Can the Navier-Stokes Equations Blow Up in Finite Time? | Prof. Terence Tao

my name is Itamar Vilna and I'm the head of the Natural Science Division of the Israel Academy of Science and I'm happy to open this year the Einstein 2015 a lectureship that will be given by professor Terence Tao from UCLA in the United States the lectureship is every year dedicated to a different discipline this year it was this year it was the term of mathematics to be the present to be the topic of presentation of the Einstein lecturer and I must say that when I asked our mathematician to select and nominate a candidate for the lecture it was very easy to find the lecturer because there was a very nice overlap by the different members who suggested Professor Tom and I would like also to say that the president of these our academy of science and humanities professor unknown and was unable to attend the ceremony and she asked me to open the ceremony I would like before introducing the lecturer I would like to ask the audience that after the end of the lecture the audience should not leave the hole because we have a very short ceremony where we donate a scroll and the medal to the light Einstein lecturer so please stay for additional few minutes I would like to invite Naga alone to introduce speaker okay it's a great pleasure to introduce say professor Terence Tao from UCLA who will deliver the Einstein lecture today theory has made the fundamental contributions in a wide range of fair mathematical areas including a harmonic analysis partial differential equations number theory and especially the distribution of primes random matrix Theory combinatorics a in more he received a essentially oil prices in the ones he didn't get it he will get a in the future and they his blog is a is a very best place to read about a mathematical news essentially in all areas he has a unique ability to explain a sophisticated the ideas a in a comprehensible way and in particular some of you may have seen him a last November in The Colbert Report where he managed to convince a coal birds it's a prime factorization of 15 is 5 times a three I want to tell you a one story about a theory is it a has very little to do with his mathematics but but I think still it's a interesting a so a few years ago a we served together in a some committee in the first native of the committee took place in Oslo and it so happened that we came together in the same flight from New York to a coastal airport for both of us this has been the first time in Oslo Airport you know when I come to a new airport you go away after passport control you way you look left and right you try to find a where is ATM machine a where is a train to the city what you should do a any needa was going through this routine a maybe after 30 seconds I was still looking for the ATM machine I noticed that a theory actually a already bought train tickets to the city for both of us in fact he managed a he bought this ticket say he forgot to ask her receipt and then you realize that he actually needs a receipt so he cancelled his transaction both the tickets again with the receipt and this is all when I'm still looking for the ATM machine I remember I was very impressed and you know on a second thought maybe it has something to do with his mathematics also so it's not only the case that he can do anything in mathematics better than anybody else it can also do it faster than anybody can say so besides all that a a Teri's also a great speaker as we find out there next and they today he will tell us about some new ideas regarding say in a via Stokes equation a Terence Tao please thank you very much no book no Duplin so yes I won't talk about the navier-stokes equations which are a system of mathematical equations motivated by a series of mathematical equations a system of mathematical equations motivated by physics and so you know in physics week we study matter so which are modelled by pipe which have systems of many many molecules or particles okay thank you but it's very hard to to study directly trillions of particles so we approximate laws of physics by continuous models and for example fluids like water you know rather than model every single atom individually we use a continuous model so so we model the fluid by various continuous field such as example the pressure so the pressure would at any given time and and position in space would be some scalar number P the question at that point and also maybe the velocity field so at any given time and position you would have that the fluid would have some velocity U of T which be some vector so this is the model we use to describe continuous media but it's not quite accurate so you know at atomic scales of course matter is not continuous it's me it's made of atoms and molecules and so you know there isn't a well-defined velocity field at every single point in space when Steve you're below the atomic scale and pressure also doesn't make any sense after a certain point but even if the the model is not accurate at atomic scales you know the hope is that still it's still good enough for accurate to be accurate at larger macroscopic scales okay so the the model that I'm interested in is is the so the basic model for incompressible fluids such as water is the navier-stokes equations and so then I mean every six equations is a system of three equations it's the main equations the first one it's basically a continuous version of Newton's first law F equals MA so this is the derivative of the velocity so that's that's the a as the acceleration this is the gradient of the pressure that's the force that's the F M has been normalized to be one but the fluid is moving and so there's an extra term coming from the transport of the of the fluid by the by the velocity field and also you assume some friction some viscosity that that some energy is being dissipated and that gives you another term in your equation and then you also assume that your your fluid is incompressible so that the effective velocity Q is divergence free and then finally you have to specify some initial conditions you specify the initial velocity you don't actually have to specify the initial pressure you can use