This $1,000,000 Problem Can Revolutionize Weather Modeling - Navier-Stokes Existence and Smoothness

[@kinertia4238]

Quick Note: At 4:15, I've accidentally mentioned ρ as viscosity instead of density. This is an accident. However, the rest of the equation is correct. For clarification, the div(T) term is an umbrella term in the Momentum Equation for Compressible fluids, and f(x,t) is a similar term accounting for other forces. I have purposely used these in order to avoid confusion and prevent explaining the other two terms.

[@sajedm313]

Another mistake, at 3:12 you replaced the del operator with a Laplacian operator.

[@freebiehughes9615]

Love your videos! I only studied maths up to Calculus I, but you inspire me to go back to school and study more!

[@the_sophile]

did you pin this?if not please pin this comment

[@dennywey9816]

Well, isn't pressure included in the stress tensor? so the equation at 4:15 shouldn't include the gradient of pressure (first term on the right hand side).

[@KnightCrawler005]

Took fluid dynamics in college during my Mech Eng program. I now work in the finance industry, but solving the Navier-Stokes equation is one of my life goals.

[@danthegypsy5871]

You solve yet or no.

[@jdhr7767]

Hows your progress?

[@GammaFZ]

@@danthegypsy5871 LMAO

[@justasaiyanfromearth5252]

Good luck.

[@theludvigmaxis1]

Solving the equation isn’t difficult really, we already have exact solutions like stoke’s first and second problem. Or the blasius solution. The million dollar question is whether the NS equations will always give a meaningful solution regardless of chosen parameters to the equation.

[@uhbayhue]

What an amazing channel, thank you for the intuitive explanation!!

[@kinertia4238]

Glad you enjoy it!

[@what_on_arth]

3:10 -∇p is the pressure gradient, not the rate of change of pressure. So not instant to instant but at every point in space. And in summers, balloons pop due to the rubber absorbing a lot of the sun's radiant heat and melting locally not because of excessive air pressure.

[@anandsuralkar8376]

Right

[@the_sophile]

then why does a baloon shrinks if it is kept in cold water?

[@sovietwizard1620]

@@the_sophile He just said, heat cause it to expand so cold - opposite. of heat, it compresses and does the opposite of expanding because hot and cold. are opposite. But it really depends on what. you mean by cold. Pressure is. also a. factor

[@David-km2ie]

Its not the rate of change with respect to time but with respect to the coordinates.

[@SpaghettiToaster]

Gradients are rates of change bro.

[@jcnot9712]

The understanding of fluids in modern computer simulations is insane. As long as more people get interested in fluids, I don’t see why there won’t be a solution for 3D Navier Stokes within the next 50 years, even if it’s messy and takes forever to write on a board lol

[@josephmarshall2030]

Thanjs kinertia, i found your site,and khutoryansky serenditiously at the same time here on youtube. The most challenging millennium prob to conceptualize is :np versus p comp science prob. Thanks again, your videos have been tremendously helpful.🙂

[@aadarsh8306]

Navier -S eq will be solved in our lifetime . The day I started working on it .😎

[@justasaiyanfromearth5252]

Good luck.

[@PaulPukite]

The book Mathematical Geoenergy solves a simplified version of Navier-Stokes (Laplace's Tidal Equations) in closed form. This likely won't win the Clay prize but it can be used to model the behavior known as El Nino oscillations accurately, which is important for climate science. A $1 million prize is peanuts in comparison.

[@m24213]

As a guy who studies fluid mechanics and Heat transfer, I wouldn't get my hopes up about solving this equation. Werner Heisenberg once said, "When I meet God, I am going to ask him two questions: why relativity? And why turbulence? I really believe he will have an answer for the first." I think you guys get the idea.

