Unstable vortices and non-uniqueness for 2D Euler and $\alpha$-gSQG

Abstract

In the sixties, Yudovich proved global existence and uniqueness of solutions for the 2D Euler incompressible equation in $L^1\cap L^{\infty }$. This result extends to the case with a force in $L_t^1(L^1\cap L^{\infty })$. Global existence in $L^1\cap L^p$, $1<p<\infty$, was obtained by R. DiPerna and A. Majda in 1987, but the problem of uniqueness, in this range of $p$, has remained open until Vishik’s work in 2018. Indeed, M. Vishik proved that there are forces in $L_t^1(L^1\cap L^p)$, $1<p<\infty$, such that at least two different solutions exist for 2D Euler starting in the same initial datum. A nice exposition of the proof of this result can be found in D. Albritton et al. (2024). In this paper, we will present an alternative approach to prove this theorem. More precisely, we will show a different way to construct unstable radial vortices with compact support which also give rise to non-uniqueness. This strategy also works for $\alpha$-gSQG, $0\le \alpha \le 1$. This family of transport equations, with velocity $\mathbf{u}=-{\nabla }^{\perp }(-\Delta {)}^{-1+\frac{\alpha }{2}}\theta$, for the scalar $\theta$, interpolates from 2D Euler ($\alpha =0$) to the surface quasi-geostrophic equation ($\alpha =1$).