Nonuniqueness of weak solutions to the SQG equation

TL;DR

This paper demonstrates that weak solutions to both inviscid and dissipative SQG equations are not unique, resolving an open problem and highlighting the complexity of these fluid dynamics models.

Contribution

It proves nonuniqueness of weak solutions for both inviscid and dissipative SQG equations, including cases with strong fractional dissipation.

Findings

Weak solutions of inviscid SQG are not unique.

Weak solutions of dissipative SQG are not unique even with strong dissipation.

Addresses an open problem in the mathematical fluid dynamics community.

Abstract

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Sz\'ekelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.