this equation to solve for the pressure once you have the velocity so we usually we don't specify initial pressure but we do specify initial velocity and then this is number nu which is the viscosity this is how Biscuits the fluid is and for navier-stokes use positive there's a closely related Euler equations which is the same equation when u is zero okay so we have initial condition velocity and then we have seen compressible fluid solving this equation and this is this the basic most basic model to study fluids and so we have this so we have this this basic mathematical problem which is actually uh which is the global regularity problem for navier-stokes so the clay institute famously of mathematics famously post seven five price problems in mathematics these are what are called the millennium prize problems they have a million dollar prize attached to them so things like p goes NP the Riemann hypothesis it's very famous questions in mathematics one of them is solved the other six are open and one of these six is the regularity problem for navier-stokes and the question is that given an initial state flow fluid so it's also given an initial velocity field which you assume to diversions free for consistency given any smooth let's say oh is it not okay so given any smooth especially localized initial conditions can it just can you always solve these equations globally in types can you can you always find a velocity field and a pressure field that exists for all future time which solve these these navier-stokes equations so this is the global regularity problem for navier-stokes alright so what is this have to do with the you know the actual physical interpretation of both of these equations so what what you can prove from sort of standard PD methods instead so you want global existence you want these fluids to exist globally globally in time but what is easy its local existence that but if given any nice smooth initial data you can always find a local solution of so you can always make the fluid exist and have a well-defined smooth pressure field and smooth velocity field up to some maximum time T star which could either be finite or infinite so your solution that you do exist for finite time or for infinite time and in either case you get a nice smooth solution which then stops if it's finite what goes on forever and if it stops there's a good reason why it stops because your solution is no longer physical so if if there is only a finite type of existence you can prove that we are blow-up occurs and what blow-up means is that the velocity of your fluid is becoming more and more infinite at a plat at at some point so that at some point in space your fluid is going faster and faster and as you approach this maximal time of existence you have attained infinite velocity so this is what you can show if there is blow-up and if if if you're there's no blow up if the time of existence is infinite then this doesn't happen that the velocity in fact goes to zero as as time goes infinite no just like you actually seen in physical reality splash water around you will get you'll get a lot of turbulence and velocity but eventually it will die down and you'll get no more normal velocity so that's the mathematical theorem but so what it tells you as a physical consequence is that if if blob occurs if the navier-stokes equations are can only be continued for a finite amount of time before moving up what that means is that the navier-stokes equation is break down as a physical model because of course real life in real life fluids cannot attain infinite velocity so if your equation is predicting an infinite velocity solution then what I mean is that your equation is is not physically actually it's not physically realistic but it is not accurately modeling your fluid and what's so and so what that means is that something is going on at them at the molecular level which we can no longer be captured by by these equations so if if if these equations blow up then we know that the navier-stokes equations are not necessarily the the correct way to describe is fluid and you need to look for more advanced connected models which we unfortunately don't have at this point conversely if you don't have blow-up if you know that that we said that these equations always have global solutions let's go global regularity then that's evidence that these equations are good physical models it's not it's not conclusive it could be that these equations have nice solutions but these solutions are still not what actually physically happens but it's it's at least a necessary condition for for physical soundness copies of these equations so that's sort of the the motivation for for this question although I think of this question really as a mathematical question is sometimes when you look at popular articles on the navier-stokes regularity problem is is that oh if you can solve this question then you can predict the weather or oh I have all these reward consequences but probably it's not so direct a more of a fundamental science question than and what one of direct applications okay so of course you can simulate these equations numerically and there's a whole field of computational fluid dynamics which is devoted to exactly that and so what we can see from experiment and numerical simulation is that global regularities appears to be true at least generically so if you pick real-world data your typical initial data typical initial conditions then whenever we want to rerun the our simulations we always do find a global smooth solution it may be hard to find may be computationally intensive to to to find the solution but it we always seem to find one although an interesting thing happens when you look at these solutions when you start with so when you start with small data just walk did which is initial velocity almost zero nothing much happens it the solution just decays to zero quite quite predictably but when you start with large data very big velocity initial conditions then there's a phenomenon of turbulence the the solution will eventually decay and go to zero but it takes its time to go to zero and before it goes to zero it moves its energy and your final