BOUNDS FOR THRESHOLD AMPLITUDES IN SUBCRITICAL SHEAR FLOWS

A general theory which can be used to derive bounds on solutions to th e Navier-Stokes equations is presented. The behaviour of the resolvent of the linear operator in the unstable half-plane is used to bound th e energy growth of the full nonlinear problem. Plane Couette flow is u sed as an example. The norm of the resolvent in plane Couette flow in the unstable half-plane is proportional to the square of the Reynolds number (R). This is now used to predict the asymptotic behaviour of th e threshold amplitude below which all disturbances eventually decay. A lower bound is found to be R Examples, obained through direct numeric al simulation, give an upper bound on the threshold curve, and predict a threshold of R-1. The discrepancy is discussed in the light of a mo del problem.