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Fundamental Axioms of Fluid Dynamics: Tangential Convergence and Normal Boundedness — An Analysis of Turbulence Finiteness, Global Flow Stability, and the Domain of N-S Blow-up Solutions

Author: Suhang Zhang

Affiliation: Luoyang School of Mathematics

Abstract

The global regularity of the three-dimensional incompressible Navier-Stokes equations remains a long-standing, unresolved core problem in mathematical physics. The recent construction of finite-time blow-up solutions has further triggered widespread controversy regarding whether singularity divergence exists in turbulent evolution. To distinguish between the mathematical solution sets of unconstrained partial differential equations and the physical solutions of real fluids in nature, this paper proposes and establishes two fundamental axioms for the evolution of real fluids: the Axiom of Tangential Convergence and the Axiom of Normal Boundedness. Through this axiomatic system, it is rigorously deduced that the tangential statistical field of real fluid flow tends toward a stable elliptical steady state over the long term, ensuring the persistent existence of a global mean flow direction. The transverse disturbance gradients of the flow field possess a globally uniform upper bound. All turbulent fluctuations, vortex breakdowns, and boundary layer disturbances evolve with finite amplitudes, inherently precluding finite-time gradient blow-up. This paper explicitly points out that artificially constructed blow-up solutions of the N-S equations are mathematical special cases within an unconstrained initial value space, and do not belong to the physical solution subspace of natural fluids. Turbulence is essentially a finite disturbance state situated between smooth steady states and mathematical blow-up singularities, possessing the inherent characteristic of "local disordered fluctuations and global topological order." While this study does not solve the full-set regularity proposition of the original Millennium Prize Problem defined by the Clay Mathematics Institute, it systematically resolves the core physical controversies derived from the Millennium Prize Problem, providing a unified fundamental theoretical framework for the finiteness of real fluid turbulence, transition mechanisms, and global flow field stability.

Keywords: Navier-Stokes equations; fundamental axioms of fluids; tangential convergence; normal boundedness; turbulence finiteness; flow field topological constraints; blow-up solutions; physical solution subspace

1. Introduction

The global regularity of the three-dimensional incompressible Navier-Stokes equations is a foundational core problem in contemporary mathematical physics and fluid mechanics. The core mathematical subject of this problem is the entire set of smooth solutions under unconstrained initial values and unconstrained external forces, aiming to determine whether the solutions are globally smooth or exhibit finite-time singularity blow-up.

In recent years, with the emergence of finite-time blow-up solution constructions, widespread cognitive confusion has arisen in academia: numerous studies directly equate the mathematical blow-up characteristics of the unconstrained N-S equations with the evolutionary trends of real fluid turbulence, defaulting that the development of high-Reynolds-number turbulence will ultimately lead to gradient divergence and singularity blow-up.

However, all natural observations, wind tunnel tests, and engineering flow phenomena present a consistent conclusion: turbulence is ubiquitous, vortex structures continuously break down and reorganize, and boundary layer transitions occur frequently; yet, physical flow fields never exhibit the real phenomenon of infinite gradients or velocity blow-up. Even in high-supersonic strong disturbances, strong shear, and transient impact flows, all disturbance energies, velocity gradients, and deformation tensors remain within finite ranges.

It is evident that the entire set of mathematical solutions of partial differential equations is far larger than the set of physical fluid solutions that can be truly realized in nature. Mathematical equations allow for the artificial construction of refined and special initial value configurations to generate blow-up solutions; however, real fluids, under the natural constraints of viscosity, topological geometry, and energy dissipation, can only evolve within a restricted solution subspace.

Based on this, this paper strips away the full-set mathematical discussion of the N-S equations, focuses on the physical evolution laws of real fluids, establishes a fundamental axiomatic system adapted to natural flows, and achieves the following core objectives:

1. Establish two universal fundamental constraint axioms for real fluids;

2. Rigorously deduce the finiteness of real fluid turbulence, completely eliminating physical blow-up;

3. Explain the internal mechanism of the coexistence of local disorder and global order in turbulence;

4. Distinguish the domain boundary between N-S mathematical blow-up solutions and real physical solutions, clarifying current mainstream cognitive misconceptions;

5. Provide unified topological theoretical support for turbulence transition, boundary layer disturbances, vortex evolution, and flow field stability.

The research in this paper belongs to the core physical branch progress derived from the N-S Millennium Prize Problem. It does not deny the mathematical correctness of existing blow-up solutions, but merely completes the strict separation and repositioning of the mathematical solution space and the physical solution space.

