An Onsager-type theorem for SQG equations

Abstract

We construct non-trivial weak solutions $\theta \in C_t^0{C}_x^{0-}$ 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 and Vicol (2015)) and resolves the Onsager conjecture for SQG equations. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced by Giri and Radu (2023).