  In the seminal work [39], Leray demonstrated the existence of global weak
solutions to the Navier-Stokes equations in three dimensions. We exhibit two
distinct Leray solutions with zero initial velocity and identical body force.
Our approach is to construct a `background' solution which is unstable for the
Navier-Stokes dynamics in similarity variables; its similarity profile is a
smooth, compactly supported vortex ring whose cross-section is a modification
of the unstable two-dimensional vortex constructed by Vishik in [43,44]. The
second solution is a trajectory on the unstable manifold associated to the
background solution, in accordance with the predictions of Jia and \v{S}ver\'ak
in [32,33]. Our solutions live precisely on the borderline of the known
well-posedness theory.
