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

TL;DR

This paper presents explicit lower bounds for advection-diffusion equations across various flow settings, with all proofs generated automatically by an AI system, demonstrating AI's potential in rigorous mathematical proof creation.

Contribution

First application of AI-generated proofs to establish explicit bounds in complex PDEs, showcasing AI's capability in producing rigorous mathematical results.

Findings

Established polynomial ot H^{-1} bounds for inviscid shears

Derived uniform positive lower bounds on mixing scales

Produced exponential L^2 bounds for oscillatory flows

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.