  Verifying nonlinear stability of a laminar fluid flow against all
perturbations is a central challenge in fluid dynamics. Past results rely on
monotonic decrease of a perturbation energy or a similar quadratic generalized
energy. None show stability for the many flows that seem to be stable despite
these energies growing transiently. Here a broadly applicable method to verify
global stability of such flows is presented. It uses polynomial optimization
computations to construct non-quadratic Lyapunov functions that decrease
monotonically. The method is used to verify global stability of 2D plane
Couette flow at Reynolds numbers above the energy stability threshold found by
Orr in 1907. This is the first global stability result for any flow that
surpasses the energy method.
