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

2026-05-19

AI Summary

This work presents the first fully AI-generated, rigorous explicit lower bounds for the advection-diffusion equation across three distinct flow regimes—non-diffusive shear flows, diffusive shear flows, and rapidly oscillating time-periodic flows—without any human intervention. Leveraging the multi-agent automated reasoning system QED, which integrates partial differential equation analysis with formal verification, the study derives data-dependent explicit constants: a polynomial Ḣ⁻¹ decay bound for non-diffusive shear flows, a uniform positive lower bound on mixing scales for diffusive shear flows, and an exponential L² decay bound for rapidly oscillating flows. This achievement marks a breakthrough in the application of artificial intelligence to complex mathematical proofs.

Abstract

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.

Problem

• advection-diffusion equations
• lower bounds
• mixing scale
• Sobolev norm
• time-periodic flows

Innovation

• advection-diffusion equations
• AI-generated proofs
• explicit lower bounds
• multi-agent proving system
• mixing scale