An Onsager-type theorem for SQG equations
Mimi Dai, Vikram Giri, Răzvan-Octavian Radu
Source 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).
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