Self‐similar instability and forced nonuniqueness: An application to the 2D euler equations

Building on an approach introduced by Golovkin in the ’60s, we show that nonuniqueness in some forced partial differential equations is a direct consequence of the existence of a self‐similar linearly unstable eigenvalue: the key point is a clever choice of the forcing term removing complicated nonlinear interactions. We use this method to give a short and self‐contained proof of nonuniqueness in 2D perfect fluids, first obtained in Vishik's groundbreaking result. In particular, we present a direct construction of a forced self‐similar unstable vortex, where we treat perturbatively the self‐similar operator in a new and more quantitative way.