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The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit

Renjun Duan, Shuangqian Liu, Robert M. Strain, Anita Yang

TL;DR

This work analyzes the Boltzmann equation in the diffusive limit for 3D Couette flow in a channel, proving that the first-order kinetic dynamics is governed by a perturbed incompressible Navier–Stokes–Fourier system around the Couette solution. By applying a Fourier transform on $\mathbb{T}^2$, employing anisotropic Chemin–Lerner spaces with Wiener algebra, and using Caflisch's decomposition, the authors establish the existence and positivity of a 3D stationary kinetic Couette flow $F_{st}^\epsilon$ and prove its exponential stability for the time-dependent problem; furthermore, in the absence of external forcing ($\Phi\equiv0$), the 3D flow converges uniformly in $\epsilon$ to the 1D planar Couette flow. The results are shown under small shear and forcing, and the analysis yields precise bounds tying the kinetic remainder to hydrodynamic quantities through a robust $L^2$–$L^\infty$ framework. Overall, the paper provides a rigorous kinetic-geometry understanding of multi-D Couette flows and their hydrodynamic limits, with techniques potentially applicable to other non-equilibrium steady states.

Abstract

In the paper we study the Boltzmann equation in the diffusive limit in a channel domain $\mathbb{T}^2\times (-1,1)$ for the 3D kinetic Couette flow. Our results demonstrate that the first-order approximation of the solutions is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid Couette flow. Moverover, in the absence of external forces, the 3D kinetic Couette flow asymptotically converges over time to the 1D steady planar kinetic Couette flow. Our proof relies on (i) the Fourier transform on $\mathbb{T}^2$ to essentially reduce the 3D problem to a one-dimensional one, (ii) anisotropic Chemin-Lerner type function spaces, incorporating the Wiener algebra, to control nonlinear terms and address the singularity associated with a small Knudsen number in the diffusive limit, and (iii) Caflisch's decomposition, combined with the $L^2\cap L^\infty$ interplay technique, to manage the growth of large velocities.

The 3D kinetic Couette flow via the Boltzmann equation in the diffusive limit

TL;DR

This work analyzes the Boltzmann equation in the diffusive limit for 3D Couette flow in a channel, proving that the first-order kinetic dynamics is governed by a perturbed incompressible Navier–Stokes–Fourier system around the Couette solution. By applying a Fourier transform on , employing anisotropic Chemin–Lerner spaces with Wiener algebra, and using Caflisch's decomposition, the authors establish the existence and positivity of a 3D stationary kinetic Couette flow and prove its exponential stability for the time-dependent problem; furthermore, in the absence of external forcing (), the 3D flow converges uniformly in to the 1D planar Couette flow. The results are shown under small shear and forcing, and the analysis yields precise bounds tying the kinetic remainder to hydrodynamic quantities through a robust – framework. Overall, the paper provides a rigorous kinetic-geometry understanding of multi-D Couette flows and their hydrodynamic limits, with techniques potentially applicable to other non-equilibrium steady states.

Abstract

In the paper we study the Boltzmann equation in the diffusive limit in a channel domain for the 3D kinetic Couette flow. Our results demonstrate that the first-order approximation of the solutions is governed by the perturbed incompressible Navier-Stokes-Fourier system around the fluid Couette flow. Moverover, in the absence of external forces, the 3D kinetic Couette flow asymptotically converges over time to the 1D steady planar kinetic Couette flow. Our proof relies on (i) the Fourier transform on to essentially reduce the 3D problem to a one-dimensional one, (ii) anisotropic Chemin-Lerner type function spaces, incorporating the Wiener algebra, to control nonlinear terms and address the singularity associated with a small Knudsen number in the diffusive limit, and (iii) Caflisch's decomposition, combined with the interplay technique, to manage the growth of large velocities.
Paper Structure (19 sections, 19 theorems, 485 equations)

This paper contains 19 sections, 19 theorems, 485 equations.

Table of Contents

  1. Introduction
  2. Problem
  3. Reformulation
  4. Steady problem
  5. Time-dependent problem
  6. Main results
  7. Literature review
  8. Strategies of the proof
  9. Notations
  10. Organization of the paper
  11. Steady problem
  12. Macroscopic estimates
  13. Lg estimates
  14. Lg estimate on Lg
  15. Lg estimate on Lg
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\gamma\in[0,1]$ and $l_\infty\gg l_2\gg4$. There exists a constant $\delta_0>0$ such that for any $\alpha,\epsilon\in (0,\delta_0)$ and for any $\Phi(z)\in C([-1,1];\mathbb{R}^3)$ with $\|\Phi(z)\|_{2}\leq \delta_0$, the steady boundary value problem st-rbe and st-obd admits a unique strong sol as well as the following things:

Theorems & Definitions (31)

  • Remark 1.1
  • Theorem 1.1: Existence of the steady solution
  • Theorem 1.2: Stability of the steady solution
  • Remark 1.2
  • Lemma 2.1
  • Corollary 2.1
  • proof : The proof of Theorem \ref{['sta-th']}
  • Lemma A.1
  • proof
  • Lemma A.2
  • ...and 21 more