118: Chapter 118 Existence and Smoothness of Navier-Stokes Equations: Proof Complete

Time in the study seemed to be stretched and frozen by an invisible force.

The piles of manuscript paper were like silent witnesses, the mathematical symbols flowing across them acting as steps into the abyss of truth.

The proof of the existence and smoothness of the Navier-Stokes Equation—this mountain, which had stood in the field of fluid mechanics for a century and where countless top scholars had met their downfall.

Chen Youhua could clearly see the finish line, but the path leading there flickered in and out of the fog of his thoughts.

Traditional mathematical tools had failed here, unable to shake that final, most solid barrier.

The essential difficulty in the mathematical description of the intense energy transfer and dissipation mechanisms between turbulent scales.

He had tried all known variational methods, energy estimates, regularization techniques...

The manuscript paper was filled with complex derivations; every calculation approached the limit, yet always hit an invisible wall at the final moment.

Sweat gathered on his brow and dripped silently onto the edge of the manuscript paper, blooming into a small dark spot.

Just as his mind was about to be swallowed by the frustration of repeated failures, the long-dormant system in his mind emitted a faint light again.

"Hum."

【Hint: Regularization. Scale separation. Energy cascade of vortex structures... Between the dissipation and nonlinear terms... find the threshold that allows for smoothness.】

This hint was not a specific formula, but an insight into the essence of the problem from a higher dimension.

It pointed directly to the core contradiction of turbulence.

How energy, in the process of cascading from large-scale vortices to small-scale vortices, could maintain the smooth characteristics of the flow while simultaneously converting energy into heat through the intense dissipation of small-scale vortices.

Chen Youhua opened his eyes abruptly, his fatigue swept away, replaced by extreme excitement after being enlightened.

"Regularization... scale separation... threshold..."

He muttered to himself, as if grasping at a straw.

He no longer fixated on those huge and complex overall proof frameworks.

Instead, he roughly pushed aside the manuscript paper filled with calculations and spread out a new, snow-white sheet of paper.

He picked up his pen and wrote with vigor.

He no longer attempted to directly tackle the original Navier-Stokes Equation itself, which was fraught with the risk of singularities.

He introduced an ingenious regularized Navier-Stokes Equation with a small viscosity correction parameter.

Based on the new regularized equation, Chen Youhua began to build a multi-scale analysis framework.

This was the most critical and most genius step!

Based on the core insight from the system hint, Chen Youhua locked his gaze onto the dynamic balance between the energy cascade rate and the viscous dissipation rate of small-scale vortices.

He picked up his pen and wrote a decisive inequality at the center of the manuscript paper.

(Power input to a certain scale by energy cascade) ≤ ν · C · (Energy dissipation rate of the vortex packet at that scale)

Where ν is the kinematic viscosity of the fluid, and C is a universal constant independent of the fluid properties and scale!

This inequality was the mathematical expression of the threshold he had been searching for so desperately!

Using this key inequality regarding the threshold, which he had proven to hold in the regularized system, he began a series of extremely precise mathematical operations.

Time passed minute by minute.

In the study, only the rustling of the pen tip racing across the paper remained, faster and more urgent!

Chen Youhua's face looked pale from extreme focus and excitement.

Sweat soaked the back of his shirt, but he was completely unaware.

Finally!

When he wrote down the final inequality and completed the final confirmation of the smoothness of the limit solution after removing the regularization, his pen tip stopped abruptly!

All derivations, all estimates, all inductive arguments closed perfectly at this moment!

Forming a self-consistent, rigorous, and impeccable logical chain!

The mystery of turbulence that had plagued humanity for a century, the ultimate question about the deepest laws of fluid motion, had been completely conquered on this snow-white piece of manuscript paper!

Chen Youhua slowly, extremely slowly, put down his pen.

It was as if he had laid down a heavy burden, or perhaps completed a long and lonely pilgrimage.

He leaned back in his chair, took a long, deep breath, as if trying to inhale all the air in the room into his lungs, and then exhaled slowly.

An indescribable sense of calm and vast satisfaction, like a warm tide, instantly submerged all his fatigue and tension.

He closed his eyes, his lips curling upward uncontrollably, forming a smile of pure perfection.

"Done."

Two words, light as a sigh, yet heavy as a thousand pounds.

This was a confirmation of truth, and a declaration of an era.

He gently put down the manuscript paper and muttered to himself.

"Mathematics can be pure, it can be timeless; a solution can define eternity, just like the debate over whether 0.999... equals 1. It pursues the flawlessness of the logical chain itself.

And physics must be rooted in this real world that we perceive, a world full of change and uncertainty, and then use mathematics, the sharpest of knives, to dissect it and define it.

The two complement each other, yet they walk on different paths."

His gaze fell back onto the thick stack of manuscript paper that condensed his countless efforts.

He reached out, his movements casual and relaxed, and pulled a few pages from that stack of manuscript paper.

These pages happened to outline several key "easily obtained" and "easily known" conclusive steps.

Then, he picked up his phone, aimed it at the mountain of manuscript paper on the table filled with symbols, adjusted the angle, and click, click, took pictures of them all.

He logged into arXiv, the world's largest preprint server.

He did not set any access restrictions.

He created a new submission with a concise and powerful title.

"Proof of the Existence and Smoothness of the Navier-Stokes Equations"

Author: Chen Youhua.

Attachment upload: Hundreds of clear photos of the simplified proof process.

Click, submit.

The page refreshed, indicating a successful submission.

The whole process was as mundane as uploading an ordinary work report.

Chen Youhua closed the webpage and sank into his chair, his body completely relaxed.

The sky outside the window was already showing a faint light; dawn was approaching.

Theory is theory, and application is application.

He had never thought about hiding anything.

Theory is the cornerstone of the world and should belong to all who seek the truth.

Newton's masterpieces, Maxwell's equations, Einstein's theory of relativity... generation after generation of sages made their thoughts public, which lit the way forward for those who followed.

And he himself, was he not standing on the shoulders of these giants to reach the peak of this difficult problem?

Truth has no boundaries.

He came from a civilization with a profound history of five thousand years.

What this history has deposited should not be narrow-minded blockades, but a broad-mindedness that embraces all rivers and the confidence to face any challenge.

Making the results public is not rashness, but the inheritance of the academic spirit, a responsibility to the common knowledge territory of humanity, and the deepest conviction in his own civilization's heritage.