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arXiv · 2507.02794

Vanishing Vertical Viscosity in Two-Dimensional Anisotropic Navier-Stokes Equations with No-Slip Boundary Conditions: An $L^p$ result

Chongsheng Cao·Yanqiu Guo
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Abstract

This paper studies the inviscid limit problem for the two-dimensional Navier-Stokes equations with anisotropic viscosity. The fluid is assumed to be bounded above and below by impenetrable walls, with a no-slip boundary condition imposed on the bottom wall. For $H^2$ initial velocity, we establish strong convergence in the $L^p$ norm to the limiting problem as the vertical viscosity approaches zero, for any $2\leq p <\infty$. The main challenge lies in the mismatch of boundary conditions - specifically, the no-slip condition in the original problem versus the slip condition in the limiting problem.

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Chongsheng Cao, Yanqiu Guo. 2025-07-03. Vanishing Vertical Viscosity in Two-Dimensional Anisotropic Navier-Stokes Equations with No-Slip Boundary Conditions: An $L^p$ result. https://arxiv.org/abs/2507.02794

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