and finer scales so let's see if I can sorry so the this is this is a very cartoon picture of what happened of course and it's shooting three dimensions but I'm not going to draw three dimensional picture here so you know initially your your your velocity field might be nice and so this is some some big full this is supposed to schematically describe some velocity fields over here the waters flowing this way over here the waters flowing this way in some incompressible fashion and it's oscillating at a very big spacial scale but often what happens when you run the equations is that is that this these large-scale initial conditions become break up into smaller scale or higher frequency if you wish hire three nc components so the energy gets split up into into into smaller and smaller patterns and then you get you could find and find a scale the energy gets pushed into fine and finer scales and you get all these whirlpools and turbulent turbulent effects but then at some point the viscosity kicks in and it damps out all these these these these fields so it eventually the velocity should die down so this is the type of dynamics that you see you see in practice when you have large initial data okay okay so you know we have all this empirical evidence that global regularity is it's a true at least generically so at least in is so because of this the global regularity problem is not really a practical issue in applications you know the people who do competition of fluid dynamics they don't stay up at night wondering whether an episode is globally regular or not because they see it in there in the simulations and so they're a reasonably confident and isn't in the accuracy of the models at least before really heavy heavy turbulence what is your very heavy turbulence the equations become very unstable but okay but at least they have but you know they sort of know when when when these these these models are accurate or not in practice but nevertheless the mathematical question of where the blow-up is at least theoretically possible even if generic data is is fine there's still the possibility of exceptional data that there are some special initial conditions which can lead to to infinite amounts of turbulence and in fact so what I believe now I can't prove it yet I would love to announce I can prove it but that's that's not that's not in the cards yet so in fact I do believe now that actually there does that do exist extremely special choices of initial condition you know which would never appear in physical practice which was so they're not physically relevant really but there should is extremely rare initial conditions which by their design do create blow-up so I now believe the mathematical question actually is its answer than the negative although I can't prove it yet but I can try to explain why I think this way so some equations blow up and some don't and what and in PD we have this classification of PDS and the different types that we tease called critical equations supercritical and subcritical and roughly speaking subcritical and critical equations other ones that don't that usually don't blow up and supercritical ones are the ones which could blow up and the navier-stokes equations have in the supercritical category but so explain this this distinction in a critical supercritical and so forth I have to do a little bit of dimensional analysis right to describes of the relative sizes of various things so again this is this is the main equation for the navier-stokes equation describes how the velocity evolves and it's it's got many terms in it but the main two terms the terms that are fighting really fighting each other here are are this term this is the transport term this is the nonlinear term coming from the fact that that the the particles travel in the direction with velocity V this is the transport term as a nonlinear term and the viscosity term this this is the term is trying to damp things down drain or the energy our system to make them lost make the velocity equal to zero so these two terms are sort of the two terms that are fighting the most the pressure is just there to keep the velocity diversions free so we know that term and heuristic aliy if you want to understand which what this equation is doing it basically it's a question of which of these two terms is dominant which one is bigger and heuristic aliy if the viscosity term is bigger if this is the big term then you can ignore these two terms and if the velocity term is the biggest then viscosity term is the biggest then your equation looks like the heat equation which is an equation we understand very well it's a nice linear equation and it it its effect is to just send all the velocity uniformly to zero and so you don't get ya so you don't get any turbulent behavior you just get nice linear global regularity this is what should happen if the viscosity don't dominates okay but if instead the transport term dominates if the transport term is the biggest then that's where you expect very nonlinear behavior that that's where all the turbulence is coming from moving energy from small scales to to large scales that doesn't happen for the heat equation but it does happen with these transport equations and so this is the regime where blowup could happen okay so the basic question is under what what circumstances is the transport on bigger and what circumstances is the viscosity term bigger and so this is this is where you start doing some dimensional analysis okay so as a model you assume that your solution is this is turbulent at some scale so there's some spatial scale of L which is and you assume that your solution is just so spinning around in little spirals at at scale and let's say that your velocity is of source on speed V so at you're moving around at some velocity V at various scales of length L and if you make that sort of answers then you can predict the sizes of various various quantities so here if you a size V and U and just oscillating a size L when you differentiate the gradient of U should be like the over L rise over run and so the transport term should be about this big V squared over L and a similar argument shows that the viscosity