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Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition

Feimin Huang, Jing Ouyang, Yong Wang

TL;DR

The work addresses rigourously deriving the compressible Euler equations from the Boltzmann equation in a 2D disk under specular reflection by constructing a multi-scale Hilbert expansion that couples interior dynamics with viscous and Knudsen boundary layers, including geometric correction due to boundary curvature. It introduces a truncated Hilbert expansion with explicit Knudsen-layer corrections that account for nonzero tangential boundary velocity, and proves exponential decay for the Knudsen layer as well as strong $L^2$–$L^\infty$ estimates for the remainder. The main contributions are the existence and decay of the geometrically corrected Knudsen layer with $u^0_{\tau}\neq0$, and a comprehensive remainder analysis that justifies the hydrodynamic limit in a curved 2D domain with specular boundary. The results extend prior half-space and flat-domain analyses to a curved disk, enhancing the understanding of boundary effects in kinetic-to-fluid limits and providing tools for multi-scale analysis in bounded domains.

Abstract

In the present paper, we concern the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition in a two-dimensional disk to the compressible Euler equations. Due to the non-zero curvature and non-zero tangential velocity of compressible Euler solution on the boundary, new difficulties arise in the construction of Knudsen boundary layer. By employing the geometric correction, and an innovative and refined $L^2-L^\infty$ method, we establish the existence and space-decay for a truncated Knudsen boundary layer. Then, by the Hilbert expansion of multi-scales, we successfully justify the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition to the compressible Euler equations in the two-dimensional disk.

Hilbert expansion of the Boltzmann equation on a 2-dimensional disk with specular boundary condition

TL;DR

The work addresses rigourously deriving the compressible Euler equations from the Boltzmann equation in a 2D disk under specular reflection by constructing a multi-scale Hilbert expansion that couples interior dynamics with viscous and Knudsen boundary layers, including geometric correction due to boundary curvature. It introduces a truncated Hilbert expansion with explicit Knudsen-layer corrections that account for nonzero tangential boundary velocity, and proves exponential decay for the Knudsen layer as well as strong – estimates for the remainder. The main contributions are the existence and decay of the geometrically corrected Knudsen layer with , and a comprehensive remainder analysis that justifies the hydrodynamic limit in a curved 2D domain with specular boundary. The results extend prior half-space and flat-domain analyses to a curved disk, enhancing the understanding of boundary effects in kinetic-to-fluid limits and providing tools for multi-scale analysis in bounded domains.

Abstract

In the present paper, we concern the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition in a two-dimensional disk to the compressible Euler equations. Due to the non-zero curvature and non-zero tangential velocity of compressible Euler solution on the boundary, new difficulties arise in the construction of Knudsen boundary layer. By employing the geometric correction, and an innovative and refined method, we establish the existence and space-decay for a truncated Knudsen boundary layer. Then, by the Hilbert expansion of multi-scales, we successfully justify the hydrodynamic limit of Boltzmann equation with specular reflection boundary condition to the compressible Euler equations in the two-dimensional disk.

Paper Structure

This paper contains 30 sections, 21 theorems, 485 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Formal Hilbert expansion
  4. Formal internal expansion
  5. Formal viscous boundary layer expansion
  6. Formal Knudsen boundary layer expansion
  7. Truncated Hilbert expansion and main result
  8. Key points of the proof
  9. The existence of Knudsen layer with geometric correction and nonzero $u^0_{\tau}\neq0$.
  10. The remainder estimate.
  11. Reformulation and boundary conditions
  12. Reformulation of internal solutions
  13. Reformulation of viscous boundary layers
  14. Reformulation of Knudsen boundary layers
  15. Boundary conditions
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $\beta>3$. Assume $\mathfrak{S}\in\mathcal{N}_0^{\perp}$ and there exists a positive constant $\sigma_0>0$ such that Suppose Then the problem K has a unique solution $f$ satisfying where $C>0$ is a constant independent of $\varepsilon$ and $\sigma \in (0,\sigma_0)$. Moreover, if $\mathfrak{S}$ is continuous in $[0,d]\times \mathbb{R}^2$ and $f_b$ is continuous in $\{\mathfrak{v}\in\mathbb{R

Figures (4)

  • Figure 1: The backward progress for Case 1
  • Figure 2: The backward progress for Case 2
  • Figure 3: Backward characteristics in unit disk.
  • Figure 4: The backward progress in a slab.

Theorems & Definitions (47)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 2.1: Schochet
  • Remark 2.2
  • Lemma 2.3: GHW-2021-ARMA
  • Remark 2.4
  • ...and 37 more