182: Chapter 182 The Report Meeting Begins

Outside the Huaxia Hall, a long red carpet had already been laid out. Massive posters stood on both sides of the main entrance, displaying the title of the report in both Chinese and English.

"On the Existence and Smoothness of Solutions to the Navier-Stokes Equations: Proof and Prospects," accompanied by a giant photograph and biography of Chen Youhua.

Mathematicians, physicists, and fluid mechanics experts from all over the world held their invitations, entering in an orderly fashion under the guidance of volunteers. The air was filled with low murmurs of conversation in various languages.

The media zone was packed with cameras and microphones; reporters stood ready, prepared to record a moment that might go down in scientific history.

The venue, capable of holding nearly a thousand people, was packed to capacity, with temporary seats even added in the back rows and aisles.

Seated in the front row were the IMU President, the Secretary-General, Fields Medalists, Academicians from various national academies, and titans of the academic world.

At exactly nine o'clock in the morning, the venue lights dimmed slightly, and a spotlight shone onto the podium.

The host delivered a brief and enthusiastic opening address, introduced the distinguished guests, and emphasized the profound significance of this report.

Then, amidst warm applause, the absolute protagonist of the day, Chen Youhua, walked calmly onto the stage.

He was wearing a well-fitted dark suit and tie today, his hair meticulously styled, with a faint smile on his face.

He walked to the center of the podium, bowed slightly to the audience, and adjusted the microphone.

Under the spotlight, he appeared exceptionally composed, showing no signs that he was a young man of only twenty-something.

"Distinguished guests, colleagues, ladies and gentlemen, good morning."

His voice carried clearly throughout the venue via the high-quality sound system, his English fluent and standard.

"It is a great honor to stand here and share with you some of my thoughts and results regarding the Navier-Stokes Equations problem."

Without much small talk, he cut straight to the point.

The massive screen lit up, displaying the first slide of the report.

It showed the title, core abstract, introduction, and a restatement of the problem.

He first briefly reviewed the historical significance of the Navier-Stokes Equations and their central position in fluid mechanics, clearly elucidating the core content required to be proven for the Millennium Prize Problem.

"In the three-dimensional case, given an initial velocity field, does there exist a solution that remains smooth and does not develop singularities?"

Using concise and precise language, he pointed out the main difficulties encountered in previous research.

"The immense challenge of the nonlinear term, and the resulting potential for energy concentration and scale cascading, are the main obstacles hindering us from proving the global regularity of the solution."

Next came the proof framework and core ideas.

He displayed the logical framework diagram of the entire proof.

This was the core of his entire report.

"The core idea of my work lies in introducing a brand-new function space framework, along with a corresponding method of refined frequency-localized energy estimation."

He switched the PPT, and a series of complex yet clearly structured mathematical definitions and symbols appeared on the screen.

He did not rush to show the most difficult parts, but instead started with physical intuition.

"We can imagine fluid motion as the superposition of countless vortices of different scales.

Traditional energy estimation methods often appear crude when dealing with the nonlinear interactions between vortices of different scales, causing estimates to diverge in finite time."

He picked up a laser pointer and pointed to a key equation.

"The breakthrough lies in constructing a set of adaptive frequency projection operators that can more precisely capture the transfer paths of energy between scales, and proving that in the vast majority of cases, energy inversely cascades from small scales to large scales, rather than the forward cascade from large scales to small scales that leads to catastrophic singularities."

He guided the audience step by step, deep into his mathematical world.

The equations on the PPT became increasingly complex, but he controlled the pace of his explanation perfectly. At every critical step, he would pause to explain the intuition behind it using more intuitive geometric images or physical analogies.

"Here, I have introduced a modified Lyapunov function with a specific damping effect, which can effectively control the malignant growth of the nonlinear term."

He displayed an extremely ingenious structure.

"Through this function, we are able to prove that even if brief, intense activity occurs in certain local areas, its intensity is not sufficient to truly form a singularity; the total turbulence of the system is always controlled by a global constant."

Off-stage, whether they were elderly scholars or young graduate students, everyone was focused intently.

Many were taking notes rapidly, occasionally furrowing their brows in deep thought, and other times nodding in sudden realization.

The top scholars in the front row wore serious expressions, their gazes sharp, following every step of his derivation closely, their brains working at high speed to evaluate its rigor and innovation.

The report entered its most hardcore section.

He elaborated in detail on several of the most critical technical breakthroughs.

Regarding the new decomposition and estimation of the nonlinear term, he showcased a new method of decomposing the nonlinear term into resonant and non-resonant parts, and performed astonishingly refined estimates for each.

He then proved that even if singularities existed under certain extreme assumptions, their Hausdorff dimension would be strictly limited to an extremely low level, thereby excluding the possibility of the formation of truly dangerous singularities from the perspective of measure theory.

He demonstrated how to integrate all the final energy inequalities with global extension and local estimates, ultimately obtaining a global, consistent a priori energy estimate, thereby allowing the solution to be extended smoothly to an arbitrary length of time.

Chen Youhua's logic was extremely rigorous; every step of the derivation seemed difficult, yet appeared natural under the guidance of profound mathematical intuition.

The entire process was like weaving an incredibly precise and magnificent tapestry of mathematics.

Although pure mathematical proofs do not rely on numerical experiments, Chen Youhua briefly presented some actual turbulence data at the end, along with numerical simulation results for specific turbulence scenarios based on his theoretical framework.

The simulations showed that under the guidance of the new mathematical tools, the accuracy and stability of numerical predictions were indeed significantly improved.

This provided support for his theory from a different perspective.

"In summary," Chen Youhua concluded.

"We believe that this provides a complete and affirmative answer to the problem of the global existence and smoothness of solutions to the three-dimensional incompressible Navier-Stokes Equations."

He paused, his gaze sweeping across the entire venue.

"Of course, mathematical rigor requires the most stringent testing by peers and time.

I have made all details of the preprint public, and I welcome all colleagues to examine, question, and discuss them."

Finally, he sincerely thanked his mentors, collaborators, Z University, the National Natural Science Foundation, and all the institutions and friends who had supported this work.