[@kinertia4238]

Hey guys, I have a pretty awesome announcement: I have officially joined Twitter! From now on, I'll be posting updates and announcements of new videos at twitter.com/Kinertia42, along with a bunch of other stuff you may find cool. If I had more than 2000 subs at this point, then I could make a common community post, but apparently KZbin doesn't allow that, so I'm gonna be posting and pinning this comment everywhere. I've been posting to this channel a lot less lately, but I find it awesome that a pretty large number of people are watching and commenting on these videos - so here's to reaching (at this time) 275 subs! Really nice, huh? Since a lot of people seem to really love this channel, I'd like to say just one thing to our small but growing community: I'm working on a ton of new stuff right now, including cool new video styles and something which will (hopefully) turn out to be awesome. Above all, thank you guys for watching!

[@the_sophile]

How do you do these without even 2000 subscribers? I know ,it takes a lot of time and effort to do these animations.They are wonderful! which app do you use to make these animations? well done!

[@TheSandkastenverbot]

4:17: that's the Cauchy momentum equation. Navier Stokes assumes a Newtonian fluid and is therefore less general

[@RichTheEngineer]

Junior year of college (1976), as part of earning my BS in chemical engineering, I took a course in fluid mechanics. Went through derivation of Navier-Stokes and then solved for specific applications, like flow of a Newtonian fluid through a pipe. I still have textbook and class notes. Still gives me a headache thinking about it.

[@RichTheEngineer]

It was also my introduction to tensors.

[@Tadesan]

All that money doesn't make you feel better? Sigh....

[@Avidcomp]

How on earth can climate models be used effectively if we haven't solved the existence smoothness problem of Navier Stokes. Turbulence is quite a vital factor on climate through moisture and gases?

[@Tadesan]

Lol Can you unpack what you mean by 'moisture and gasses'?! Lol

[@OmnipresentPotato]

Don't you think the P vs. NP problem will become obsolete once we devise quantum technology?

[@TheTimeDilater]

Awesome Channel, Really amazing explanations !!!

[@CRehm]

At 3:13 it is not the gradient but rather gradient squared of p (according to the symbol)

[@ashishsharma-og4nl]

It will take 20-39 years

[@turbothrottletrouble4217]

Just wow, the way you just explained it, just wow. Amazing

[@ashwnicoer]

wonderful information ..

[@mealynleonor1504]

Please,tell me more about it.

[@isaacsaxton-knight7708]

Mistake at 4:17 where the symbol rho is labelled as viscosity when it is density

[@JimBobe]

Imaging knowing with 100% confidence what the weather in 3 weeks would be? Solving this problem would mean mastery of weather

[@joda7697]

No. Even if the existence gets proven, the boundary conditions can never truly be known, making an actual modelling impossible. Also, the equations are not designed to handle quantum mechanics and (general)relativity.

[@JimBobe]

@@joda7697 Solving turbulance is key

[@vishwanathaeducationalchan7994]

Nice explanation. But 4 equations each terms explained means good.you make another video like above with equation explanation

[@compellingpoint7802]

I would ask how the Navier-Stokes equations are useful, or what purpose they serve. After all, there is no model that correctly predicts turbulence in any fluid system at all, and so I cannot see how anything can be said about existence and smoothness. But, I suppose that is a question of semantics: if one wishes to discuss interesting solutions or smoothness in the Navier-Stokes equations, then perhaps these questions can be meaningfully addressed. But this seems like an uninteresting topic for discussion. I would ask why it is that the Navier-Stokes equations have precisely this form, and not a different one? After all, we can trivially choose to represent any given fluid system with an arbitrary number of equations. For example, instead of three equations for the u_i components and v_j components of velocity in 3D space (which are impossible to solve exactly), we could use four or five or ten such component variables. Why do these particular numbers arise naturally? Why do they arise at all? It seems to me that the Navier-Stokes equations arise because of a series of choices made by engineers and scientists, which led eventually to our current formulation. In particular, it seemed reasonable at various points along this path to choose between several possible fluid models (e.g., whether or not viscosity is zero), and then later on we found ourselves with these three component variables. I would ask why we have chosen to represent fluids with equations of this particular form, rather than some other. And then I would go on to point out that the Navier-Stokes equations are not unique in this regard. For example, the diffusion equation arises from a similar series of choices: whether or not there is convection (or various ways to treat it), whether or not there is a magnetic field (which affects the velocity vector in an interesting way), etc. I would ask why it is that the diffusion equation has precisely this form and not some other. And then I would point out that there are also many other equations that could describe diffusion, such as for example the k-epsilon model in 1D or 2D space.

