[Paper Review] The Steady Boltzmann and Navier-Stokes Equations

Kazuo Aoki, François Golse+1 more|arXiv (Cornell University)|Apr 10, 2015

Gas Dynamics and Kinetic Theory33 references3 citations

TL;DR

This paper investigates the mathematical parallels and disparities between the steady Boltzmann and incompressible Navier-Stokes equations in bounded domains. It identifies that viscous heating—dependent on the Froude-to-Mach number ratio—determines whether solutions to the Boltzmann equation converge to the Navier-Stokes-Fourier system. Crucially, while the Navier-Stokes equations with smooth forces always admit smooth solutions, the Boltzmann equation may fail to have any solution under the same conditions unless the force is zero, a breakdown linked to temperature rise in numerical simulations of the evolution problem.

ABSTRACT

The paper discusses the similarities and the differences in the mathematical theories of the steady Boltzmann and incompressible Navier-Stokes equations posed in a bounded domain. First we discuss two different scaling limits in which solutions of the steady Boltzmann equation have an asymptotic behavior described by the steady Navier-Stokes Fourier system. Whether this system includes the viscous heating term depends on the ratio of the Froude number to the Mach number of the gas flow. While the steady Navier-Stokes equations with smooth divergence-free external force always have at least one smooth solutions, the Boltzmann equation with the same external force set in the torus, or in a bounded domain with specular reflection of gas molecules at the boundary may fail to have any solution, unless the force field is identically zero. Viscous heating seems to be of key importance in this situation. The nonexistence of any steady solution of the Boltzmann equation in this context seems related to the increase of temperature for the evolution problem, a phenomenon that we have established with the help of numerical simulations on the Boltzmann equation and the BGK model.

Motivation & Objective

• To analyze the mathematical similarities and differences between steady Boltzmann and incompressible Navier-Stokes equations in bounded domains.
• To determine under what scaling limits the steady Boltzmann equation asymptotically approaches the steady Navier-Stokes-Fourier system.
• To investigate why the Boltzmann equation may fail to admit solutions under smooth, non-zero external forces, unlike the Navier-Stokes equations.
• To explore the physical origin of this nonexistence, particularly the role of viscous heating and temperature increase in time-evolving solutions.
• To establish a connection between nonuniqueness in steady Navier-Stokes solutions and numerical bifurcations observed in the Boltzmann equation.

Proposed method

• Formal derivation of two variants of the incompressible Navier-Stokes-Fourier system from the steady Boltzmann equation under distinct scaling assumptions.
• Analysis of the periodic boundary setting to compare solution existence between the Boltzmann and Navier-Stokes equations.
• Use of Gaussian averages and orthogonality relations in velocity space to compute moments of collisional operators.
• Application of tensor decomposition techniques to evaluate statistical moments of velocity-dependent fields such as $ A_{ij} $, $ B_i $, and $ C_{jkl} $.
• Numerical simulations of the time-dependent Boltzmann equation and BGK model to study temperature evolution and viscous heating effects.
• Use of functional analysis tools, including the linearized collision operator $ ilde{L} $, to analyze the structure of solutions and their regularity.

Experimental results

Research questions

• RQ1 Under what scaling limits does the steady Boltzmann equation converge to the steady Navier-Stokes-Fourier system?
• RQ2 Why does the steady Boltzmann equation fail to admit solutions under non-zero smooth external forces, while the Navier-Stokes equations always do so?
• RQ3 How does the ratio of Froude to Mach number determine whether viscous heating appears in the hydrodynamic limit of the Boltzmann equation?
• RQ4 What is the role of viscous heating in the nonexistence of steady solutions for the Boltzmann equation with non-zero forces?
• RQ5 How do numerical simulations of the time-evolving Boltzmann equation reveal the physical mechanism behind solution breakdown?

Key findings

• The steady Navier-Stokes equations with smooth, divergence-free external forces always admit at least one smooth solution, regardless of the domain.
• In contrast, the steady Boltzmann equation with the same external force in a torus or bounded domain with specular reflection may fail to have any solution unless the force is identically zero.
• The presence or absence of viscous heating in the hydrodynamic limit depends critically on the ratio of the Froude number to the Mach number of the gas flow.
• Numerical simulations of the time-evolving Boltzmann equation and BGK model show a significant rise in temperature, suggesting this heating effect as a key cause of solution nonexistence in the steady regime.
• The nonexistence of steady solutions for the Boltzmann equation under non-zero forces is physically linked to the unbounded temperature increase observed in the time evolution, which prevents steady-state formation.
• Tensor moment computations reveal that $ u = \frac{1}{15}\langle|v|^4 \alpha(|v|)\rangle > 0 $, confirming the positivity of the viscosity coefficient derived from the linearized collision operator.