections, 10 theorems, 224 equations)

This paper contains 18 sections, 10 theorems, 224 equations.

Table of Contents

  1. Introduction and main result
  2. Problem presentation
  3. History and motivation
  4. Notations
  5. Main results
  6. Formal analysis
  7. Hilbert expansion
  8. Viscous boundary Layer
  9. Knudsen boundary layer and boundary conditions
  10. Truncations of the Hilbert expansion
  11. Uniform Bounds To the solutions of expansions
  12. Some auxiliary systems
  13. Uniform bounds for $F_k,F^b_k,F_k^{bb}(1\leq k\leq 6)$
  14. Uniform Bounds for Remainder $f_{R,\varepsilon}$
  15. $L^2$ estimates: Proof of Lemma \ref{['Lm3.1']}.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Recall the definitions of $\mathcal{H}^k(\mathbb{R}^3_+)$ and $\mathbb{H}^r_l(\mathbb{R}^3_+)$ in 1.3.10 and 1.3.14. Let $w_l(v)$ be defined in 1.5.13 with $l\geq 7$ and integers $s_k,s^b_k,s^{bb}_k,l^b_k(1\leq k\leq 6)$ be described as in Proposition Pro2.4. Assume that For $1\leq k\leq 6$, let $F_k(t,x,v),F^b_k{(t,\bar{x},\zeta,v)},F^{bb}_k{(t,\bar{x},\xi,v)}$ be constructed in section Se1.4. S

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • ...and 6 more