Nontrivial integrable weak stationary solutions to active scalar equations with non-odd drift
Nicholas Gismondi
TL;DR
The paper addresses the existence of nontrivial weak stationary solutions to a class of active scalar equations with a non-odd nonlocal drift. It develops a convex integration scheme using intermittent building blocks to construct a velocity–scalar pair $(u,\theta)$ with $u=T\theta$ that lie in $\bigcap_{0<\epsilon<1}(\dot{B}^{-\epsilon}_{\infty,\infty}(\mathbb{T}^d) \cap L^{2-\epsilon}(\mathbb{T}^d))$, and defines $\mathbb{P}_{\not=0}(\theta u)$ as a paraproduct in $\dot{H}^{-s}$ to give a meaningful weak formulation. The method handles the obstruction from non-odd Calderón–Zygmund multipliers by decomposing the nonlinear term into a mean and its projection, then inverting divergence to cancel leading errors, with intermittency controlled to ensure convergence. The result extends prior stationary constructions by achieving integrable weak solutions in a Besov–Lebesgue intersection setting and clarifying the role of intermittency and paraproducts in defining weak solutions. This advances understanding of non-uniqueness and regularity thresholds in stationary active scalar dynamics and highlights the potential applicability to a broader class of nonlocal drift operators.
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.