STABILITY FOR TWO-DIMENSIONAL PLANE COUETTE FLOW TO THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS WITH NAVIER BOUNDARY CONDITIONS

摘要

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 is an element of 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.