Lower Bounds for Advection-Diffusion Equations: An Exploration with AI-Generated Proofs

We establish explicit lower bounds for advection-diffusion equations in three settings: a polynomial ${\dot{H}}^{-1}$ bound for inviscid shears with $u\in L_t^{\mathrm{\infty }}{W}_y^{1,1}$, a uniform positive lower bound on the mixing scale for diffusive shears, and an exponential $L^2$ bound for rapidly oscillating time-periodic flows. All constants are explicit in the data. The proofs were generated entirely by a multi-agent math proving system, QED, without expert human intervention, serving as a test of AI’s capability to produce rigorous mathematics.