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Non-Hydrodynamic Solutions to the linear Density-dependent BGK equation

Florian Kogelbauer

TL;DR

This work investigates the linear density-dependent BGK equation in $d$ dimensions to distinguish hydrodynamic and non-hydrodynamic regimes. It combines spectral analysis of the BGK operator with an explicit Laplace–Fourier solution to obtain a time-domain formula for the macroscopic density, revealing a critical wave number $k_c = \frac{1}{\tau}\sqrt{\frac{\pi}{2}}$ that separates regimes. For $|k|<k_c$ the macroscopic dissipation follows Chapman--Enskog scaling, yielding $\Delta[\hat{\rho}](k,t,\tau) \sim \tau |k|^2$ as $\tau\to0$, while for $|k|\ge k_c$ the dynamics are governed by the essential spectrum and produce a dissipation $\Delta[\hat{\rho}](k,t,\tau) \sim \frac{Z'(\hat{\zeta}_\beta)}{\tau}$ (with $\beta=\tau|k|$ and $Z(\hat{\zeta}_\beta)=i\beta$), causing a $1/\tau$ divergence. These results demonstrate non-hydrodynamic exact solutions and highlight the nonuniform nature of hydrodynamic limits, with implications for how Closures and reduced-order models may fail to capture certain fast kinetic fluctuations.

Abstract

We prove the existence of non-hydrodynamic solutions to the linear density-dependent BGK equation in $d$ dimensions. Specifically, we show the existence of an initial condition for any Knudsen number $τ$ for which the dissipation rate of the macroscopic mass density diverges $\sim 1/τ$. Our results rely on a detailed spectral analysis of the linear BGK operator, an explicit solution formula for the time-dependent problem using a combination of Fourier series with the Laplace transform and subsequent contour integration arguments from complex analysis.

Non-Hydrodynamic Solutions to the linear Density-dependent BGK equation

TL;DR

This work investigates the linear density-dependent BGK equation in dimensions to distinguish hydrodynamic and non-hydrodynamic regimes. It combines spectral analysis of the BGK operator with an explicit Laplace–Fourier solution to obtain a time-domain formula for the macroscopic density, revealing a critical wave number that separates regimes. For the macroscopic dissipation follows Chapman--Enskog scaling, yielding as , while for the dynamics are governed by the essential spectrum and produce a dissipation (with and ), causing a divergence. These results demonstrate non-hydrodynamic exact solutions and highlight the nonuniform nature of hydrodynamic limits, with implications for how Closures and reduced-order models may fail to capture certain fast kinetic fluctuations.

Abstract

We prove the existence of non-hydrodynamic solutions to the linear density-dependent BGK equation in dimensions. Specifically, we show the existence of an initial condition for any Knudsen number for which the dissipation rate of the macroscopic mass density diverges . Our results rely on a detailed spectral analysis of the linear BGK operator, an explicit solution formula for the time-dependent problem using a combination of Fourier series with the Laplace transform and subsequent contour integration arguments from complex analysis.
Paper Structure (18 sections, 1 theorem, 89 equations, 3 figures)

This paper contains 18 sections, 1 theorem, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and Basic Definitions
  3. General Definitions
  4. The Plasma Dispersion Function
  5. The Linear BGK Equation with Density Dependence
  6. Preliminaries
  7. The Chapman--Enskog Series
  8. The Main Theorem
  9. Laplace--Fourier Analysis of the Main Equation
  10. The Spectral Theory of $\mathcal{L}_k$
  11. Explicit Solutions in Time-Frequency Space
  12. Hydrodynamic and Non-Hydrodynamic Scaling Laws: Proof of the Main Theorem
  13. Hydrodynamic Solutions
  14. Non-Hydrodynamic Solutions
  15. Summary, Discussion and Further Perspective
  16. ...and 3 more sections

Key Result

Theorem 3.2

Define the critical wave number as and consider the initial condition for some amplitude $A>0$ and some frequency $k_0\in\mathbb{Z}^d$. There exist two separate scaling regimes for equation maineqfrequency: In particular, for every Knudsen number $\tau$, there exists a solution to maineq whose dissipation rate of the mass density is or order $\tau^{-1}$.

Figures (3)

  • Figure 1: The integration contour $\Gamma_S$, contained in the upper half-plane, encircling the zero $\hat{\zeta}$ once in the positive direction.
  • Figure 2: The function $y\mapsto \mathrm{i} Z(\mathrm{i} y)$ (black solid) together with its asymptotics $y\mapsto -2\sqrt{\pi/2}e^{y^2/2}$ (red dashed).
  • Figure 3: The integration contour $\tilde{\Gamma}_S$, contained in the lower half-plane, encircling the zero $\hat{\zeta}_{\beta}$ once in the negative direction.

Theorems & Definitions (6)

  • Remark 3.1
  • Theorem 3.2: Hydrodynamic Versus Non-Hydrodynamic Solutions
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6