The Boltzmann Equation for 2D Taylor-Couette Flow

TL;DR

This paper proves the existence of non-equilibrium steady solutions to the Boltzmann equation modeling 2D Taylor-Couette flow with small shear, revealing polynomial velocity tails and stability under radial perturbations.

Contribution

It develops a novel framework for constructing steady Boltzmann solutions in a geometrically complex setting with shear, including uniform estimates and stability analysis.

Findings

Existence of steady solutions for small shear rate.

Polynomial tail behavior at large velocities.

Stability of solutions under radial perturbations.

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 $\alpha$. 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 $F(r,v_{r},v_{\phi}-\alpha\frac{r-r_{1}}{1-r_{1}},v_{z})$, where the translation angular velocity $\alpha\frac{r-r_{1}}{1-r_{1}}$ 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 $\alpha$ 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 $\alpha$ 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 $(\epsilon,\sigma)$-approximation argument for the construction of solutions. In particular, we obtain uniform macroscopic dissipation estimates in the absence of mass conservation for $\sigma \in [0,1)$ 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.