On Pseudospectral Bound for Non-selfadjoint Operators and Its Application to Stability of Kolmogorov Flows

We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient nu is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale O(nu(-1/2)) for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale O(nu(-1)). Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows.