Vanishing viscosity limit of the 2D stationary Navier-Stokes equations outside a rotating disc and its application

TL;DR

This paper proves the vanishing viscosity limit for 2D stationary Navier-Stokes equations outside a rotating disc, constructing solutions with boundary layer analysis and addressing an open problem in mathematical hydrodynamics.

Contribution

It establishes the vanishing viscosity limit with boundary layer expansion for rotating boundary conditions, partially solving an open problem by Yudovich.

Findings

Valid boundary layer expansion constructed as viscosity approaches zero

Asymptotic behavior at infinity characterized

Existence of solutions under large boundary perturbations confirmed

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 $\frac{A}{r} e_\theta$ 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.