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

TL;DR

This paper investigates the stability of two-dimensional plane Couette flow under Navier boundary conditions, extending known results from no-slip conditions to more general boundary conditions with specific stability criteria.

Contribution

It establishes new stability results for plane Couette flow with Navier boundary conditions, including cases with mixed boundary conditions and varying parameters.

Findings

Existence of exponentially stable Couette flows under certain Navier boundary conditions.

Extension of classical stability results from no-slip to Navier boundary conditions.

Conditions on Navier coefficients and boundary velocities ensuring stability.

Abstract

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.