Unstable vortices, sharp non-uniqueness with forcing, and global smooth solutions for the SQG equation

TL;DR

This paper demonstrates non-uniqueness of weak solutions for the forced $ ext{SQG}$ equation in certain regularity regimes, constructs unstable vortices, and proves the existence of global smooth solutions without forcing.

Contribution

It establishes non-uniqueness in supercritical Sobolev spaces, constructs unstable vortex solutions, and shows global smooth solutions for the unforced $ ext{SQG}$ equation.

Findings

Non-uniqueness of weak solutions in supercritical regimes

Construction of smooth, compactly supported vortices with instability

Existence of global smooth solutions without forcing

Abstract

We prove non-uniqueness of weak solutions to the forced $\alpha$-SQG equation with Sobolev regularity $W^{s,p}$ in the supercritical regime $s < \alpha + \frac{2}{p}$, covering the 2D Euler equation ($\alpha = 0$), the Surface Quasi-Geostrophic equation ($\alpha = 1$), and the intermediate cases. A key step is the construction of smooth, compactly supported vortices that exhibit non-linear instability. As a by-product, we show existence of global smooth solutions to the (unforced) $\alpha$-SQG equation that are neither rotating nor traveling.