 All right, if everyone's had their coffee, we can start on the final talk of this afternoon's session with Professor Sinai, also from Princeton University, Professor Sinai. Yeah, OK, thank you. So I would like to thank the organizing committee for the invitation to participate in this conference and to deliver the talk. I have several people who helped me in this. The first one is Don Klee, my co-author with whom we wrote together something like 10 papers. And many of them will be used during this talk. Then my colleague from Princeton, Professor Nicholas Templier, who helped me with a translation into French, several phrases, which originally were in English. And then I would like to thank Carla Bodrigini, who is here. And he worked together with his colleagues from the University, the Camarino, Frigio and Maponi. And his results will be also just essential part of the talk, so maybe at the end he will make several just additional comments. So let me say a few words which somehow can be considered as a continuation of discussion, which we had yesterday after the talk by Professor Chaim Brzeziz. So I was surprised to hear about the months and the years of silence around the papers by Poincare. So it was certainly not true if you consider the works of Poincare in Ergodic theory. Every textbook in Ergodic theory began with Poincare recurrence lemma, which contained not only the formulation, but also the proof belonged to Poincare. And also just the sections of vector fields which were proposed by Poincare. This is very useful tool which allows to prove for vector fields several results which are reduced to problems with discrete time. And actually there were some extension of this result which were proven by Ambrose and Cacutani. And they just constructed the Poincare sections in the category of mirror preserving flows. So just I want to say that sometimes it is very difficult to interpret the events which happened something like 100 years ago. And let me give you just one example. Some of you probably know that Maxwell equations appeared in the United States 20 years later than they became known in Europe. The reason was the following. Maxwell was invited many times to the United States. And he was there with many visits. But he did not like his derivation of Maxwell equation. So all the time he gave talks on something like different subjects, statistical mechanics, optics, and all that. And then it took, as I said, 20 years of American scientists to understand the derivation and the meaning of Maxwell equation. So when we talk about the story of Poincare, it might happen certainly that Poincare visited Germany many times. But he just wanted to give a talk in the presence of Klein. And he thought that the most appropriate talk was about the automorphic forms, just discrete groups, and all that. And it is quite possible that Hilbert even did not know that Poincare worked on the Dirichlet problem. So now it is hard to understand. But this can explain the absence of references in the works of Hilbert. On the other hand, Poincare considered himself as a very great mathematician. And the absence of just reference to his paper was a non-essential detail for him. At that time, there were no just citation index and all that. So it was not so important what his paper quoted or not. So now I just want to come closer to the subject of my talk. It concerns fluid dynamics. And presumably fluid dynamics was one of favorite topics of Poincare. Actually, he gave just a theory of turbulence was in the list of lecture course which were given by Poincare. And also he has a book or booklet on the theory of vortices, which also belongs to fluid dynamics. And therefore, there are some reasons to believe that fluid dynamics was just one of his favorite fields. So the results and the methods which I shall discuss in my talk are concerned somehow with a clay problem about the existence or non-existence theorems for Navier-Stokes system. And there are many people also in France who just proved remarkable results on this existence theorems in various spaces. Two-dimensional case was solved completely by La Dijon-Ské. And as you know, recently there was a series of papers by Delales and Schochely-Hidde, where they constructed surprising solutions of Euler equation. And it is not the set of problems which will be discussed in this talk. What I want to discuss is just connected with the quotation of Poincare, especially with the second phrase, which is here. I'm sorry. I'm sorry. Oh. Doesn't it work? Oh, I'm sorry. Maybe I must do it here. Can you see it now? I'm sorry. I'm sorry. Maybe I must do it here. Can you see it now? No, but OK. So I shall discuss in this talk several problems which appeared from the analysis of nature. And the most notable example is just solutions of type of tornado. So tornado is just the solution for Navier-Stokes system where at some point of the space, there appear very strong rotation, having huge energy. For example, in Princeton, once a tornado appeared in the center of a big growth, and the result of tornado was that all trees fall down, the electricity broke down, and just many roads just were broken with this damage. And it took several days before everything was repaired. So but we believe that the solution of equation of fluid dynamics, which is connected with this phenomena, is actually close to solutions which have blown up. So the energy is huge. And from just mathematical point of view, it is more natural to consider solutions with infinite amount of energy. So we believe that such solution exists. And the problem of mathematician is to construct model or just study the cases when the solution actually do appear. We shall not consider the temperature, which sometimes is needed to analyze this problem. And this means that we shall consider a tornado of purely kinematic origin. So they just appear because of the motion which is governed by equation of fluid dynamics. So now I want to go further to equations which we shall discuss in this talk. So actually, this equation can be written in the spaces of arbitrary dimension d. They are written here. From this, you see that the coefficient near Laplacian equals to 1. That means that the viscosity equals to 1. And then there is the pressure term minus the gradient of the pressure. And the second equation, just the gradient of u equals to 0, is incompressibility condition. Incompressibility condition means that if you consider the vector field, which is created by moving this fluid, then it is just vector field preserve the Lebesgue mirror. And just from another, just it means that if we study this kind of dynamics, then there cannot be empty space without the fluid. So fluids feel all the volume which she can feel. Does everybody agree with this? OK, good. So the next step will be to analyze. I'm sorry. We shall also consider a close relative of the equation, Navier-Stokes equation, which were written before, and which are called Burgers system. After the great, just Holland physicist, Burgers, who spent his last years in the United States, it differs from Navier-Stokes system by the absence of pressure and the absence of incompressibility condition. And some people believe that this Burgers system is integrable. This is not true because integrability is valid only for solutions which are gradients of some function. In other words, if the vector u is a gradient of some function phi, then this representation is