A direct approach to nonuniqueness and failure of compactness for the SQG equation

TL;DR

This paper provides a new proof of nonuniqueness of weak solutions to the SQG equation and shows that any smooth, compactly supported scalar field conserving the integral can be approximated by solutions.

Contribution

It introduces an alternative, scalar-level proof of nonuniqueness and demonstrates the realization of certain scalar fields as weak limits of SQG solutions.

Findings

Reproves nonuniqueness of weak solutions to SQG

Shows any smooth, compactly supported scalar can be approximated by solutions

Provides a scalar-level approach to SQG analysis

Abstract

We give an alternative proof of the nonuniqueness of weak solutions to the surface quasigeostrophic equation (SQG) first shown in [Buckmaster-Shkoller-Vicol, '16]. Our approach proceeds directly at the level of the scalar field. Furthermore, we prove that every smooth scalar field with compact support that conserves the integral can be realized as a weak limit of solutions to SQG.