A proof of Onsager's Conjecture for the SQG equation

TL;DR

This paper proves the generalized Onsager conjecture for the SQG equation by constructing solutions with critical regularity that do not conserve the Hamiltonian, advancing understanding of energy conservation thresholds.

Contribution

It provides the first construction of solutions at the critical regularity threshold for SQG that fail Hamiltonian conservation, confirming the conjecture.

Findings

Solutions with critical regularity do not conserve Hamiltonian.

The result confirms the threshold regularity for energy conservation in SQG.

Advances understanding of Onsager's conjecture in the context of SQG.

Abstract

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG.