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<p>In this paper, we establish the vanishing viscosity limit of the 2D Navier-Stokes equations in a horizontally periodic strip. In the vertical direction, the horizontal component of the velocity is subject to two different types of boundary conditions: at the lower boundary, we give the degenerate zero boundary condition, while at the upper boundary, a small smooth perturbation of a non-zero constant is prescribed. Due to this different boundary condition setting, the boundary layer effects are different near the lower and upper boundaries, which result in different thicknesses of boundary layers and different leading-order boundary-layer equations. We construct an approximate solution to the 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 Couette flow (<i>Ay</i>, 0) for some suitable constant <i>A</i>, which is determined by using the principle of the Prandtl-Batchelor theory.</p>

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Vanishing viscosity limits of the Navier-Stokes equations in a horizontally periodic strip

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • Mingwen Fei,
  • Xinghong Pan,
  • Jianfeng Zhao

摘要

In this paper, we establish the vanishing viscosity limit of the 2D Navier-Stokes equations in a horizontally periodic strip. In the vertical direction, the horizontal component of the velocity is subject to two different types of boundary conditions: at the lower boundary, we give the degenerate zero boundary condition, while at the upper boundary, a small smooth perturbation of a non-zero constant is prescribed. Due to this different boundary condition setting, the boundary layer effects are different near the lower and upper boundaries, which result in different thicknesses of boundary layers and different leading-order boundary-layer equations. We construct an approximate solution to the 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 Couette flow (Ay, 0) for some suitable constant A, which is determined by using the principle of the Prandtl-Batchelor theory.