AI自动生成严格证明，给出三种流体方程的精确下界。
Lower Bounds for Advection-Diffusion Equations: An Exploration with AI-Generated Proofs
- 用AI多智能体系统生成数学证明，无需人工干预。
- 对无粘剪切流、扩散剪切流和周期振荡流分别给出下界结果。
- 结果常数显式可算，适合验证AI数学能力的研究者参考。
我们针对三类情形建立了标量输运-扩散方程的显式下界：对于无粘剪切流（速度场 $u\in L^\infty_t W^{1,1}_y$），给出了 $\ ext{dot}H^{-1}$ 空间中多项式阶的下界；对扩散剪切流，得到混合尺度的统一正下界；对快速振荡的周期性时变流，获得 $L^2$ 范数的指数级下界。所有常数均显式依赖于问题数据。全部证明由名为 QED 的多智能体数学证明系统在无专家人工干预下生成，作为测试人工智能生成严谨数学的能力。
原文摘要 · Abstract (English)
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.
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