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

This paper concerns with the stability of the plane Couette flow resulted from the motions of boundaries that the top boundary ${\mathrm{\Sigma }}_1$ and the bottom one ${\mathrm{\Sigma }}_0$ 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$\alpha<0$butviscosity$\mu$ 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.