An Onsager-type theorem for SQG

TL;DR

This paper proves an Onsager-type theorem for the SQG equations by constructing weak solutions that violate Hamiltonian conservation, using Nash iteration and linear decoupling methods, thus resolving a key conjecture.

Contribution

It establishes the Onsager conjecture for SQG by constructing non-trivial weak solutions with specific regularity properties.

Findings

Constructed weak solutions with compact support in time.

Demonstrated violation of Hamiltonian conservation.

Resolved the Onsager conjecture for SQG.

Abstract

We construct non-trivial weak solutions $\theta\in C_t^0C_x^{0-}$ to the surface quasi-geostrophic (SQG) equations, which have compact support in time and, thus, violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class $C_{t,x}^0 \subset L_{t,x}^3$ (Isett-Vicol, 2015) and resolves the Onsager conjecture for SQG. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced in Giri-Radu (2023).