Communications in Mathematical Sciences

Volume 18 (2020)

Number 5

Stability for two-dimensional plane Couette flow to the incompressible Navier–Stokes equations with Navier boundary conditions

Pages: 1233 – 1258



Shijin Ding (South China Research Center for Applied Mathematics and Interdisciplinary Studies, and School of Mathematical Sciences, South China Normal University, Guangzhou, China)

Zhilin Lin (School of Mathematical Sciences, South China Normal University, Guangzhou, China)


This paper concerns with the stability of the plane Couette flow resulting from the motions of boundaries such 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 $\alpha \lt 0$ 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.


incompressible Navier–Stokes equations, stability, plane Couette flow, Navier boundary condition

2010 Mathematics Subject Classification

35Q30, 76E05, 76N10

Ding’s research is supported by the National Natural Science Foundation of China (No.11371152, No.11571117, No.11871005 and No.11771155) and Guangdong Provincial Natural Science Foundation (No.2017A030313003).

Received 12 March 2018

Accepted 13 February 2020

Published 23 September 2020