Lower bounds for advection–diffusion

The problem asks for constructive lower bounds preventing excessively rapid mixing of mean-zero solutions on the two-dimensional torus. A 2026 preprint titled Lower Bounds for Advection-Diffusion Equations addresses all three requested regimes; no poser or earlier attribution is identified.

May 2026 constructive resolution

Theorem 1.1 proves, for inviscid shears, $\|\theta(t)\|_{\dot H^{-1}}\geq c_{*}/(1+t^{2})$. Theorem 1.2 gives, for bounded diffusive shears, $\|\rho(t)\|_{L^{2}}\geq\|\rho_{0}\|_{L^{2}}e^{-c_{2}t}$ and a uniform positive lower bound for $\|\rho\|_{\dot H^{-1}}/\|\rho\|_{L^{2}}$. Theorem 1.3 gives explicit $A_{0},c_{A},C>0$ such that $\|\rho(t)\|_{L^{2}}\geq Ce^{-c_{A}t}$ for $A>A_{0}$ in rapidly oscillating time-periodic flows, ruling out superexponential decay. The paper states that its proofs were generated by the named system QED, with wording edited by the authors.

Current status (as of August 2026): The three requested lower bounds are stated and proved in an arXiv preprint; no further cases remain open within the stated formulation.