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The Boltzmann Equation for 2D Taylor-Couette Flow

Renjun Duan, Weiqiang Wang, Yong Wang

TL;DR

This work establishes the existence and stability of a non-equilibrium Taylor–Couette type steady state for the steady Boltzmann equation in a 2D annulus between coaxial rotating cylinders under a small angular shear. It develops a geometrically corrected reformulation, employs Caflisch decomposition and a two-parameter $(\varepsilon,\sigma)$-approximation to construct solutions, and proves uniform $L^{\infty}$ and $L^{2}$ estimates to close the nonlinear fixed-point argument. A key contribution is the handling of boundary curvature through a $\frac{1}{1-\eta}$ geometric correction and the demonstration of polynomial velocity tails in the steady profile, along with exponential large-time stability under radial perturbations to ensure non-negativity. The paper also extends to an unsteady problem, delivering local well-posedness and robust a priori estimates that underpin global behavior and the steady-state justification. Overall, it provides a rigorous kinetic-theoretic framework for weakly sheared Taylor–Couette flows of hard potentials with diffuse boundaries.

Abstract

In this paper, we investigate the existence of 2-D Taylor-Couette flow for a rarefied gas between two coaxial rotating cylinders, characterized by differing angular velocities at the outer boundary $\{r=1\}$ and the inner boundary $\{r=r_{1}>0\}$, with a small relative strength denoted by $α$. We formulate the problem using the steady Boltzmann equation in polar coordinates and seek a solution invariant under rotation. We assume that the steady state has the specific form $F(r,v_{r},v_φ-α\frac{r-r_{1}}{1-r_{1}},v_{z})$, where the translation angular velocity $α\frac{r-r_{1}}{1-r_{1}}$ is linearly sheared along the radial direction. With this ansatz, the problem is reduced to solve the nonlinear steady Boltzmann equation with geometric correction, subject to an external shear force of strength $α$ and the homogeneous non-moving diffuse reflection boundary condition. We establish the existence of a non-equilibrium steady solution for any small enough shear rate $α$ through Caflisch's decomposition, complemented by careful uniform estimates based on Guo's $L^{\infty} \cap L^2$ framework. The steady profile displays a polynomial tail behavior at large velocities. For the proof, we develop a delicate double-parameter $(ε,σ)$-approximation argument for the construction of solutions. In particular, we obtain uniform macroscopic dissipation estimates in the absence of mass conservation for $σ\in [0,1)$ getting close to 1. Additionally, due to the non-trivial geometric effects, we develop subtle constructions of test functions by solving second-order ODEs with geometric corrections to establish macroscopic dissipation. Furthermore, we justify the non-negativity of the steady profile by demonstrating its large-time asymptotic stability with an exponential convergence rate under radial perturbations.

The Boltzmann Equation for 2D Taylor-Couette Flow

TL;DR

This work establishes the existence and stability of a non-equilibrium Taylor–Couette type steady state for the steady Boltzmann equation in a 2D annulus between coaxial rotating cylinders under a small angular shear. It develops a geometrically corrected reformulation, employs Caflisch decomposition and a two-parameter -approximation to construct solutions, and proves uniform and estimates to close the nonlinear fixed-point argument. A key contribution is the handling of boundary curvature through a geometric correction and the demonstration of polynomial velocity tails in the steady profile, along with exponential large-time stability under radial perturbations to ensure non-negativity. The paper also extends to an unsteady problem, delivering local well-posedness and robust a priori estimates that underpin global behavior and the steady-state justification. Overall, it provides a rigorous kinetic-theoretic framework for weakly sheared Taylor–Couette flows of hard potentials with diffuse boundaries.

Abstract

In this paper, we investigate the existence of 2-D Taylor-Couette flow for a rarefied gas between two coaxial rotating cylinders, characterized by differing angular velocities at the outer boundary and the inner boundary , with a small relative strength denoted by . We formulate the problem using the steady Boltzmann equation in polar coordinates and seek a solution invariant under rotation. We assume that the steady state has the specific form , where the translation angular velocity is linearly sheared along the radial direction. With this ansatz, the problem is reduced to solve the nonlinear steady Boltzmann equation with geometric correction, subject to an external shear force of strength and the homogeneous non-moving diffuse reflection boundary condition. We establish the existence of a non-equilibrium steady solution for any small enough shear rate through Caflisch's decomposition, complemented by careful uniform estimates based on Guo's framework. The steady profile displays a polynomial tail behavior at large velocities. For the proof, we develop a delicate double-parameter -approximation argument for the construction of solutions. In particular, we obtain uniform macroscopic dissipation estimates in the absence of mass conservation for getting close to 1. Additionally, due to the non-trivial geometric effects, we develop subtle constructions of test functions by solving second-order ODEs with geometric corrections to establish macroscopic dissipation. Furthermore, we justify the non-negativity of the steady profile by demonstrating its large-time asymptotic stability with an exponential convergence rate under radial perturbations.
Paper Structure (18 sections, 17 theorems, 590 equations, 1 figure)

This paper contains 18 sections, 17 theorems, 590 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Reformulation of problem
  4. Asymptotic analysis
  5. Main results
  6. The linearized approximate systems
  7. A priori $L^{\infty}$ estimate for approximate systems （2.9)--(2.10)
  8. A priori $L^{2}$ estimates for approximate systems (2.9)--(2.10)
  9. Existence of the linearized approximate system
  10. The limit $\sigma\to 1$
  11. The limit $\epsilon\to 0$
  12. Existence for the nonlinear system: Proof of Theorem \ref{['thm1']}
  13. Unsteady problem
  14. Local-in-time existence of （7.2）
  15. A priori $L^{\infty}$ estimates
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $0\leq\gamma\leq 1$ and $\ell_{\infty}\gg 4$. For any given total mass $M_{0}>0$, there exist positive constants $\alpha_{*}$ and $C_{0}$ such that for any $\alpha\in (0,\alpha_{*})$, the steady boundary value problem 1.1--1.2${\rm (}$equivalently 1.16--1.17${\rm )}$ and tm admits a unique solut where $\mathfrak{u}_{\phi}=\frac{\alpha(\eta_{1}-\eta)}{\eta_{1}}$, $\tilde{M}_{0}$ is determined b

Figures (1)

  • Figure 3.1: The schematic of backward characteristic lines

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1
  • Corollary 5.2
  • ...and 9 more