  In this expository work, we present Vishik's theorem on non-unique weak
solutions to the two-dimensional Euler equations on the whole space,
  \[ \partial_t \omega + u \cdot \nabla \omega = f \, , \quad u =
\frac{1}{2\pi} \frac{x^\perp}{|x|^2} \ast \omega \, , \] with initial vorticity
$\omega_0 \in L^1 \cap L^p$ and $f \in L^1_t (L^1 \cap L^p)_x$, $p < \infty$.
His theorem demonstrates, in particular, the sharpness of the Yudovich class.
An important intermediate step is the rigorous construction of an unstable
vortex, which is of independent physical and mathematical interest. We follow
the strategy of Vishik but allow ourselves certain deviations in the proof and
substantial deviations in our presentation, which emphasizes the underlying
dynamical point of view.
