Non-unique weak solutions of forced SQG

We construct non-unique weak solutions $\theta \in C_t^0{C}_x^{0-}$ for forced surface quasi-geostrophic (SQG) equation. This is achieved through a convex integration scheme adapted to the sum-difference system of two distinct solutions. Without external forcing, non-unique weak solutions $\theta$ in space $C_t^0{C}_x^{\alpha }$ with $\alpha <-\frac{1}{5}$ were constructed by Buckmaster, Shkoller and Vicol, and Isett and Ma.