A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation

TL;DR

This paper presents a simplified proof of Vishik's nonuniqueness theorem for the forced 2D Euler equation in certain vorticity classes, using an alternative unstable vortex construction.

Contribution

It introduces a more straightforward construction of an unstable vortex, simplifying Vishik's original proof of nonuniqueness in the 2D Euler equations.

Findings

Established nonuniqueness in the specified vorticity class.

Constructed a piecewise constant unstable vortex as part of the proof.

Simplified the proof structure compared to Vishik's original approach.

Abstract

We give a simpler proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation in the vorticity class $L^1\cap L^p$ with $2<p<\infty$. The main simplification is an alternative construction of a smooth and compactly supported unstable vortex, which is split into two steps: Firstly, we construct a piecewise constant unstable vortex, and secondly, we find a regularization through a fixed point argument. This simpler structure of the unstable vortex yields a simplification of the other parts of Vishik's proof.