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

We establish new non-uniqueness results for the forced inviscid surface quasi-geostrophic equation, via an alternating formulation of convex integration techniques. Our results imply non-uniquenesss in the class of weak solutions with $\mathrm{\mid }\mathrm{\nabla }{\mathrm{\mid }}^{-1}\theta \in C_t{C}_x^{\alpha }$, for any $\alpha <1$.