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arXiv · 2404.15995

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

Ángel Castro·Daniel Faraco·Francisco Mengual·Marcos Solera
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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.

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Ángel Castro, Daniel Faraco, Francisco Mengual, Marcos Solera. 2024-04-24. A proof of Vishik's nonuniqueness Theorem for the forced 2D Euler equation. https://doi.org/10.1515/crelle-2025-0025

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