计算机辅助的三维周期纳维-斯托克斯流非线性族全局正则性
Computer-assisted global regularity across nonlinear families of three-dimensional periodic Navier-Stokes flows
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中文总结 AI 辅助
本文提出计算机辅助框架，结合有限参考轨迹与公共误差界，为三维周期纳维-斯托克斯流连续族建立全局正则性，并通过数值实验和神经算子验证其有效性。
中文摘要 AI 辅助
数值模拟揭示了涡旋如何拉伸和传递能量，但建立光滑演化需要超出模拟分辨率的界。本文开发了一个计算机辅助框架，为三维周期纳维-斯托克斯流的连续族建立全局正则性。其核心构造将有限参考轨迹与覆盖中心场区间和无穷多个光滑扰动模式的公共误差界相结合。该方法在谱截断前保留完整的非线性残差，并控制演化直至粘性耗散保证所有后续时刻的正则性。应用于循环剪切、Arnold-Beltrami-Childress和三分量Taylor-Green场，得到显式扰动半径，并包含直接Fourier-Wiener小性判据之外的初始条件。参数一致扩展覆盖了非Beltrami Taylor-Green中心的连通族，无需对单个参数值重复证明。由4,096个配置组成的集成，辅以1,600条细化轨迹和公共湍流数据，将数学可观测量与谱传递和涡旋几何联系起来。匹配的神经算子实验表明，物理信息训练改善了物理预测，同时也揭示这些增益不一定改善证明极限初始条件的发现。总之，这些结果为在指定流族上建立正则性提供了可重用方法，并为评估学习预测如何辅助严格计算提供了定量设置。
英文摘要
Numerical simulations reveal how vortices stretch and transfer energy, but establishing smooth evolution requires bounds that remain valid beyond the simulated resolution. Here I develop a computer-assisted framework that establishes global regularity for continuous families of three-dimensional periodic Navier-Stokes flows. Its central construction combines finite reference trajectories with a common error bound that covers an interval of centre fields and infinitely many smooth perturbation modes. The method retains the complete nonlinear residual before spectral truncation and controls the evolution until viscous decay guarantees regularity for all subsequent times. Applications to cyclic-shear, Arnold-Beltrami-Childress and three-component Taylor-Green fields yield explicit perturbation radii and include initial conditions outside the direct Fourier-Wiener smallness criterion. A parameter-uniform extension covers a connected family of non-Beltrami Taylor-Green centres without repeating the proof for individual parameter values. An ensemble of 4,096 configurations, supplemented by 1,600 refinement trajectories and public turbulence data, connects the mathematical observables to spectral transfer and vortex geometry. Matched neural-operator experiments show that physics-informed training improves physical prediction, while also revealing that these gains do not necessarily improve the discovery of proof-limiting initial conditions. Together, these results provide a reusable method for establishing regularity across prescribed flow families and a quantitative setting for evaluating how learned predictions can assist rigorous computation.
发表机构
- Federal University of Bahia（巴伊亚联邦大学）
机构由 AI 辅助整理，请以论文原文为准。
