An Onsager-type theorem for SQG

We construct non-trivial weak solutions θ∈ Cₜ⁰Cₓ⁰⁻ 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 Ct,x⁰ ⊂ Lt,x³ (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).