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

TL;DR

This paper constructs nontrivial weak stationary solutions for a class of active scalar equations with non-odd nonlocal drift using convex integration, revealing complex solution structures in specific function spaces.

Contribution

It introduces a convex integration method to produce weak solutions with non-odd drifts, expanding understanding of solution behaviors in active scalar equations.

Findings

Solutions lie in specific Besov and Lebesgue spaces.

Use of oscillatory corrections with decreasing intermittency.

Construction demonstrates existence of complex stationary solutions.

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.