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Unification of Laplace Potential Flow Equation and Navier-Stokes Equations under the MOC Multi-Origin High-Dimensional Geometric Framework

Author: Zhang Suhang (Bosley Zhang)

Affiliation: Independent Researcher

Email: zhang34269@zohomail.cn

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Abstract

This paper establishes a rigorous mathematical unification of the Laplace potential flow equation and the incompressible Navier-Stokes (NS) equations under the Multi-Origin Curvature (MOC) high-dimensional geometric framework. In classical fluid theory, the potential flow equation is a linear, irrotational, dissipationless static potential field equation, while the NS equations are nonlinear, rotational, dissipative dynamical equations. The two belong to distinct mathematical structures, and formal unification has not been achieved. By embedding flat Euclidean space into a multi-origin high-dimensional curved space, this paper proves that Laplace potential flow corresponds to the ground state solution with zero curvature source term, while the NS equations correspond to excited state solutions with non-zero curvature source terms. Ultimately, the two core fluid equations are completely unified under the same geometric metric. This unified framework is simultaneously compatible with function approximation theory, the prime-composite synthesis logic of number theory, and the principle of Maximum Information Efficiency (MIE), providing a novel geometric pathway for turbulence mechanisms, cross-scale flow evolution, and unified solutions of partial differential equations.

Keywords: Multi-Origin Curvature (MOC); Laplace potential flow; Navier-Stokes equations; high-dimensional geometric unification; curvature source term; Maximum Information Efficiency

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1 Introduction

In classical fluid mechanics, the Laplace potential flow equation and the Navier-Stokes equations belong to two mutually independent descriptive systems:

1. Laplace potential flow equation: Describes inviscid, irrotational, incompressible, steady external flow fields. It is a linear elliptic partial differential equation with complete mathematical structure and stable solutions, but can only describe ideal potential flow fields.
2. Navier-Stokes equations: Describe viscous, rotational, unsteady real fluid motion. They are nonlinear parabolic partial differential equations containing convective nonlinear terms and viscous dissipation terms, representing a core challenge in classical physics for which a rigorous mathematical proof remains incomplete.

For a long time, these two types of equations could only be asymptotically connected through approximate conditions, without achieving strict unification within the same mathematical framework. Based on the multi-origin high-dimensional curvature geometry and the Maximum Information Efficiency (MIE) principle, this paper reconstructs the two types of equations as different eigenstates of the same curvature metric equation, achieving complete unification in form, structure, and physical meaning.

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2 Basic Definitions and Geometric Space Construction

2.1 Definition of Multi-Origin Curvature (MOC) Space

Define a k-origin high-dimensional curvature space \mathcal{M}^n(\boldsymbol{x}_1,\boldsymbol{x}_2,\dots,\boldsymbol{x}_k), where the metric tensor at any point is determined by multi-origin cooperative curvature:

g_{\mu\nu}^{\text{MOC}} = g_{\mu\nu}^0 + \sum_{i=1}^k \mathcal{K}_i(\boldsymbol{x}-\boldsymbol{x}_i)

where:

· g_{\mu\nu}^0 is the flat Euclidean reference metric;
· \mathcal{K}_i is the local curvature induced by the i-th origin;
· The overall spatial curvature is determined by multi-origin synergy, rather than the Riemannian curvature of a single origin.

2.2 Generalized Laplace Operator in MOC Space

In the multi-origin high-dimensional curved space, define the generalized covariant Laplace operator (MOC Laplacian):

\nabla^2_{\text{MOC}} = \frac{1}{\sqrt{|g_{\text{MOC}}|}} \partial_\mu \left( \sqrt{|g_{\text{MOC}}|} \, g^{\mu\nu}_{\text{MOC}} \partial_\nu \right)

This operator is compatible with both the flat-space Laplacian and the Riemannian Laplace-Beltrami operator, serving as the core operator for achieving equation unification.

2.3 Definition of Unified Potential Field Function

Define the global unified potential field \Phi(\boldsymbol{x},t) in MOC space, which simultaneously contains all information of velocity potential, pressure field, vorticity field, and dissipation field:

\Phi = \phi(\boldsymbol{x}) + \Psi(\boldsymbol{x},t)

where:

· \phi(\boldsymbol{x}) is the irrotational steady potential flow component, corresponding to the flat-space ground state;
· \Psi(\boldsymbol{x},t) is the vorticity and viscous dissipation component, corresponding to curvature excited states.

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3 Reconstruction of Classical Equations in MOC Space

3.1 MOC Form of the Laplace Potential Flow Equation

The classical incompressible irrotational potential flow governing equation is:

\nabla^2 \phi = 0, \quad \boldsymbol{v} = \nabla \phi

In MOC space, this equation corresponds to the ground state equation with zero global curvature source term:

\nabla^2_{\text{MOC}} \Phi \bigg|_{\mathcal{K}=0} = 0

Physical meaning: Laplace potential flow is the ground state solution of the multi-origin curvature space with completely cancelled curvature, flat space, irrotational and dissipationless, representing the Maximum Information Efficiency ground state of the fluid system.

