Resolvent estimates for plane Couette flow

We discuss the problem of deriving estimates for the resolvent of the linear operator associated with three dimensional perturbations of plane Couette flow, and determining its dependence on the Reynolds number $R$. Depending on the values of the parameters involved, we derive estimates analytically. For the remaining values of the parameters, we prove that deriving estimates for the resolvent can be reduced to estimating the solutions of a 4th order linear homogeneous ordinary differential equation with non-homogeneous boundary conditions. We study these boundary value problems numerically. Our results indicate the $L_2$ norm of the resolvent to be proportional to $R^2$.