which comes from insult Ossian is if I something like V over l squared and so these are sort of the dimensions of these quantities and so the protection from this analysis is that if your velocity is not too big if your velocity is less than the inverse of the scale if the velocity is less than these spatial frequency then viscosity should win and you should see linear behavior okay so things should behave very nicely when the velocity is less than this this is threshold and conversely if you have a large velocity of large loss is much bigger than the reciprocal of the length then you should expect normally behave turbulence transport all kinds of Poznan possible potentially blow up so that's the dimensional analysis mystic okay but just your fluid whatever it is it can only have a finite amount of energy so there's limitation as to how much velocity you can have in any given region of space because of energy conservation so you know energy per particle is half MV squared so in info for fluid the an analogous energy the kinetic energy it's one-half of the integral of the velocity squared there's a master because you're incompressible you can just scale it up you can think of the masses being one so this is the energy and it's decreasing in time that's easy to see so the no matter how far you go along your energy is always bounded and because it is bounded so it's bounded from above on the other hand if your velocity is if your velocity is moving you know so if you have some ball radius L and your solution is moving around as a velocity V the in three dimensions the volume of the boys is about L cubed and the and the velocities is V so they're the the imaging just coming from a single one of these little war balls the force is at least these are the energies at least V squared L cubed it could be a lot bigger because there could be many many of these are these little wall box but you have at least this one lower bound may be all the energy maybe you're all the energy is stuck in one little wall ball so if it's lower bound and the energy is bounded above so that gives you an upper bound on the velocity so at any given scale L the velocity cannot exceed Eltham minus three-halves because this has to be bounded okay so so that's how big the velocity could get in principle it may not it may be are smaller than this but if all the energy gets squished into one little ball this is this is how large the velocity could get and the point is that this bound is bigger than the the threshold between linear and nonlinear behavior this is what it means for Navisworks be a super critical equation that the the size of the solution can be so big that it crosses the threshold between linear and nonlinear behavior at small scales when L is small L the minus three-halves is bigger than L 4 minus 1 and so what that means is that if you can manage to push all the energy in this equation into a single scale into a single ball then you can create very very nonlinear behavior in principle and so energy conservation is not strong enough to rule out very very nonlinear behavior and this is a special feature of being in three or more dimensions you can also write down I mean of course reality is this is three dimensional but but you know math additions can work any information they please you know so you can write down the navier-stokes equations and say two dimensions and if you do the same analysis in two dimensions you find that it's it's a yeah the energy bound is now b squared l squared of B squared L cubed and the equation now becomes critical it's it's now very hard for the energy to actually all concentrate into into so much that that you get nonlinear behavior and in fact in this case it's been known just for many years that you don't have to much Melanie behavior this is so a finite amount of nonlinear nonlinear behavior that can happen and your solution eventually becomes linear and dies down okay so two dimensions we understand everything was this critical three dimensions it's super critical at this this is why we don't understand this this equation and so basically one of the reasons mathematically why we care about this problem is that the navier-stokes is a very good example of a supercritical equation I can give me the ice on lecture theatre really famous example of the supercritical equation is the ice and equations gravity but they're much harder than ever Stokes to solve but you know so potentially we understand every Stokes better we might also understand the only Center questions but I have nothing interesting to say about the answer equations so I unfortunately didn't prepare anything about that okay so that's the dimensional analysis so if you want to actually make the the fluids blow up what I make these equations blow up we now have sort of a scenario a potential scenario that you would like to to make happen and to make blow up so the you want to make the velocity as large as possible for given scale and we saw that the most efficient way to do that is to push all the energy into it one in the single ball not spread it around although many many balls so the type of scenario scenario blow-up scenario that should be easiest to to make happen is as follows at so at any should be some special times t1 where at a certain time t1 all the velocities should be focused in a fairly small ball so there'll be some special scale l1 some small number small radius and inside this ball if you have pushed all the energy of the fluid so you're spinning around quite fast l1 of the minus three hops alone is small so this is a big number you have a fairly fast velocity field here and so this is what your solution should look like at some intermediate time T and then what you would like is you should somehow design your solution so that at a slightly later time say T 2 is the solution manages to compress this energy further so you have this this energy pattern some sort of vortex here and you want you want to somehow generate a push for the energy interested into a smaller vortex so somehow you should someone use the nonlinear effects that the transport and effects of your equation and the pressure to some well create a smaller vortex like