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

TL;DR

This paper discusses Vishik's theorem on the non-uniqueness of weak solutions to the 2D Euler equations in vorticity form, highlighting the sharpness of the Yudovich class and constructing an unstable vortex.

Contribution

It provides an expository presentation of Vishik's theorem with some deviations, emphasizing the dynamical perspective and including the construction of an unstable vortex.

Findings

Demonstrates non-uniqueness of weak solutions in the Yudovich class

Constructs an unstable vortex with physical and mathematical significance

Highlights the sharpness of the Yudovich class for 2D Euler equations

Abstract

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.