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

Vanishing viscosity as a selection principle for the Euler equations: The case of 3D shear flow