A proof of a resolvent estimate for plane flow by new analytical and numerical techniques

Abstract: This thesis concerns stability of plane Couette flow in three space dimensions for the incompressible Navier-Stokes equations. We present new results for the resolvent corresponding to this flow. Previously, analytical bounds of the resolvent have been derived in parts of the unstable half-plane. In the remaining part, only bounds based on numerical computations in an infinite parameter domain are available. Due to the need for truncation of this infinite parameter domain, these results are mathematically insufficient.We obtain a new analytical bound of the resolvent at s = 0 in all but a compact subset of the parameter domain. This is done by deriving approximate solutions of the Orr-Sommerfeldt equation and bounding the errors made by the approximations. In the remaining compact set, we use standard numerical techniques to obtain a bound. Hence, this part of the proof is not rigorous in the mathematical sense.In the thesis, we present a way of making also the numerical part of the proof rigorous. By using analytical techniques, we reduce the remaining compact subset of the parameter domain to a finite set of parameter values. In this set, we need to compute bounds of the solution of a boundary value problem. By using a validated numerical method, such bounds can be obtained. In the last part of the thesis, we investigate a validated numerical method for enclosing the solutions of boundary value problems.