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 $$\bigcap _{0<\hat{I}\mu <1}{\dot{B}}_{\infty ,\infty }^{-\hat{I}\mu }({\mathbb{T}}^d)\cap L^{2-\hat{I}\mu }({\mathbb{T}}^d)$$ for $d\ge 2$. 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.

25 pages