[@claudioviotti295]

NS equations express conservation of momentum and mass. They can be derived from these first principles plus the Newtonian fluid hypothesis, but - like the diffusion equation - they can also be derived from statistical particles models such as the Boltzmann equations in the continuum limit (ie assuming particles are very small). Although not a physics first-principle, they are by no means 'made up', and are fully validated experimentally within their realm of validity. The mathematical challenge of demonstrating existence and smoothness is hardly relevant to any practical application. We have solved NS both by numerical and analytical methods (where possible) in a variety of contexts regardless.

[@canhakan]

You may find Évariste Galois interesting.

[@pawankhanal8472]

Nope. Non of these people select these variable by their own . No this is not the way universe works .

[@RichTheEngineer]

In order to solve for turbulence, you need to model complete behaviour of each and every molecule in fluid. Like many-body problem, there is most likely no analytical solution and numerical solution is massive, beyond our capabilities to solve in a timely manner. After all, a weather forecast for tomorrow does no good if it takes two days to solve.

[@Tadesan]

Ooo You sound smart.............

[@spoicydeemer985]

What kind of highschool student knows how to solve for air resistance? Except if they know differential equations, or given the EOM before hand.

[@rossyburns4392]

navier stokes requires riemann zeta. is my current theory. the zeta function i will try to explain in the entry essay iwant to write. I tackled the zeta function. and shifted my directed thoughts over to black hole research. and found some new ways to examine a Planck length.well I will save some of this for the entrance essay. up and atom did help by the animations. sincerely from a fellow scientist

[@seenybeany]

I enjoyed this. You've made some good quality videos - I'm surprised you don't have more of a following. Keep it up. What is your background in physics/math?

[@kinertia4238]

Thank you! I'm actually just a high school student interested in math- great to know that you found the video useful!

[@calicoesblue4703]

Nice😎👍

[@abz998]

The NS problem needs a trillion dollar prize instead of a measly one million. Although I know that money doesn't interest those that are solving this but the implications of a solution would be beyond ground breaking. We'd be instantly moving towards a tier 2 civilizations since fusion would be solved instantly.

[@anandsuralkar8376]

Exactly

[@GammaFZ]

a trillion is too much, maybe around 100 million or a billion is reasonable.

[@byte7645]

a trillion? okak uhh time to solve

[@tranhap6348]

The best mathematician in the world win Clay Millennium award in 2019

[@Virtuoso80]

If my coffee is going faster than light, then it won't orbit Mars. Although that's probably the least of our problems at that point.

[@InderjeetSingh-bd9gi]

Great video man , Keep it up !

[@alimohammadzade5906]

I've got a problem. You may have seen that some people throw playing cards through the air and it travels a long distance. I'm investigating effective parametres. My question is that "Is Navier Stokes equation helpful to solve this problem"?

[@kinertia4238]

It should, yes, although a parabolic approximation would be accurate immediately after the throw. Once the card has slowed down due to air resistance, modelling air turbulence would help you. The Navier-Stokes equations as they are at present would be useful if you were throwing the card in, say, a cylindrical wind channel (remember that Navier-Stokes would model the wind, not the card). That said, the problem is non-trivial - chaos sets in very soon after the card throw, and extremely tiny differences in measurement of atmospheric conditions would alter your flight trajectory. I would guess that it's impossible to model the entire trajectory of a card throw without knowing atmospheric conditions to a T.

