Une #IA #AI de #Google résout un #problème #mathématique réputé #impossible Une nouvelle #famille de #singularités des équations de #́Navier-Stokes a émergé, confirmée ensuite par des #mathématiciens de #NewYork #University et #Stanford. L' #IA devient co-auteur www.cointribune.com/une-ia-de-go...

IA : Google DeepMind franchit ...

Basics of Numerical Weather Prediction (NWP):

1. THE HORIZONTAL MOMENTUM EQUATION:
\[
\frac{d\mathbf{V}}{dt} + f\hat{k} \times \mathbf{V} = -\nabla \phi + \frac{\sigma}{p_s} \frac{\partial \phi}{\partial \sigma} \nabla p_s + \mathbf{F}
\]

2. THE CONTINUITY EQUATION:
\[
\frac{\partial p_s}{\partial t} + \nabla \cdot (p_s \mathbf{V}) + \frac{\partial}{\partial \sigma}(p_s \dot{\sigma}) = 0
\]

3. THE THERMODYNAMIC ENERGY EQUATION:
\[
\frac{1}{R} \frac{d}{dt} \left[ \sigma \frac{\partial \phi}{\partial \sigma} \right] + \frac{RT}{C_p p} \left[ p_s \dot{\sigma} + \sigma\dot{p_s} \right] = -Q
\]

4. HYDROSTATIC EQUATION:
\[
\frac{\partial \phi}{\partial \sigma} = -\frac{RT_v}{\sigma}
\]

5. SURFACE PRESSURE TENDENCY EQUATION:
\[\displaystyle
\frac{\partial p_s}{\partial t} = -\int_{0}^{1} \nabla\cdot (p_s \mathbf{V}) \, d\sigma
\]

6. MOISTURE EQUATION:
\[\displaystyle
\frac{\partial}{\partial t} (p_s q) + \nabla\cdot (p_s q \mathbf{V}) + \frac{\partial}{\partial \sigma} (p_s q \dot{\sigma}) = p_s S
\]

The six primary unknowns are: \(\mathbf{V}\) (horizontal wind velocity), \(p_s\) (surface pressure), \(T\) (temperature), \(q\) (specific humidity or moisture), \(\phi\) (geopotential), and \(\dot{\sigma}\) (sigma velocity or vertical velocity in \(\sigma\)-coordinates).

#NWP #Weather #NumericalWeatherPrediction #Meteorology #Climate #ClimateScience #Earth #EarthScience #ClimateChange #ClimateSciences #Science #WeatherPrediction #Humidity #Moisture #Pressure #Velocity #SurfacePressure #HydrostaticEquation #WeatherPrediction #Ocean #Atmosphere #AOS #ClimateDynamics #WeatherDynamics #Geopotential #SigmaVelocity #VerticalVelocity #MoistureEquation #Thermodynamics #Dynamics #NavierStokes

Imagine being a brilliant physicist/mathematician and still avoiding the most important problems because your career depends on publishing frequent papers, not solving the biggest mysteries in the world.

That's why you can't do things like this in academia.

https://english.elpais.com/science-tech/2025-06-24/spanish-mathematician-javier-gomez-serrano-and-google-deepmind-team-up-to-solve-the-navier-stokes-million-dollar-problem.html

#NavierStokes #GoogleDeepMind #DeepMind #MillenniumProblems #Existence #Smoothness #Fluid #FluidDynamics #Turbulence #Dynamics #TurbulentFlows #Research #Engineering #Physics #Math #Maths #Mathematics #UnsolvedProblems #BiggestMystery #Flows #MillionDollarProblem

The Secret Order of Pi - Terence Tao on Lex Fridman #navierstokes #pi #turbulence

Blow-Up or Not? - Terence Tao on Lex Fridman

#navierstokes #blowup #millenniumprize

🌊 Scientists have advanced in understanding turbulence, a long-standing puzzle for physicists. This progress aids in solving the Navier–Stokes equations, a major challenge in math and physics, and a Millennium Prize Problem by the Clay Mathematics Institute. Recent developments are crucial for fluid dynamics, impacting engineering, meteorology, and more.

@goodnews

#GoodNews #Physics #Turbulence #NavierStokes #ScienceBreakthrough
https://edition.cnn.com/2025/02/06/science/turbulence-physics-oldest-unsolved-problem/index.html

Researchers claim to have solved Hilbert’s sixth problem by unifying three theories of #FluidDynamics at different levels of granularity:

+ Newton’s laws of motion at the microscopic level where fluids are composed of particles - little billiard balls bopping around and occasionally colliding

+ The Boltzmann equation at the mesoscopic level where the equation considers the likely behavior of a typical particle

+ Euler and #NavierStokes equations at the macroscopic level where the fluids are a single continuous substance

https://www.scientificamerican.com/article/lofty-math-problem-called-hilberts-sixth-closer-to-being-solved

Preprint https://arxiv.org/abs/2503.01800

An Article in the Annual Review of Condensed Matter Physics on Turbulence by KR Sreenivasan and J Schumacher
https://www.annualreviews.org/content/journals/10.1146/annurev-conmatphys-031620-095842