this where the energy is now constituting a smaller scale l2 maybe you say 1/2 the size of the original scale and so you push all the energy into this smaller scale and so now that the velocity field is is moving even faster now it's kind of 2 to the minus 3 halves rather than l1 of the - to be hot and how long we should it take to get from this pattern this vortex to this pattern well you are contracting your you're velocity field go from l1 to l2 and you're moving at a certain speed at 1/4 minus 3 halves so distance is time over velocity so the time that in principle that the time it should take to get from the first pattern the second it's just l1 divided by the velocity jutsu which is something like L 1 2 5 halves ok so in principle at least you should able to take a solution of fluid which is rotating at a certain velocity and somehow manage to compress it to a smaller vortex and then hope and then you can hope that this process repeats that maybe if its vortex is just off the right type it compresses to an even smaller vortex and an even smaller scale maybe half the previous scale and so on and so forth so you can hoping in this hierarchy of ever decreasing vortices so if this happens then this deliciousness will eventually lead to infinite velocity because the time taken to get from each vortex to the next is some positive power of the blood scale of one to five half cell to the five-halves and so forth and if each length scale is half the previous one this is a convergent geometric series and so this will converge to a finite finite time so in principle this would create a finite time lower okay and so you're pushing all your energy into into into a single point very very fast the solutions getting faster and faster and concentrating on edges at a single point because this is not what this is not what happens when you actually see water okay what it doesn't do this when you see it in real life but but but that's because we know we you're not specifying specifically wall shows and data it could be that if you really choose a specific special initial condition that you would see this this phenomenon okay so basically this solution we could be behaved like a self-similar solution and in this scenario the viscosity forces is negligible it's tiny compared to the transport so this should just be a transport effect that's just the effect of transport and incompressibility should create it should create as this law okay so this is an example of what we can essentially be like a self-similar solution so it won't exactly self-similar we know that in fact navier-stokes equations doesn't have exactly similar solutions would be approximately sub similar solutions okay so this is a scenario it's not it's consistent with the dimensional analysis is consistent with energy conservation but it may not actually exist we're going to actually find a solution that does this so so that's that's the problem can we actually create a solution that actually has this predicted behavior so I don't know I I believe so so what I can actually prove is that if you allow yourself to cheat and this is a very big cheat that the cheat is to change laws of physics that if you if you allow yourself to change the equation of the navier-stokes basically you replaced the the most difficult term which is this transporter by an averaged version of this of this transport term it's and averaged in some boolean multiplier sense which I don't want to get into but you can use you can use the Fourier transform to split up this this this nonlinear expression into many many pieces different interactions between one frequency another frequency all and so on and so forth and what I did was that I observe that that if you can if you allow yourself the freedom to turn some of these interactions off that you don't allow certain physical interactions to actually happen you just loop by change laws of physics at certain points so yeah I can make this effect this Melanie arity slightly smaller in a very carefully-designed way still in a way that preserves energy and preserves all the other things we know about the equation but if we turn off some of the non-linearity then for this sort of averaged version of the equation then you can create solutions that do exactly what I said before evades they start at a single scale as a single vortex and then they collapse to a slightly smaller vortex and collapsed and even smaller vortex always the boss sees increasing and the time between collapses it could goes to zero and you get this blow-up and the reason why I can do this is that is that once you allow the freedom to turn off certain interactions and so you want to keep a small number of interactions it the the equations of motion are then something that you can engineer you can then sort of build in into loop laws of motion various interactions which which behave like you know kinda like resistors and capacitors and so forth they they play behave like like in a very predictable fashions and then you can sort of engineer solutions that take advantage of all these some of these circuits that some well now inherent in the equation to build little machines that are designed to do exactly what this blower so that's the strategy so let me I can try to explain so I want to kick show you I mean that if there's lots of equations but I'll try to show you pictures how this this mechanism is supposed to to to to run okay so there's a certain loop that this is sort of a program that that you'll feel fluid is now executed so your fluid is now basically trying to be a computer or actually a program I mean the equation is the computer but the fluid is is sort of less than one run in one time instance of this computer so suppose that you've managed to somehow reach a certain scale so it's so your fluid is now circulating at a high velocity in a little bowl like this so normally this this could be somewhat unstable that the fluid will leak out and do all kinds of developed turbulence here but what we do is that we turn off all the possible interactions that so so we turn opponent interactions that would disintegrate this this loop and we just keep solutely just keep circulating and