Resolvent estimates for 2-dimensional perturbations of plane Couette flow

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 $L_2$ 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. Submitted September 24, 2002. Published October 27, 2002. Math Subject Classifications: 76E05, 47A10. Key Words: Couette flow; resolvent estimates