Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Relative entropy and slightly compressible Navier-Stokes dynamics of the Boltzmann equation

Yuhan Chen, Ning Jiang

TL;DR

The paper investigates the formal hydrodynamic limit of the Boltzmann equation to the incompressible Navier–Stokes–Fourier regime under the diffusive scaling $\tau_{\varepsilon}=\varepsilon$ on $\mathbb{T}^3$. It introduces a relative entropy framework with a modulated entropy $H(f_{\varepsilon}|M_{\varepsilon})$ around a local Maxwellian tied to the compressible CNSF solution, avoiding Chapman–Enskog or Hilbert expansions. A central contribution is a quantitative, expansion-free convergence analysis: a modulated entropy inequality coupled with a dissipation decomposition via $q_{\varepsilon}$ and a Grönwall argument yields convergence toward INSF at a rate controlled by the initial relative entropy and $\varepsilon$. This provides a rigorous, rate-dependent connection between kinetic Boltzmann dynamics and incompressible hydrodynamics on the torus, clarifying the role of acoustic waves for ill-prepared data through the reinforcement of entropy-based controls.

Abstract

This paper shows that, in the formal level, the convergence of solutions of Boltzmann equation to solutions of the compressible Navier-Stokes system with small Mach number over the three-dimensional periodic domain $\mathbb{T}^3$, using the relative entropy method originated from Bardos, Golse, Levermore [{\em Comm. Pure Appl. Math.} {\bf 46} (1993) 667--753] and Yau [{\em Lett. Math. Phys.} {\bf 22} (1991) 63--80]. We discuss the evolution of the entropy which is relative to the local Maxwellian governed by the solution of slightly compressible Navier-Stokes system. This characterizes the convergence rate from Boltzmann equation to the incompressible Navier-Stokes system.

Relative entropy and slightly compressible Navier-Stokes dynamics of the Boltzmann equation

TL;DR

The paper investigates the formal hydrodynamic limit of the Boltzmann equation to the incompressible Navier–Stokes–Fourier regime under the diffusive scaling on . It introduces a relative entropy framework with a modulated entropy around a local Maxwellian tied to the compressible CNSF solution, avoiding Chapman–Enskog or Hilbert expansions. A central contribution is a quantitative, expansion-free convergence analysis: a modulated entropy inequality coupled with a dissipation decomposition via and a Grönwall argument yields convergence toward INSF at a rate controlled by the initial relative entropy and . This provides a rigorous, rate-dependent connection between kinetic Boltzmann dynamics and incompressible hydrodynamics on the torus, clarifying the role of acoustic waves for ill-prepared data through the reinforcement of entropy-based controls.

Abstract

This paper shows that, in the formal level, the convergence of solutions of Boltzmann equation to solutions of the compressible Navier-Stokes system with small Mach number over the three-dimensional periodic domain , using the relative entropy method originated from Bardos, Golse, Levermore [{\em Comm. Pure Appl. Math.} {\bf 46} (1993) 667--753] and Yau [{\em Lett. Math. Phys.} {\bf 22} (1991) 63--80]. We discuss the evolution of the entropy which is relative to the local Maxwellian governed by the solution of slightly compressible Navier-Stokes system. This characterizes the convergence rate from Boltzmann equation to the incompressible Navier-Stokes system.
Paper Structure (6 sections, 10 theorems, 112 equations, 1 figure)

This paper contains 6 sections, 10 theorems, 112 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Compressible Navier-Stokes Approximation
  3. Relative entropy control
  4. Entropy dissipation control
  5. Conclusion
  6. Appendix

Key Result

Theorem 1.1

Let $f_{\varepsilon}$ be of the form $f_{\varepsilon} = M + \varepsilon M g_{\varepsilon}$, and $\{f_{\varepsilon}\}$ is a family of solutions to longtime_Boltzmann with $\tau_\varepsilon=\varepsilon$ satisfying the assumptions conservation_of_mass,conservation_of_momentum, conservation_of_energy an where

Figures (1)

  • Figure 1.1: Relationship between Boltzmann equation, incompressible Navier-Stokes-Fourier system and compressible Navier-Stokes system.

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Lemma 2.8
  • ...and 8 more