  We show that for a certain family of initial data, there exist non-unique
weak solutions to the 3D incompressible Euler equations satisfying the weak
energy inequality, whereas the weak limit of every sequence of Leray-Hopf weak
solutions for the Navier-Stokes equations, with the same initial data, and as
the viscosity tends to zero, is uniquely determined and equals the shear flow
solution of the Euler equations. This simple example suggests that, also in
more general situations, the vanishing viscosity limit of the Navier-Stokes
equations could serve as a uniqueness criterion for weak solutions of the Euler
equations.
