Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Note on Non-Hydrodynamic Solutions of Kinetic Systems

Florian Kogelbauer, Ilya Karlin

TL;DR

The paper investigates how kinetic theory relates to fluid dynamics by analyzing a linear, one-dimensional, three-component Grad system. Through a spectral decomposition into slow (hydrodynamic) and fast (non-hydrodynamic) manifolds, it demonstrates the existence of non-hydrodynamic solutions that diverge as the Knudsen number $\varepsilon$ vanishes, while the slow manifold closures recover Euler dynamics at leading order and Navier–Stokes corrections at $O(\varepsilon)$. This reveals a breakdown of the Chapman–Enskog expansion for a subset of kinetic solutions and supports viewing hydrodynamics as dynamics constrained to a slow invariant manifold. The work highlights the limitations of traditional hydrodynamic closures and motivates extending the invariant-manifold perspective to understand convergence in kinetic-to-fluid transitions, potentially across more complex or nonlinear models.

Abstract

We show that the one-dimensional three-component Grad system admits solutions that violate the Chapman--Enskog scaling in Knudsen number. In particular, there exist solutions that do not converge to the analogues of the Euler and Navier--Stokes equations for vanishing Knudsen number. These non-hydrodynamic solutions correspond to a fast spectral manifold in kinetic phase space.

A Note on Non-Hydrodynamic Solutions of Kinetic Systems

TL;DR

The paper investigates how kinetic theory relates to fluid dynamics by analyzing a linear, one-dimensional, three-component Grad system. Through a spectral decomposition into slow (hydrodynamic) and fast (non-hydrodynamic) manifolds, it demonstrates the existence of non-hydrodynamic solutions that diverge as the Knudsen number vanishes, while the slow manifold closures recover Euler dynamics at leading order and Navier–Stokes corrections at . This reveals a breakdown of the Chapman–Enskog expansion for a subset of kinetic solutions and supports viewing hydrodynamics as dynamics constrained to a slow invariant manifold. The work highlights the limitations of traditional hydrodynamic closures and motivates extending the invariant-manifold perspective to understand convergence in kinetic-to-fluid transitions, potentially across more complex or nonlinear models.

Abstract

We show that the one-dimensional three-component Grad system admits solutions that violate the Chapman--Enskog scaling in Knudsen number. In particular, there exist solutions that do not converge to the analogues of the Euler and Navier--Stokes equations for vanishing Knudsen number. These non-hydrodynamic solutions correspond to a fast spectral manifold in kinetic phase space.

Paper Structure

This paper contains 5 sections, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Three-Component Grad System: Spectral Analysis in Frequency Space
  3. Hydrodynamics on the Slow Manifold and Divergence in Decay Rates
  4. Conclusions
  5. Acknowledgment

Figures (2)

  • Figure 1: Spectrum of the linear Grad system \ref{['eqGrad']} for $\varepsilon=0.1$ and various wave numbers as derived in, e.g., kogelbauer2020slow. The fast real modes (star symbol on the real axis) accumulate at $-\frac{5}{9\varepsilon}$ (vertical dashed line at $\Re{\lambda} = -50/9$), while the slow complex conjugated modes (bell-shaped star symbol) accumulate at $\Re\lambda = -\frac{2}{9\varepsilon}$ (dashed line at $\Re{\lambda} = -20/9$) as $k\to\infty$. The smaller the Knudsen number, the more the axis of accumulation of the fast modes moves the left.
  • Figure 2: The functions $A$ (solid) and $B$ (dashed) in dependence of $\varepsilon k$.

Theorems & Definitions (4)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2