Non-unique weak solutions of forced SQG

Abstract: We construct non-unique weak solutions $\theta\in C_t<sup>0C_x<sup>{0-}$ 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 $C_t<sup>0C_x<sup>{\alpha}$ with $\alpha&lt;-\frac15$ were constructed by Buckmaster, Shkoller and Vicol, and Isett and Ma.