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 ${\mathbb{T}}^d$. 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 $C_t^0{C}_x^{\alpha }$ with $\alpha <\frac{1}{2d+1}$. We note that in 1D, the regularity $\alpha <\frac{1}{3}$ is sharp as the energy identity is satisfied for solutions in $C^{\alpha }$ with $\alpha >\frac{1}{3}$. Without external forcing, Isett and Vicol constructed non-unique weak solutions for such active scalar equations with spatial regularity $C_x^{\alpha }$ for $\alpha <\frac{1}{4d+1}$.

30 pages