[@alimohammadzade5906]

@@kinertia4238 Thanks a lot! Actually, I did'nt understand how the turbulance appears? (The air is constant and there is no wind)

[@kinertia4238]

The spin of the card causes changes in the wind.

[@viktorols340]

Very good video! Keep up the good work!

[@amidsid3482]

beautifully done man

[@alimohammadzade5906]

This video is wonderful!

[@dylanparker130]

beautifully made video

[@trevorholton1835]

Awesome bro

[@iamkrishradha]

Hey , I really liked your videos... How can I contact you?

[@schronos4479]

I think It Will Take At least 30 more years To Solve,. and I can TellIt Because I Will Be Big In That Age And I Can Solve It... so Yes I Will Try to Solve It As I Am Just 6 now.. When I Get 36 I Will Try to solve It. I Am Really trying from now.. Hope I Do It.....

[@PainDGod-dt3iq]

Do your best young man : )

[@arnavdayal9778]

can you make a video on p vs np problem

[@jamesbra4410]

If it were to be solved then I presume it would be in the realm of Symplectic Topology. Also considering that an interference between gravitational forces and the entropy of the fluids cannot be ruled out since understanding in the realm of gravity and thermodynamics are also nonlinear.

[@mrnobody2344]

Great video! Love your channel I hope you grow and can do this as a job :)

[@suspendedsuplexchannel1000]

Was it solved by a Kazakhstani mathematician, pzz reply me , I have found some videos like this in KZbin

[@kinertia4238]

I believe that the general consensus is that Atiyah is mistaken. While no doubt a brilliant mathematician, he's almost 90 years old and has been prone to making several mistakes recently, including the claim that he had also solved the Riemann hypothesis. That said, if I recall correctly there was some technical difficulty in understanding his paper because it was entirely in Russian.

[@Akkhinus]

7 seconds times 10^9

[@shalvagang951]

I THINK WE HAVE TO TAKE A DIFFRENT UNIT TO SOLVE NON LINEAR DIFFRENTIAL EQUATION A ASSUMPTION

[@kamnasingh5380]

I've already solved the Riemann hypothesis problem

[@philipphoehn3883]

same

[@douglasstrother6584]

... but the margin is too small to contain it.

[@imironman5763]

😂😂😂😂😂😂

[@rafisk159]

hii there, if u have time, will u make a video on Einstein gravity theory and newtons gravitational force one is saying gravity trapped us inside the earth and the second one says gravity is the force which is attracting us which one is true if any one of them is true why we are still fallowing two of them

[@oswalmartin4888]

I will solve the equation next year OZM

[@luisantonioarosmarquez2968]

45 weeks left buddy

[@anandsuralkar8376]

Times up

[@BELLAROSE21212]

Solved

[@HDQuote]

I have a truly marvelous proof, but this commentsection is too small to contain it

[@bersano4572]

nice

[@singh3549]

i will solve this

[@vishalpande6599]

Great

[@hacodecode6248]

You should upload this type of video persistently. Ok sir

[@FranFerioli]

I have discovered a truly remarkable solution of this problem which this comment section is too small to contain. Trolling like a boss...

[@trekdiaries3540]

Hey, ur description is on the topic is 👍👍👍

[@shawn980]

1:55-2:28 Huh... Anyone else seeing some similarities between describing fluids and describing quantum mechanics?

[@gustavoalejandromorletavil426]

Statistical mechanics, maybe?

[@shawn980]

@@gustavoalejandromorletavil426 randomness. Think about this in terms of fluctuations in vacuum energy.

[@mohtashimnizamani]

I would solve it!

[@samtipler4390]

Wow really enjoyed this

[@austingrey6641]

hodge conjecture next!!! amazing video :-)

[@thatoneguy8941]

Around 3years

[@priyoranjan1420]

Hi appreciate your hard work..you have stated infinite blow up of this equation.i would really appreciate an explanation regarding this..