instead we turn on one tiny interaction so we keep one little nonlinear interaction whose purpose is to drain a little bit of the energy of this loop into a nearby loop over about the same scale so if is this big loop of fluid moving really fast and you you allow one little nonlinear interaction to allow some of the energy here to so slowly start so you write that this this portion of fluid starts off stationary but then it slowly starts rotating by a little bit okay now that by itself we can't use very much but then what you have to do next is that you have to use this so slowly increasing pull vertex to then trigger a second nonlinear interaction so we so then this you make a you take a third patch of space and you use this little this little portion of of energy to start to start draining energy into and to a third vortex and the point here is that different from the nature of the way that you can define these nonlinear interactions you can make this this non-linearity grow exponentially the velocity here turns out that you can only make it grow linearly but this one you can create exponential growth so it starts up a very small velocity but the velocity over time just shoots up exponentially so you know when you plot the velocities in before a certain period time that's basically zero and then suddenly just turns on it become this step is coming and analog to digital converter this summit and analog signal is the digital signal just so it turns on and so the nothing happens for why suddenly they start spinning and then finally what you do is that you you introduce into your laws of physics a third nonlinear interaction which the moment that this interaction becomes significant what it what it will do is that it will it will compress this one so you you add in the third nonlinear term whose whose purpose is to take so whenever this mode is activated what it does is it takes it very quickly takes this large mode and since squishes all the energy of this large mode into a smaller mode so so so when you run this equation there'll be a time delay where what you have to set up a second loop and then suddenly it squishes this this energy in into into em very suddenly into a smaller scale okay so that's that's what that's so you design the laws of physics to do exactly this and then and then you you just repeat the process okay so this smaller mode now does exactly the same thing it slowly drains a little bit energy into into into other oxygen modes which at some point this mode activates and it squishes this mode into an even smaller mode and then and then you have to wait a while for the next stage to set up and then it squishes it again and so you get this very discrete self-similar sort behavior I mean this there's some leftover energy here so the energy these loops still still hang around but they're slower these guys move faster and faster and at smaller and smaller timescales and so it turns out that these guys you can forget about it after a certain point and so you can actually add so by carefully choosing certain interactions that you like and then suppressing all the interactions that you don't like you can create a system with your based energy conservation in all the dimensional analysis but it has this blow finite right hand blow up okay so that is something this is what you can do if you cheat so literally the million-dollar question is what what do you do if you do not allow the cheat so basically you want to mimic this type of behavior where you're not allowed to turn off all the interaction so you actually have to use sort of the physical laws of motion fluids but you want to somehow simulate on this sort of idealized interactions that we had before so you want to create basically a fluid computing you want to create patterns of fluid which behave like a machine like I could like a robot which you have which and you program it to do things like you know that is if it's if your robot is moving in a big loop you want to suffer drain a little bit energy to create a little loop over here and then another loop over here and then at some point this this auxiliary robot that you can structure can go back and push the energy in your initial robot to a smaller scale and so basically it becomes in a no longer a mathematical task so much more of an engineering task you have to build a machine but purely out of fluid peeling out of water so I don't know how to do this so you know I mean the there is fluid computing in in your life can people use it for avionics and some other specialized applications but you know if you if you have fluids together with something else your like pipes and valves and turbines okay some solids as well as fluids then you can make it useful machines and you know you can make very simple machines but you can also make these gates like amplifiers and and things things like logic gates or gates and gates and in principle you could make I mean no no one does this but you could make a instead of mechanical computer you can make a fluidic computer out of big pipes in your and and and just you know full of water rather than electricity yeah so I mean computing is possible if you allow yourself some additional material than just your fluid so what I but I don't see any conceptual problem you know just it's just a humongous engineering problem but you know in principle you could maybe use the water itself you know maybe some vortex sheets or something as your pipes and and so forth or we can play of solar tones or whatever you may be able to you know there's nothing in principle that stops you from building logic gates and amplifiers and and and and all these things that were used to in other engineering computing paradigms you know if you could be able to to build these things purely out of out of water and I don't know how to do this but if you could do this then in principle you can you can create machines that that that can do arbitrary computation you you know somebody instances to incomplete and fluid machines and if you can you know happy so programmable computers may purely out of fluid then there doesn't seem to be anything stopping in principle at least the creation of this sort of