Vanishing viscosity limit of the stationary Navier-Stokes equations with the Navier-slip boundary and its application

IF 2.4 2区数学 Q1 MATHEMATICS

Journal of Differential Equations Pub Date : 2025-06-12 DOI:10.1016/j.jde.2025.113548

Xinghong Pan , Jianfeng Zhao

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引用次数: 0

Abstract

In this paper, we consider the vanishing viscosity limit of the stationary Navier-Stokes equations with the total Navier-slip boundary condition in a horizontally periodic strip. We will show that as the viscosity approaches to zero, there exist a sequence of solutions of the Navier-Stokes equations that approach that of the limiting Euler system. Moreover, we construct this sequence of solutions to keep the same cross-section flux with that of the limiting Euler, which is independent of the viscosity. Such construction of flux-conserved solutions is not easy to achieve in the case of the no-slip boundary condition. Due to the principle of the Prandtl-Batchelor theory, the limiting Euler solution can only be the Couette flow

(A y + B, 0)

for some suitable constants A and B. The constant A is determined by the boundary condition and the constant B is determined by the given flux.

As an application, we can show the structure stability of any Couette flow

(A y + B, 0)

for the stationary Navier-Stokes equations with fixed viscosity and suitably large flux when equipped with the total Navier-slip boundary condition.

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具有navier -滑移边界的平稳Navier-Stokes方程的消失黏度极限及其应用

本文考虑了在水平周期条上具有总navier -滑移边界条件的平稳Navier-Stokes方程的消失粘度极限。我们将证明，当粘度趋于零时，存在一系列接近极限欧拉系统的纳维-斯托克斯方程的解。此外，我们构造了这个解序列，以保持与极限欧拉方程相同的截面通量，这与粘度无关。在无滑移边界条件下，这种通量守恒解的构造是不容易实现的。由于Prandtl-Batchelor理论的原理，对于某些合适的常数A和B，极限欧拉解只能是Couette流（Ay+B,0），常数A由边界条件决定，常数B由给定的通量决定。作为一个应用，对于具有固定粘度和适当大通量的平稳Navier-Stokes方程，我们可以在具有总navier -滑移边界条件时，证明任意Couette流（Ay+B,0）的结构稳定性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Differential Equations 数学-数学

CiteScore

4.40

自引率

8.30%

发文量

543

审稿时长

9 months

期刊介绍： The Journal of Differential Equations is concerned with the theory and the application of differential equations. The articles published are addressed not only to mathematicians but also to those engineers, physicists, and other scientists for whom differential equations are valuable research tools. Research Areas Include: • Mathematical control theory • Ordinary differential equations • Partial differential equations • Stochastic differential equations • Topological dynamics • Related topics