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 for the incompressible Navier-Stokes equations in a two dimensional slab domain. If one imposes Dirichlet boundary condition on the top boundary and Navier boundary condition on the bottom boundary with Navier coefficients $\alpha$, there always exists a plane Couette flow which is exponentially stable for nonnegative $\alpha$ and any positive viscosity $\mu$, or, for $\alpha<0$ but viscosity $\mu$ satisfies some conditions, see 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 flow) which is exponentially stable provided that any one of two conditions on $\alpha_0,\alpha_1$ and $\mu$ in Theorem 1.2 holds. Therefore, the known results on the stability of incompressible Couette flow to Dirichlet boundary value problems are extended to the Navier boundary value problems.