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We establish new non-uniqueness results for the forced inviscid surfacequasi-geostrophic equation, via an alternating formulation of convexintegration techniques. Our results imply non-uniquenesss in the class of weaksolutions with $|\nabla|^{-1}\theta\in C_tC_x^\alpha$, for any $\alpha<1$.

Non-uniqueness up to the Onsager threshold for the forced SQG equation