machine that replicates itself um I mean these machines have a name the global phenomena machines if it's proposed we're going to call it I say Mars as the cheapest thing to do is to send one probe up to Mars which has the capability to to use his environment to create another copy of itself I'm just hoping thank you two copies ourselves better okay and then you and then you you keep your awaiting that but you know and in other context we know how to do this so for example for cellular automata no this is very famous Conway's Game of Life which is has very simple rules and much simpler than navier-stokes but we know in that you know in that in the rules of that universe you can construct logic gates and then we construct computers and people have constructed huge fun learning machines which you run the game of life they just they create another copy of themselves and in principle you can do that for you know all you need is some artists these basic gates enough to make things true and complete and of course you need was error correction and huge technical obstacles but but there's no physical obstruction to creating this this sort of machine which would demonstrate these equations blow up and in fact you know I now believe that any system which is sufficient is what anti-system urge is supercritical and and which is so sufficiently complex that it would support computation you know you should be able to design solutions that blow up there's no physical reason why why this shouldn't happen but actually demonstrating this rigorously is is going to be an amazingly difficult task but I hope that can gather some more evidence that you can do this I sort of my my research points Linux to a few years okay thank you very much we still have time for a question or two or more yeah you describe two regimes when either the knee turn or the victim is dominate okay one way the one with the Mito connected that was simple put that video showing the other is you you have no need equal to zero namely like that and a bespoke sequential thought so doesn't that make situations in principle yeah in principle yes yeah so if the equation collapses to the Euler equations what you see yes yes I would know this list it is with other one this order well there's several types of Euler equations is this all there's other kids with vorticity yeah so yeah so the equations are simpler this actually makes things harder because the Euler equations have some additional conservation laws which had to learn so for navier-stokes basically the only important conservation was energy the Euler equations once you remove or tissa T there so there's a few more conservation laws its own called hillier city and if you are the angular momentum a couple of other things which make it a little bit harder to actually design flows that do what you want like if you want to compress a big vortex into it into a small vortex it is still consistent that you can do this with all the known conservation laws but it's harder to find an actual mechanism but yeah the other equations they're simpler but they're still not so simple that you can solve them exactly but they're not they're nonlinear still so it's but you're right the viscosity is no longer for this paradigm of blow-up the viscosity is basically irrelevant this is usability that's yeah so I'm constructing blow up at basically the maximum rate that turns out to be easiest my analysis because that means that that makes it the easiest to ignore the effects of viscosity if you're blowing up at a slower rate then viscosity could could could potentially ruin my blow-up solution so yeah so for technical reasons I'm creating solutions that blow up at as fastest as possible but that it may be possible to also construct solutions that blow up slower so in two dimensions no any equation this type that I that I write down has has what global solutions that's a theorem basically yeah every time you you pass one scale to another you a little bit of energy leaks and you lose a little bit energy from participation and in two dimensions energy is dimensionless and the amount of energy you lose at each each stage is a constant and so you can't actually make this process run for more than a finite number of steps but because of the weight dimensions work in three dimensions the the amount of energy that you lose at each stage decreases geometrically and so it there's nothing that you to go around to to make the ball so let's say think a carry again and I will read the nomination in Hebrew as well as in English academia Israeli team a team outside Albert Einstein academia lumen de tomate am with Cobell Atlantic to dazu the professor of Terence Tao neurotic era la Tasha submission abduction I'm a zephyr Albert Einstein will it's even follow from a torsion Albert Einstein - Pato lure us their physical theoretically vada in esto el principio Omega freedom Baku Mohammed avakov rock the corruption Terence Tao of Hemi Shiro a Tejada the term ooh la la the who of the four shall under fashion an opportune brutal non-atomic vulnera super surf at the bottom of their table in english the albert einstein memorial lecture these are an academy of sciences and humanities hereby expresses its appreciation to professor of Terence Tao for his presentation of the albert einstein memorial lecture for 2015 this occasion serves to commemorate Albert Einstein the man and his accomplishments under understand interests and influence will not only restricted to theoretical physics but to embark in a spectrum of areas of profound significance to science and society in this spirit of Albert Einstein Terence Tao has enriched science and society in many ways would have not president if I'm a will not chairman of the casualties I'm not sure what I'm supposed to say exactly the ceremony but there is also it may talents at first better you see since prom very nice so here by a give Lisa me attitude okay okay thank you very much

Channel: The Israel Academy of Sciences and Humanities

Views: 52,395

Rating: 4.9215684 out of 5

Keywords: Terence Tao (Academic), Mathematics (Field Of Study)

Id: DgmuGqeRTto

Length: 52min 24sec (3144 seconds)

Published: Tue Mar 24 2015