Self-similar instability and forced nonuniqueness: an application to the 2D Euler equations

TL;DR

This paper demonstrates that nonuniqueness in certain forced PDEs, including 2D Euler equations, stems from self-similar unstable eigenvalues, using a novel approach that simplifies the proof and constructs explicit unstable vortex solutions.

Contribution

It introduces a new method linking self-similar instability to nonuniqueness in forced PDEs, providing a direct and quantitative proof for 2D Euler equations.

Findings

Nonuniqueness arises from self-similar unstable eigenvalues.

Constructs explicit forced self-similar unstable vortex.

Provides a simplified proof of Vishik's nonuniqueness result.

Abstract

Building on an approach introduced by Golovkin in the '60s, we show that nonuniqueness in some forced PDEs is a direct consequence of the existence of a self-similar linearly unstable eigenvalue: the key point is a clever choice of the forcing term removing complicated nonlinear interactions. We use this method to give a short and self-contained proof of nonuniqueness in 2D perfect fluids, first obtained in Vishik's groundbreaking result. In particular, we present a direct construction of a forced self-similar unstable vortex, where we treat perturbatively the self-similar operator in a new and more quantitative way.