Vanishing viscosity non-unique solutions to the forced 2D Euler Equations

The forced 2D Euler equations exhibit non-unique solutions with vorticity in $L^p$, $p>1$, whereas the corresponding Navier-Stokes solutions are unique. We investigate whether the inviscid limit $\nu \to 0^{+}$ from the forced 2D Navier-Stokes to Euler equations is a selection principle capable of “resolving” the non-uniqueness. We focus on solutions in a neighborhood of the non-uniqueness scenario discovered by Vishik; specifically, we incorporate viscosity $\nu$ and consider $O(\epsilon )$ size perturbations of the initial datum. We discover a uniqueness threshold $\epsilon \sim {\nu }^{{\kappa }_{\mathrm{c}}}$, below which the vanishing viscosity solution is unique and radial, and at which there are viscous solutions converging to non-unique, non-radial solutions.