Vanishing viscosity limit of the stationary Navier-Stokes equations in an expanding annulus
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 (Ar+B1/r)eθ 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
@misc{pan2026vanishing,
title = {{Vanishing viscosity limit of the stationary Navier-Stokes equations in an expanding annulus}},
author = {Xinghong Pan and Chuankai Zhao and Jianfeng Zhao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17754},
eprint = {2610.17754},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17754},
}
RIS
TY - UNPB
TI - Vanishing viscosity limit of the stationary Navier-Stokes equations in an expanding annulus
AU - Xinghong Pan
AU - Chuankai Zhao
AU - Jianfeng Zhao
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17754
UR - https://arxiv.org/abs/2610.17754
AB - 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 (Ar+B1/r)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.
ER -