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

Xinghong Pan; Jianfeng Zhao

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arXiv · 2511.12953

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

Xinghong Pan·Jianfeng Zhao

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Abstract

We justify the vanishing-viscosity limit for 2D stationary Navier-Stokes flows in the exterior of a rotating disc. The boundary data are a small perturbation of a nonzero rigid rotation, and the velocity vanishes at spatial infinity. Guided by Prandtl-Batchelor theory, we select the limiting Euler flow $\frac{A}{r}e_θ$, where $A$ is fixed by the Batchelor-Wood formula. In the regime of small viscosity, we construct stationary Navier-Stokes solutions through a higher-order matched asymptotic expansion and prove the validity of the corresponding boundary-layer expansion. We also identify the far-field asymptotics of the Navier-Stokes solution. This gives a partial resolution of Problem 11b in Yudovich [Eleven great problems of mathematical hydrodynamics, Mosc. Math. J. 3 (2003), no. 2, 711--737]. As an application, for fixed viscosity we prove the existence of stationary solutions in the same exterior domain under a large perturbation of a fast rigid rotation at the boundary.

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Xinghong Pan, Jianfeng Zhao. 2026-09-20. Vanishing viscosity limit of the 2D stationary Navier-Stokes equations outside a rotating disc and its application. https://arxiv.org/abs/2511.12953

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