[@omarsaleh8855]

B4 next yr cause im working in it

[@imironman5763]

😅😅😅

[@sourabhsfb8656]

Hey yputuber! I love physics and now I also started loving your channel due to your hard work.

[@anasagha5972]

1 year

[@EliteBeast]

Only 1 million dollars? Lmao. This problem leads to billions of dollars in future innovations.

[@ottahmatthew2151]

its possible

[@ashishsharma-og4nl]

Which country are you from? India?

[@alinaxie8378]

who came here after watching the movie gifted? XD

[@philippstephan459]

mee

[@vinayc12]

Epic Video. Keep it up.

[@jimtwisted1984]

Water is really simple? It sounds as if water is really difficult. You think it is simple because you are very familiar with it.

[@summertime8525]

why dont they include turbulence in the equation?.. or incorporate chaos theory

[@ashishsharma-og4nl]

Because those are a consequence born out of it

[@fritzmartintorrescabual2550]

I really want to answer this problem. But I feel like my brain can't handle it. hahahaha I really like physics.

[@binayakthakur5122]

In a nutshell F=MA for water

[@jeremiahmullikin]

How about a video about gravitoelectromagnetism?

[@sentinalcry5725]

Good one buddy!

[@MrHARSH-jt9ei]

It will be proved on 4 March 2022....

[@davidmejiazambrano]

Good job

[@lifeeverythingandtheuniver5409]

I know why turbulence occurs. There is a basic fundamental difference between fluids and solids that not many people know about. I don't know to express it with math BUT I know when this difference is taken into account then the turbulence goes away.

[@alexilaiho8534]

What will happen it is solved?

[@matiasdanieltrapagliamansi3109]

u great

[@diegoag85]

Just make it a billion USD and we will see Navier Stokes getting solved real fast!! 😂

[@aryanbhatia1729]

If it would have worked that way we have would be a space faring civilization right now

[@diegoag85]

Aryan Bhatia impossible is nothing!! It’s just that there are easier ways to make a million! When money won’t be subject resources will and maybe we will evolve! So far this is not the case.. unfortunately!

[@aryanbhatia1729]

@@diegoag85 you think if newton and Einstein were give a whole lot of money would they have given their laws and theory's before they came out in this timeline?

[@diegoag85]

Aryan Bhatia chances are yes! Unfortunately..

[@aryanbhatia1729]

@@diegoag85 then my friend I think you are wrong you can't buy thoughts with happiness This is just my opinion

[@prajwalchunarkar]

Great video brother loved the way you teach but be somewhat slow just to make us understand

[@tatijaki3320]

Hi guy reymel quemner have to solution navier stokes equation I love equation quemnerianas

[@KingHim00]

Not long

[@imironman5763]

It will take only 36 hrs to SOLVE

[@lewishaustein8552]

Now

[@bizepsmizeps]

there are a few mistakes in your video! For instance, rho in the Navier Stokes equation is the density and not viscosity.

[@kinertia4238]

Thanks for catching that! Can you tell me if there are others? I will correct them. I don't have much experience in this field, I'm only in high school so would love some feedback!

[@mr.goodsir3410]

I came here from reading a light novel... I don’t understand shit

[@gibbysure964]

{50 Cent}

[@fabulusinvictus2198]

How many time? This year. Or the next, no more than that

[@emilywong4601]

Theorem provers

[@imironman5763]

Within 2020

[@marcosainte6114]

This is L-It will take approximately three years.

[@whichwitchswitchedtheswiss]

Arabic translation pls 😖😖

[@oofusmcdoofus]

This problem will have to wait until i grow up

[@johannrajan6357]

stupid how humans build everything upon weak bases. Which, is the reason why we have to go back and try solving a mistake. Build on a strong base, so we don't go solve the problem after building up high.

[@wuxi8773]

new way of math needed, not calculus