What is the turbulence problem, and when can we say it’s solved? 🌪️ This deep dive by Sreenivasan & Schumacher explores the math, physics, and engineering challenges of turbulence—from Navier-Stokes equations to intermittency and beyond. A must-read for anyone fascinated by chaos, complexity, and the unsolved mysteries of fluid dynamics! 🌀

A summary of the talk presented by KR Sreenivasan in December 2023 at the International Center for Theoretical Sciences (ICTS-TIFR) in Bengaluru, as part of a program on field theory and turbulence.
https://www.youtube.com/watch?v=fwVSBYh-KC4

"Field Theory and Turbulence" program link: https://www.icts.res.in/discussion-meeting/ftt

#FluidDynamics #Physics #NavierStokes #UnsolvedMystery #Mechanics #Dynamics #FluidMechanics #Science #Chaos #TurbulentMotion #Randomness #Chaotic #Fluid #ClassicalMechanics
#Turbulence

Only a genderfluid creature can hope to solve the Navier-Stokes Equations existence and uniqueness problem.

#NavierStokes #genderfluid

So, I mentioned already that we cannot really model #lava flows. The main reasons for that is that we don't actually know how lava behaves, at least not in sufficient detail.

Of course, lava is a fluid, and a (very) viscous one at that, so we know that it follows the Navier–Stokes equations. We also know that its behavior is heavily dependent on temperature, so we know that we also need the heat equation, with both kinds of boundary conditions (conduction to ground, and radiation on the free surface).

And that's all we know. Seriously.

OK, not really, but everything else is extremely uncertain. When modeling a viscous fluid (like lava, or any other geophysical flow for the matter), the first thing you need to know is what the viscosity is. And for lava, we don't know. There's a lot of things we do know, but not enough.
For example, we know that the viscosity depends on temperature, chemical composition, degree of crystalization, amount and types of volatiles in the melt, and so on and so forth. But we don't exactly know the laws relating the viscosity to all of these chemical and physical properties.

2/

#NavierStokes #NavierStokesEquations

So there's apparently a #metal song mentioning the #NavierStokes equations now.

https://youtu.be/Jy99ywCp3qU

#music #CFD #meteorology #parody

In the concluding article of his "Navier-Stokes in Chapel" series, Jeremiah Corrado demonstrates how a Chapel program with size and complexity like Python performs and scales competitively with a more complex C++/MPI/OpenMP port.

https://chapel-lang.org/blog/posts/bns4/

Thanks to the @labarba group for creating and maintaining the CFD Python tutorials upon which this series was based! https://lorenabarba.com/blog/cfd-python-12-steps-to-navier-stokes/

#ChapelLang #CFD #NavierStokes #HPC

In his latest article studying Navier-Stokes computations in Chapel, Jeremiah Corrado demonstrates how a 4-line change turns the multicore version from his previous article into a distributed, scalable computation.

Read all about it here: https://chapel-lang.org/blog/posts/bns3/

Based on Step 9 of "12 Steps to Navier-Stokes" from @labarba.

#ChapelLang #CFD #NavierStokes #HPC

We’ve just published the second in Jeremiah Corrado’s series of blog posts in which he translates key steps of the Lorena A. Barba group’s great “12 steps to Navier-Stokes” series into Chapel, demonstrating the parallel performance that can be achieved.

https://chapel-lang.org/blog/posts/bns2/

#ChapelLang #NavierStokes

Dive into the world of fluid dynamics and discover the power of the Chapel programming language in scientific computing applications in our latest blog post, 'Navier-Stokes in Chapel'!

Check it out: https://chapel-lang.org/blog/posts/bns1/

#ScientificComputing #FluidDynamics #NavierStokes #opensource

One of my colleagues recently created a #blog post that introduces #NavierStokes simulation with #chapelLang. I particularly appreciated the gentle introduction to computational fluid dynamics. It ports #python code and makes one simple change to make the simulation run #multicore!

https://chapel-lang.org/blog/posts/bns1/

#science #hpc

These fluid sim bugs ain't gonna fix themselves. #watercolor #simulation #BigWetPixels #procedural #DebugView #algorithmic #NavierStokes

I'm midway through a (major, long overdue) overhaul of the fluid transport layer of my watercolor simulation. Made some good progress over the weekend. But more importantly: I made lots of new bugs. Glorious, unrepentant, face-eating bugs. I almost don't have the heart to squash them.

https://vimeo.com/cassidy/big-wet-pixels-10

#BigWetPixels #watercolor #simulation #DebugView #generative #algorithmic #procedural #NavierStokes

#ThisMonthInFluiddyn it is. Let's go 😎

🔹@PierreAugier and friends are finishing up an article, so as a side project they released #formattex and #formatbibtex based on #TexSoup and #BibtexParser

https://pypi.org/project/formattex/
https://pypi.org/project/formatbibtex/

> a simple and uncompromising #Latex code formatter

🔹Version 0.7.4 of #fluidsim and fluidsim-core were released containing a refactored energy spectra for #NavierStokes solvers and other bug fixes

https://pypi.org/project/fluidsim/

#fluiddyn

Predicting weather is a complex dance of physics and math. The Navier-Stokes equations help us model fluid motion, which is essential for forecasting storms and understanding climate patterns. #WeatherForecasting #NavierStokes