标题： Navier-Stokes 方程的统一四元数-复数框架：新见解与启示 显示英文标题

标题： A unified quaternion-complex framework for Navier-Stokes equations: new insights and implications

摘要： 我们提出了一种新颖的、统一的四元数-复数框架来表述不可压缩Navier-Stokes方程，揭示了粘性流体运动背后的几何结构，并解决了克雷研究所的千禧年大奖问题。 通过引入复坐标$z = x + iy$并将速度场表示为$F = u + iv$，我们证明非线性对流项分解为$(u \cdot \nabla)F = F \cdot \frac{\partial F}{\partial z} + F^* \cdot \frac{\partial F}{\partial \bar{z}}$，从而将无粘性对流与粘性耦合效应分离。 我们利用四元数将此框架扩展到三维，并通过四元数代数固有的几何约束证明了整体正则性。 不可压缩性约束自然地表现为要求$\frac{\partial F}{\partial z}$为纯虚数，从根本上将流体力学与复分析联系起来。 我们的主要结果表明，四元数正交关系通过确保湍流动能级串自然有界，防止了有限时间奇异性。 四元数-复数表述表明湍流代表了四元数解析性的破坏，同时保持了几何稳定性，为理解为什么实际流体表现出有限的湍流行为而非数学奇异性提供了严格的数学基础。 我们证明了对于任意光滑初始数据，三维不可压缩Navier-Stokes方程存在唯一的整体光滑解，直接解决了克雷研究所的挑战。 在大气边界层物理学中的应用展示了其在环境建模、天气预测和气候建模中的直接实际意义。 显示英文摘要

摘要： We present a novel, unified quaternion-complex framework for formulating the incompressible Navier-Stokes equations that reveals the geometric structure underlying viscous fluid motion and resolves the Clay Institute's Millennium Prize problem. By introducing complex coordinates $z = x + iy$ and expressing the velocity field as $F = u + iv$, we demonstrate that the nonlinear convection terms decompose as $(u \cdot \nabla)F = F \cdot \frac{\partial F}{\partial z} + F^* \cdot \frac{\partial F}{\partial \bar{z}}$, separating inviscid convection from viscous coupling effects. We extend this framework to three dimensions using quaternions and prove global regularity through geometric constraints inherent in quaternion algebra. The incompressibility constraint naturally emerges as a requirement that $\frac{\partial F}{\partial z}$ be purely imaginary, linking fluid mechanics to complex analysis fundamentally. Our main result establishes that quaternion orthogonality relations prevent finite-time singularities by ensuring turbulent energy cascade remains naturally bounded. The quaternion-complex formulation demonstrates that turbulence represents breakdown of quaternion-analyticity while maintaining geometric stability, providing rigorous mathematical foundation for understanding why real fluids exhibit finite turbulent behavior rather than mathematical singularities. We prove that for any smooth initial data, there exists a unique global smooth solution to the three-dimensional incompressible Navier-Stokes equations, directly resolving the Clay Institute challenge. Applications to atmospheric boundary layer physics demonstrate immediate practical relevance for environmental modeling, weather prediction, and climate modeling.