Vanishing viscosity limit of the 2D stationary Navier-Stokes equations outside a rotating disc and its application
Xinghong Pan, Jianfeng Zhao
TL;DR
This work analyzes the stationary 2D Navier–Stokes equations in the exterior of the unit disk with a boundary rotation perturbed by $\omega+\delta f(\theta)$ and analyzes the vanishing viscosity limit as $\varepsilon\to 0$. The authors identify the limiting Euler flow as a tangential Taylor–Couette profile $u^e(r)=\dfrac{A}{r}$, with $A$ determined by the Batchelor–Wood formula $A=\sqrt{\dfrac{1}{2\pi}\int_0^{2\pi}(\omega+\delta f)^2\,d\theta}$, and justify this limit by constructing a high-order boundary-layer expansion near $r=1$ and proving its validity. The analysis leverages matched asymptotics, a vorticity–stream function formulation, Fourier decomposition, and sophisticated weighted $\dot{H}^k$ energy estimates to control the error; a contraction mapping argument then yields a rigorous solution that converges to the Euler limit with quantified decays at infinity. Additionally, the paper shows an existence result for fixed viscosity outside the disk under large perturbations of the boundary rotation, linking to Yudovich’s Problem 11b on viscous flows past a rigid body. These results advance the understanding of the inviscid limit in exterior domains and provide a framework for high-precision boundary-layer justification in stationary settings.
Abstract
In this paper, we establish the vanishing viscosity limit result of the 2D stationary Navier-Stokes equations outside a rotating disc. On the boundary of the disc, the fluid is subjected to a small perturbation of a non zero rotation of rigid body. While at the spacial infinity, the fluid stays at rest. Due to the Prandtl-Batchelor theory, the limiting Euler solution is chosen to be the rotation flow $\frac{A}{r} e_θ$ for some suitable constant $A$, which is determined by the Batchelor-Wood formula. When the viscosity approaches to zero, we will construct a solution to the 2D Navier-Stokes equations by using higher order asymptotic approximation and show the validity of the boundary layer expansion. Also the asymptotic behavior of the solution at spacial infinity is obtained. Our result partially answers one of the open problems (Problem 11b) raised by V. I. Yudovich in [Eleven great problems of mathematical hydrodynamics, Mosc. Math. J. 3 (2003), no. 2, 711--737]. As an application, we can show an existence result to the 2D stationary Navier-Stokes equations with fixed viscosity outside a disc when the fluid is subjected to a large perturbation of a fast rotation of rigid body at the boundary.