Lower Bounds for Advection-Diffusion Equations: An Exploration with AI-Generated Proofs
For mathematicians and researchers in PDEs, this work demonstrates AI's capability to produce rigorous mathematical proofs, though the results themselves are incremental extensions of known techniques.
The paper establishes explicit lower bounds for advection-diffusion equations in three settings, with all constants explicit. The proofs were generated entirely by an AI system without expert human intervention.
We establish explicit lower bounds for advection-diffusion equations in three settings: a polynomial $\dot H^{-1}$ bound for inviscid shears with $u\in L^\infty_t W^{1,1}_y$, 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.