  An important open problem in the theory of the Navier-Stokes equations is the
uniqueness of the Leray-Hopf weak solutions with $L^2$ initial data.
  In this paper we give sufficient conditions for non-uniqueness in terms of
spectral properties of a natural linear operator associated to scale-invariant
solutions recently constructed in \cite{JiaSverak}. If the spectral conditions
are satisfied, non-uniqueness and ill-posedness can appear for quite benign
compactly supported data, just at the borderline of applicability of the
classical perturbation theory. The verification of the spectral conditions
seems to be approachable by relatively straightforward numerical simulations
which involve only smooth functions.
