Nontrivial integrable weak stationary solutions to active scalar equations with non-odd drift

Abstract: In this paper we construct nontrivial weak solutions to a class of stationary active scalar equations with a non-odd nonlocal operator in the drift term using a convex integration scheme. We show our solutions lie in $$ \bigcap_{0 < ε< 1} \dot{B}^{{-ε}_{\infty,\infty}(\mathbb{T}^d)} \cap L^{{2-ε}(\mathbb{T}^d)} $$ for $d \geq 2$. The key ingredient of the construction is the use of highly oscillatory corrections with a variable degree of intermittency, which is arranged to decrease to zero at higher stages of the iteration procedure.