Resolvent Estimates for 2-Dimensional Perturbations of Plane Couette Flow

Abstract

We present results concerning resolvent estimates for the linear operator associated with the system of differential equations governing 2 dimensional perturbations of plane Couette flow. We prove estimates on the L2 norm of the resolvent of this operator showing this norm to be proportional to the Reynolds number R for a region of the unstable half plane. For the remaining region, we show that the problem can be reduced to estimating the solution of a homogeneous ordinary differential equation with non-homogeneous boundary conditions. Numerical approximations indicate that the norm of the resolvent is proportional to R in the whole region of interest.