Instability and nonuniqueness for the $2d$ Euler equations in vorticity form, after M. Vishik

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.