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

TL;DR

This paper introduces a quaternion-complex framework for Navier-Stokes equations that reveals geometric structures, proves global regularity, and resolves the Clay Millennium Prize problem, with implications for turbulence understanding and environmental modeling.

Contribution

It develops a unified quaternion-complex formulation of Navier-Stokes equations, proving global regularity and providing new geometric insights into turbulence and fluid stability.

Findings

Proves existence of unique global smooth solutions for 3D Navier-Stokes.

Demonstrates turbulence as breakdown of quaternion-analyticity.

Links incompressibility to complex analysis constraints.

Abstract

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.