  This paper introduces a new method for proving global stability of fluid
flows through the construction of Lyapunov functionals. For finite dimensional
approximations of fluid systems, we show how one can exploit recently developed
optimization methods based on sum-of-squares decomposition to construct a
polynomial Lyapunov function. We then show how these methods can be extended to
infinite dimensional Navier-Stokes systems using robust optimization
techniques. Crucially, this extension requires only the solution of
infinite-dimensional linear eigenvalue problems and finite-dimensional
sum-of-squares optimization problems.
  We further show that subject to minor technical constraints, a general
polynomial Lyapunov function is always guaranteed to provide better results
than the classical energy methods in determining a lower-bound on the maximum
Reynolds number for which a flow is globally stable, if the flow does remain
globally stable for Reynolds numbers at least slightly beyond the energy
stability limit. Such polynomial functions can be searched for efficiently
using the SOS technique we propose.
