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 $$ \bigcap_{0 < \epsilon < 1} \dot{B}^{-\epsilon}_{\infty,\infty}(\mathbb{T}^d) \cap L^{2-\epsilon}(\mathbb{T}^d) $$ 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.