Vanishing viscosity limit of the stationary Navier-Stokes equations in an expanding annulus

Xinghong Pan; Chuankai Zhao; Jianfeng Zhao

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arXiv · 2610.17754

Vanishing viscosity limit of the stationary Navier-Stokes equations in an expanding annulus

Xinghong Pan·Chuankai Zhao·Jianfeng Zhao

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Abstract

In this paper, we establish vanishing viscosity limit of the 2D stationary Navier-Stokes equations in an expanding annulus with a fixed width. The flow is subjected to a large perturbation of a rotation of rigid body at one boundary and stays at rest at another boundary. Due to this different boundary condition setting, the boundary layer effects are different near the interior and exterior boundaries, which result in different thickness of boundary layer and different leading order boundary layer equations. We will construct an approximate solution to this 2D Navier-Stokes equations by using higher order asymptotic approximation and show the validity of the boundary layer expansion. The leading order of the Euler solution is the Taylor-Couette flow $\left(Ar+B\frac{1}{r}\right)e_θ$ for some suitable constant $A$ and $B$, which is determined by using the principle of the Prandtl-Batchelor theory. The perturbation of the tangential velocity becomes large as the annulus expands, which causes heavy difficulties when we show the convergence of the approximate solution. Variously weighted energy estimates up to third-order derivatives of the error solution are constructed to close the energy of the nonlinear system.

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BibTeX RIS

Xinghong Pan, Chuankai Zhao, Jianfeng Zhao. 2026-10-13. Vanishing viscosity limit of the stationary Navier-Stokes equations in an expanding annulus. https://arxiv.org/abs/2610.17754

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