This paper concerns with the stability of the plane Couette flow resulted from the motions of boundaries that the top boundary $\Sigma_1$ and the bottom one $\Sigma_0$ move with constant velocities $(a,0)$ and $(b,0)$, respectively. If one imposes Dirichlet boundary condition on the top boundary and Navier boundary condition on the bottom boundary with Navier coefficient $\alpha$, there always exists a plane Couette flow which is exponentially stable for nonnegative $\alpha$ and any positive viscosity $\mu$ and any $a, b \in \mathbb{R}$, or, for $\alpha<0$ but viscosity $\mu$ and the moving velocities of boundaries $(a,0), (b,0)$ satisfy some conditions stated in Theorem 1.1. However, if we impose Navier boundary conditions on both boundaries with Navier coefficients $\alpha_0$ and $\alpha_1$, then it is proved that there also exists a plane Couette flow (including constant flow or trivial steady states) which is exponentially stable provided that any one of two conditions on $\alpha_0,\alpha_1$, $a, b$ and $\mu$ in Theorem 1.2 holds. Therefore, the known results for the stability of incompressible Couette flow to no-slip (Dirichlet) boundary value problems are extended to the Navier boundary value problems.

中文翻译：

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二维平面 Couette 流到具有 Navier 边界条件的不可压缩 Navier-Stokes 方程的稳定性

本文关注的是由边界运动引起的平面库埃特流的稳定性：顶部边界 $\Sigma_1$ 和底部边界 $\Sigma_0$ 以恒定速度 $(a,0)$ 和 $(b,0 )$，分别。如果在顶部边界施加 Dirichlet 边界条件，在底部边界施加 Navier 边界条件，Navier 系数为 $\alpha$，则总是存在一个平面 Couette 流，该流对于非负 $\alpha$ 和任何正粘度 $\mu 都是指数稳定的$ 和任何 $a, b \in \mathbb{R}$，或者，对于 $\alpha<0$ 但粘度 $\mu$ 和边界 $(a,0), (b,0)$ 的移动速度满足定理 1.1 中规定的一些条件。然而，如果我们在具有 Navier 系数 $\alpha_0$ 和 $\alpha_1$ 的两个边界上施加 Navier 边界条件，那么证明也存在一个指数稳定的平面库埃特流（包括恒流或平凡稳态），条件是$\alpha_0、\alpha_1$、$a、b$和$\mu的两个条件中的任意一个定理 1.2 中的 $ 成立。因此，不可压缩库埃特流稳定性的已知结果到无滑移 (Dirichlet) 边值问题被扩展到 Navier 边值问题。