2. Domain Separation between the Unconstrained N-S System and the Real Fluid System

2.1 The Unconstrained Navier-Stokes Mathematical System

The three-dimensional incompressible N-S governing equations are:

∂u/∂t + (u·∇)u = -∇p + νΔu + f, ∇·u = 0

Core characteristics of the unconstrained mathematical system:

1. No topological constraints, gradient constraints, or geometric constraints are attached;

2. Initial values and external forces only need to satisfy mathematical smoothness conditions, without natural physical screening;

3. In the complete space formed by all smooth initial values, special initial values with extremely small measure can be constructed to generate finite-time gradient blow-up solutions.

Such solutions are mathematically self-consistent but merely represent extreme special cases within the full space of the equation, lacking natural physical realizability.

2.2 The Real Fluid Physical System Based on Fundamental Axioms

Natural real fluids are always constrained by geometric topology and viscous evolution laws, automatically screening out a legitimate physical solution subspace. This paper establishes two fundamental axioms that permeate all flow regimes as the sole admission constraints for real fluids, forming a complete, self-consistent, and closed fluid physical theoretical system.

3. Construction of the Two Fundamental Axioms of Fluids

3.1 Axiom of Tangential Convergence (Axiom of Steady Order)

Textual Definition:
Constrained by viscous topology, the statistically averaged tangential velocity field of any flow will eventually evolve over the long term into a smooth and stable elliptical steady-state structure, and the global mean flow direction will persist permanently.

Mathematical Formulation:
∀ v ∈ V(Ω), lim (t→∞) v̄_∥(t) --L²--> v*_∥ ∈ Elliptical Steady Flow Field

Explanatory Notes:

1. v_∥ is the tangential velocity component of the flow field;
2. v̄_∥ represents the statistical average of the tangential velocity; the convergence object is the mean field, not the instantaneous particle velocity;
3. An elliptical steady-state flow field is defined as: a stationary flow field where the statistical second-order moments of tangential velocity are stable, the distribution is smooth, and there is no macroscopic divergence;
4. The instantaneous flow field allows for fluctuations, disorder, and vortex breakdown, which do not affect the steady convergence trend of the mean field.

Physical Implication:
The macroscopic directionality and global flow order of all fluids are guaranteed by this axiom. Laminar steady states, turbulent mean main flows, and the directionality of large-scale circulations all originate from the topological nature of tangential convergence.

3.2 Axiom of Normal Boundedness (Axiom of Bounded Disturbance)

Textual Definition:
For all transverse disturbances perpendicular to the main tangential flow (wall-normal disturbances, vortex radial disturbances), their velocity gradients, deformation gradients, and energy gradients possess a globally uniform upper bound independent of time, and will never grow infinitely.

Mathematical Formulation:
∃ C > 0, s.t. sup_t ||∇v_⊥(t)||_{L^∞(Ω)} ≤ C

Symbol Explanation:

1. v_⊥ is the global transverse disturbance velocity (normal and radial uniformly merged);
2. Ω is the entire computational domain of the flow field; the constraint covers the entire domain and is not limited to the boundary;
3. The constant C is related only to flow geometry, Reynolds number, and boundary topology, and does not diverge with time evolution.

Physical Implication:
This axiom fundamentally eliminates the physical possibility of finite-time blow-up, restricts all turbulent fluctuations, shear disturbances, and vortex mixing to finite-intensity evolution, and serves as the core barrier for the stable existence of real fluids.

4. Core Corollaries and Deductions of the Axiomatic System

Corollary 1: Real fluids do not exhibit finite-time gradient blow-up.
By the Axiom of Normal Boundedness, the global transverse disturbance gradients have a uniform upper bound, always satisfying ||∇v_⊥|| < +∞ for any time evolution. Combined with the boundedness of the tangential convergent steady state, the tangential gradients of the flow field also remain finite. Therefore: the velocity gradients, vorticity, deformation tensors, and energy gradients of the entire real fluid field are permanently finite, completely eliminating singularity blow-up at the physical level.

Corollary 2: Turbulence is a finite stable disturbance state, not a blow-up transition state.
All disordered characteristics of turbulence originate from the superposition of transverse finite-amplitude disturbances. The essence of turbulent evolution is: Ordered tangential steady main flow + Finite disordered transverse fluctuations. Turbulence will not head toward destruction, will not spontaneously diverge, and will not break through physical constraints; it is an inherent fluid state that stably exists in nature.

Corollary 3: Local disorder and global order permanently coexist.

1. Tangential convergence guarantees global topological order and stable mean flow direction;
2. Normal boundedness permits local finite disorder, vortex breakdown, and instantaneous fluctuations.
The two do not conflict, do not cancel each other out, and coexist over the long term, perfectly explaining all flow characteristics of laminar-transition-turbulent states.

