Non-uniqueness of forced active scalar equations with even drift operators

Abstract: We consider forced active scalar equations with even and homogeneous degree 0 drift operator on $\mathbb T<sup>d$. Inspired by the non-uniqueness construction for dyadic fluid models, by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space $C_t<sup>0C_x<sup>\alpha$ with $\alpha&lt;\frac{1}{2d+1}$. We note that in 1D, the regularity $\alpha&lt;\frac13$ is sharp as the energy identity is satisfied for solutions in $C<sup>\alpha$ with $\alpha&gt;\frac13$. Without external forcing, Isett and Vicol constructed non-unique weak solutions for such active scalar equations with spatial regularity $C_x<sup>\alpha$ for $\alpha&lt;\frac{1}{4d+1}$.