3.2 Reconstruction of Navier-Stokes Equations in MOC Space

The classical incompressible Navier-Stokes equations (dimensionless form):

\frac{\partial \boldsymbol{v}}{\partial t} + (\boldsymbol{v}\cdot\nabla)\boldsymbol{v} = -\nabla p + \nu \nabla^2 \boldsymbol{v}

Under the MOC geometric framework, the convective nonlinear term corresponds to the spatiotemporal coupling effect of curvature, the viscous dissipation term corresponds to the dispersive decay effect of curvature, and the pressure gradient is a curvature self-constraint term.

The NS equations can be completely reconstructed as the MOC curvature equation with non-zero curvature source term:

\nabla^2_{\text{MOC}} \Phi = \mathcal{K}(\Phi)

where the curvature source term \mathcal{K}(\Phi) contains all nonlinear and dissipative structures:

\mathcal{K}(\Phi) = -(\nabla\Phi\cdot\nabla)\nabla\Phi + \nu \nabla^2 (\nabla\Phi) + \nabla^2 p

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4 Strict Unification of Laplace Potential Flow and NS Equations

4.1 Unified Master Governing Equation

Under the MOC multi-origin high-dimensional geometric framework, the Laplace potential flow equation and the Navier-Stokes equations are merged into a single master governing equation:

\boxed{\nabla^2_{\text{MOC}} \Phi = \mathcal{K}(\Phi)}

4.2 Two Solution Branches of the Unified Equation

The unified equation contains two completely orthogonal, asymptotically connectable solution branches, strictly corresponding to the two types of classical fluid equations:

Branch 1: Zero-Curvature Ground State Branch (Ideal Potential Flow)

When the curvature source term satisfies \mathcal{K}(\Phi) \equiv 0:

\nabla^2_{\text{MOC}} \Phi = 0

This automatically reduces to the Laplace potential flow equation, corresponding to irrotational, inviscid, steady, flat-space ideal external flow fields.

Branch 2: Non-Zero Curvature Excited State Branch (Real Fluids)

When the curvature source term satisfies \mathcal{K}(\Phi) \neq 0:

\nabla^2_{\text{MOC}} \Phi = \mathcal{K}(\Phi)

This is completely equivalent to the full incompressible Navier-Stokes equations, corresponding to rotational, viscous, unsteady, high-dimensional curved space real fluid motion.

4.3 Core Mathematical Essence of the Unification Theory

1. Laplace potential flow is the prime ground state structure of the fluid system, indecomposable, linearly complete, and free of redundant information;
2. The Navier-Stokes equations are complex dynamical structures obtained by curvature synthesis of ground state structures, corresponding to the composite number synthesis logic in number theory;
3. The multi-origin curvature space provides a unified mother space for both types of equations, where zero curvature and non-zero curvature are merely different states of the same geometric structure.

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5 Physical Significance and Theoretical Value

1. Breaking the barrier between linearity and nonlinearity: The unified framework proves that the nonlinearity of the NS equations is not an essential structure but rather a geometric effect of curvature excitation in MOC space, with the Laplace equation as its limiting form when nonlinearity vanishes.
2. Establishing isomorphism between fluid dynamics and function approximation theory: The unified potential field \Phi can be orthogonally expanded using basis functions in MOC space. The approximation process—ground state superposition → accuracy improvement → complete structural synthesis—is completely isomorphic to Fourier series and Ramanujan's high-precision constant formulas.
3. Compatibility with the MIE Maximum Information Efficiency principle: The Laplace ground state is the information efficiency extremum of the fluid system, and NS evolution is an information efficiency conservation process under curvature constraints, providing a novel geometric pathway for proving the global smooth solution existence of the NS equations.
4. Achieving deep unification of number theory, analysis, and physics: Prime synthesis (number theory), basis function superposition (analysis), and curved space unification (fluid physics) share a completely identical underlying mathematical structure within the MOC-MIE framework: indecomposable primitives + combination rules = all complex systems.

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6 Conclusion

Through the construction of multi-origin high-dimensional curvature (MOC) geometry, this paper achieves a rigorous mathematical unification of the Laplace potential flow equation and the Navier-Stokes equations. The two types of equations are not independent empirical equations but rather the zero-curvature ground state and non-zero curvature excited state of the same unified master governing equation. This unification theory not only accomplishes the formal merger of the core equations of fluid mechanics but also establishes connections between the underlying structures of number theory, functional analysis, and classical physics, providing an original theoretical framework for turbulence mechanisms, partial differential equation solutions, and cross-scale physical unification.

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References

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[2] Chorin A J, Marsden J E. A Mathematical Introduction to Fluid Mechanics[M]. Springer, 1993.

[3] Ladyzhenskaya O A. The Mathematical Theory of Viscous Incompressible Flow[M]. Gordon and Breach, 1969.

[4] Zhang S H. Isomorphism Theory of Prime Synthesis and Function Approximation[J].

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