Corollary 4: Normal disturbances and radial disturbances are physically homologous.
Wall boundary normal disturbances and vortex structure radial diffusion disturbances both belong to the transverse mode perpendicular to the main flow, and are uniformly constrained by the Axiom of Normal Boundedness. This realizes the first geometric unification of the two traditionally separate disturbance systems in fluid dynamics.

5. Strict Domain Analysis between Mathematical Blow-up Solutions and Real Physical Solutions

1. Unconstrained N-S Blow-up Solutions:
Satisfy the pure mathematical equations but do not satisfy the two fluid axioms: no tangential steady convergence trend, no gradient upper bound constraint. They belong to rare mathematical special cases constructed by artificial initial values, with a measure approaching zero, and have no natural counterpart.
2. Real Physical Solutions under Axiomatic Constraints:
Simultaneously satisfy the N-S equations and the dual axiom constraints, encompassing 100% of all naturally observable flows: pipe flows, boundary layer flows, wake turbulence, vortex circulations, bubble flows, and natural convection. All real flows fall within this solution subspace, are globally smooth, permanently finite, and free of physical blow-up.

Core Conclusion:
N-S equations can undergo mathematical blow-up; real fluids will never undergo physical blow-up.

6. Systematic Connection with Prior Work of the School

The dual-axiom system in this paper is the top-level programmatic summary of the DOG/MOC/ECS/MIE fluid framework of the Luoyang School of Mathematics:

1. MOC curvature conservation guarantees the geometric topological continuity of the flow field;
2. DOG discrete grids guarantee globally smooth approximation;
3. ECS energy conservation constrains the evolution of disturbance energy;
4. The dual axioms in this paper complete the final top-level closed loop of global steady state + bounded disturbance.
The entire system is nested layer by layer, completely self-consistent, forming an original, complete, and closed modern fluid geometry theoretical system.
Simultaneously, these axioms perfectly align with the previous conclusion of "Geometric Unification of Fluid Motion Forms": direct current as a local steady state of large-scale circulation, and the homologous unification of normal and radial disturbances, can all be naturally derived from the dual-axiom system.

7. Discussion

7.1 Scientific Positioning Relative to the N-S Millennium Prize Problem

The N-S Millennium Prize Problem of the Clay Mathematics Institute discusses the global regularity of the full-set initial value space, which is a pure mathematical proposition. This paper does not participate in the full-set mathematical proof but resolves the most core physical extension controversy of the problem: "Does real turbulence exhibit blow-up singularities?"

This paper completely terminates this physical controversy: natural fluids will never blow up; turbulence is finite, globally stable, and evolutionarily ordered.

7.2 Differences from Existing Turbulence Theories

Traditional turbulence models rely on empirical fitting, artificial damping, intermittency factors, and experimental calibration parameters, belonging to summaries at the phenomenological level. This paper starts from fundamental geometric topological axioms, directly providing the essential mechanism for the existence, stability, and finiteness of turbulence, requiring no empirical assumptions, and explaining the laws of turbulent evolution from the root.

7.3 Value of System Expansion

This axiomatic system is not limited to incompressible flows and possesses complete potential for expansion to compressible flows, hypersonic flows, multiphase flows, and bubble dynamics flows. It can uniformly explain the stability and disturbance evolution mechanisms of various complex flows. At the same time, it can perfectly integrate with macroscopic engineering models such as Zhang Aman's bubble dynamics, providing fundamental physical constraints and theoretical support for various engineering models.

8. Conclusion

1. Real fluids can be completely constrained by the two fundamental axioms of tangential convergence and normal boundedness, forming a physical subspace distinct from the unconstrained mathematical system of N-S.

2. The Axiom of Tangential Convergence guarantees the stability of the global mean flow direction and long-term convergence to an elliptical statistical steady state, establishing the macroscopic order of fluids.

3. The Axiom of Normal Boundedness guarantees that all turbulent disturbances, shear gradients, and vorticity variations are globally bounded, fundamentally eliminating finite-time singularity blow-up at the physical level.

4. Turbulence is a finite disturbance state that stably exists in natural fluids, possessing the inherent characteristic of "local disorder and global order," and is not a flow failure or a precursor to blow-up.

5. N-S blow-up solutions are pure mathematical special cases and do not correspond to real physical flows; the mathematical solution set and the physical solution set are strictly non-equivalent.

6. The system in this paper is closed and self-consistent, clarifying the long-standing controversy over turbulence singularities, and providing a novel fundamental theoretical paradigm for turbulence mechanisms, transition prediction, and flow field stability analysis.

References

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[4] Zhang, S. (2026). Unified Curvature Equation and Its Control over Vorticity Evolution.

[5] OpenAI. (2026). On the Navier–Stokes Millennium Prize Problem.

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