93: Chapter 93 Existence and Smoothness

After Chen Youhua finished handling matters regarding enlightenment and the A-705 Laboratory, he returned to the unique, quiet atmosphere of his study.

He sat down at his large desk and casually switched on the ultrasonic humidifier.

"Hum..."

A faint, subtle buzzing sound arose.

From the top of the humidifier, a clear stream of water was dispersed and lifted by an invisible force, transforming into a fine, smoke-like mist that rose in curling wisps.

Chen Youhua's gaze followed the lightly dancing mist droplets, watching them chase, collide, merge, and separate in the air, outlining ever-changing trajectories.

This seemingly chaotic flow faintly revealed a hidden order, a dynamic equilibrium constrained by physical laws yet filled with infinite possibilities.

His thoughts, much like the mist, slowly sank into another, more vast and abstract world of flow—the Navier-Stokes Equation.

For the past few days, he had reviewed all the achievements made by his predecessors.

Thick stacks of printed papers and monographs were scattered across his desk, bookshelves, and even the floor, their covers bearing the logo of the Clay Mathematics Institute and the names of those renowned in the mathematical world.

Klainerman, Terence Tao, Ruth Williams... each paper was a brave exploration of human intellect at the edge of the abyss of turbulence.

Strictly speaking, the Navier-Stokes Equation is clearly defined; it is formed by combining the fluid's mass conservation equation and its momentum conservation equation.

It is these two seemingly basic equations, derived from the ironclad laws of the physical world, that, when combined, form a monster that the mathematical world fears.

A system of nonlinear partial differential equations.

The key lies in the nonlinearity.

Chen Youhua's fingertips traced unconsciously across the desk, as if sketching invisible formulas.

Imagine trying to describe the motion of every single droplet in a rushing river.

A linear equation is like describing a calm stream.

The flow direction is clear, the velocity is uniform, and it is easy to predict the future position of each droplet.

All of this assumes that the principle of superposition holds: the motion of one droplet does not affect another, and the overall flow is the sum of its parts.

But the Navier-Stokes Equation describes a rushing waterfall or a vortex.

Here, the motion of every droplet violently affects the motion of the countless droplets around it, and is in turn affected by them.

The squared terms of velocity, the product terms of velocity and velocity gradients... these nonlinear terms are like countless invisible hooks, binding the fates of droplets that could otherwise be calculated independently into a tight entanglement.

A tiny disturbance in one droplet can, through this complex mutual coupling, be amplified infinitely, eventually leading to a drastic change in the morphology of the entire flow field.

You cannot study a single droplet in isolation; you must consider its interaction with the entire violent flow simultaneously.

This is the essence of nonlinearity, and also the fundamental reason why solving and analyzing the Navier-Stokes Equation is like climbing a sheer cliff.

And what Chen Youhua was focusing on at this moment was one of the Millennium Prize Problems, hanging at the top of this cliff, with a one-million-dollar bounty from the Clay Mathematics Institute.

The problem of the existence and smoothness of solutions to the three-dimensional incompressible Navier-Stokes Equation.

Simply put, this problem asks:

Given an initial moment, the fluid is in a smooth initial state in space without any singularities, and assuming the fluid is incompressible.

Then, after any length of time, does a solution to this equation always exist? Is it unique? And does it remain smooth throughout the entire time domain?

For the two-dimensional case, mathematicians have already provided a definitive answer: the solution exists, is unique, and is smooth.

Chen Youhua had carefully studied those proofs; sophisticated mathematical tools were employed with masterful skill, and the logical chains were as tight as a precision Swiss watch, displaying the ultimate beauty of human reason.

He chewed over the argumentative processes of those papers repeatedly, filled with admiration.

But the three-dimensional world is the home turf of the real physical world.

And here, the problem remains unresolved.

The difficulty of the proof increases exponentially.

The A4 paper in front of Chen Youhua was already covered in symbols, arrows, and short keywords.

He tried to take those exquisite mathematical tools from the two-dimensional proof.

Energy estimates, Sobolev space embeddings, regularity lifting.

Carefully transplanting the tools from the two-dimensional realm into the soil of the three-dimensional one.

He chose a classic entry point: energy estimates.

Chen Youhua tried to apply the same inequality to three dimensions.

He wrote down the inequality and substituted the three-dimensional settings.

The derivation on the paper began to become complex.

He introduced an intermediate norm, attempting to establish a connection.

However, when he substituted the inequality into the energy equation, a problem arose.

The upper bound generated by the nonlinear term depended on a higher-order derivative norm, and this norm itself was not directly controlled in the energy equation!

Chen Youhua's pen tip stopped, hovering above that unclosable equation.

He stared at the symbol representing the higher-order derivative, his brows furrowed tightly.

Time passed silently in contemplation.

The mist from the humidifier continued to drift leisurely, but Chen Youhua's mind felt trapped in a maze constructed of equations.

He had already walked to the end of several of the most reasonable and promising main paths, and before him lay a thick, impenetrable theoretical barrier.

Just as he stared at the dead end on the paper, his thinking momentarily stalled.

"Hum."

An extremely faint, almost imperceptible vibration acted directly upon the depths of his consciousness.

This was the sign that his long-dormant system had been triggered.

It never made a clamor, only giving the most crucial hints when the host encountered a genuine cognitive dilemma.

There were no flashy sound or light effects, nor any lengthy explanations.

In his mind, a stream of incredibly clear information instantly sprouted.

【Hint: Focus on the disturbance propagation threshold; extreme nonlinear behavior may stem from an imbalance between the accumulation of local disturbance energy at specific scales and the viscous dissipation capacity. Search for a mathematical representation that characterizes the critical conditions of this local energy imbalance.】

The hint was concise and precise, pointing directly to the core contradiction of the problem and proposing an entirely new direction of thought, even hinting at the tools that might be needed, which belonged to another branch of mathematics.

The system did not provide the answer; it merely pushed open a window he had not noticed before, allowing him to see another landscape outside the maze.

He did not immediately feel ecstatic, nor did he fall into deeper confusion.

The experiences of this period allowed him to calm down quickly.

He picked up his pen and drew a heavy circle around the place where his derivation had been blocked, labeling it "Nonlinear-Higher-Order Derivative Coupling."

Then, on a fresh blank space of the paper, he wrote down a few new keywords.

His thoughts suddenly cleared.