preserved in time. And for the potential of phi, there is a simple equation. So what I'm explaining here is just called Hope-Call substitution. And in the space of gradient-like solution, the dynamics really is very simple. On the other hand, if we just consider just general solutions which are not gradient-like, then they can be much more complicated. And later I shall show you some pictures from which you can see how complicated this solution can be. On the other hand, I want to mention that for Burgers system, there was a general theorem proven by just Russian mathematician Ladyshevsky, which just gives existence and uniqueness solution to the Burgers equation in some natural spaces. In other words, for Burgers equation, we don't have problems of constructing solutions on finite time. On the other hand, you will see that if we change a little bit the problem, then the situation will be absolutely different. So the method which I shall discuss in this talk is connected with the, which is called denormalization group method. And this method appeared for the first time in quantum field theory and in statistical mechanics. And then sometime later, Mitchell Feigenbaum from New York just proposed the application of this method to some problems in dynamics. And later it just led to the appearance of the whole just new field in dynamics. And here are some names of people who worked in this field. In connection with this, I just want to make some remark that many people try to construct strange attractors for equations of fluid dynamics and even for Burgers equation. But actually these attempts were not successful because we don't know so much about the structure of solutions of equations of fluid dynamics. On the other hand, you will see that there are questions and problems where this hyperbolicity, which is needed for analysis of this property appears more or less automatically. This is just, I anticipate what we'll say later. So the first step will be to make Fourier transform for Navier-Stokes system. We make Fourier transform for the following reason. We want to study solutions which have singularities. If the solution has singularity, then it is not clear what other derivative it has to be specially defined and this is some complication in general formulation. When we make Fourier transform, then this equation does not have just derivations and it is just a sum of the first term, which is linear term corresponding in some sense to viscosity and then nonlinear term. The role of nonlinear term is to spread any information about initial condition over just the whole phase space, in case space and therefore the meaning of this solution is the following. It is the sum of linear term which describes some damping and nonlinear term, which is responsible for extending the solution to infinity and the question is who is winning in this process and then I shall show to you that there are cases when nonlinearity is just stronger than linear term and this is the class of solution which we shall discuss later. In this equation there is just an operator p sub k where p sub k is the projection to some space orthogonal to k for each k and this equation in Fourier space is considered in the space of function v which are orthogonal to k. Somehow this assumption replaces incompressibility condition. So real value solution of Fourier transform just corresponds to complex value solution of the initial equation and if we want to study a real value solution then it has the following difference compared to the usual equation. Namely if we begin with equations of real value equations of fluid dynamics then they always satisfy to the so-called energy inequality. Energy inequality means that if we consider the initial energy for our equation then this time this energy can only decrease. This is not true for Burgers equation but nevertheless just for Burgers equation there do not appear any singularities. So what we shall do is was somehow inspired by Poincare and it is just now methods which was used in many cases. Namely we shall consider one parameter families of initial condition not a single solution but just solution depending on the parameter a. a is a real number and we just assume that the initial condition is a times v of k0 and then we want to write down the expansion in this parameter a and it is easy to see from the equation that one can write just a series with the powers of parameter a and it is just written here. So this is just this is just the linear term times a and then there are other terms in this sum the parameter p is greater than 1 and we have at least four more series whose coefficients are g of p of ks and this equation is equivalent to the initial Navier-Stokes equation but for the coefficient g of p of ks we have a system of recurrent equations. So the recurrent relations are written here and in principle if we have just initial condition g1 we certainly can reproduce all coefficient g of p. The first and the last term corresponds to somehow to the initial conditions and the main part is just played by the sum over p1 plus p2 equals to p and then rather complicated integral. It is a complicated integral but nevertheless I shall explain to you that it is possible to study this integral using some general remarks about the structure of recurrent equation. The main property is just the following. Assume that the initial condition is concentrated on a set C. Then it is easy to check that the support of the coefficient g sub k is just the set C plus C plus and so on plus C k times and it just leads... I hope you didn't break it. I hope I didn't break it. So just... it just means that in some sense the coefficient g sub k can be considered as k times convolution of initial coefficient j0. If this is so then we just choose initial number k0 introduce the vector k sub r where only the last coordinate grows with r and then consider normalized variable k near the point k sub r and what is important is that it is normalized with the coefficient square root of rk0. So it is just the same normalization as in probability theory and we can expect that in a typical situation the values of y are numbers of order of 1. They do not grow. So it shows how the memory of initial condition spreads in the space k. So this is the picture. You see from this picture that in the vertical directions the dependence on k grows linearly with r but in the perpendicular directions the dependence grows like square root of r so this is the domain which has the corresponding form and then another remark is that here one can make another just change of variables connected with the coefficients s1 and s2. I don't want to discuss it in more detail. It plays an important role but later you will see a little bit of that in the form. So after all this change of variables we come to the simplified recurrent equation which is written here and some of you might think that it is rather a complicated equation. Actually it is not and it has many remarkable properties and we can study the behavior of this equation as p tends to infinity. So actually the answer is written here. You will just write our coefficients h in this form of two functions h1 and h2 then for the limiting function we have this recurrent equation. This recurrent equation is nothing else like Gauss equation for the Gaussian distribution which is central in probability theory or it is just the equation for the fixed point in renormalization group theory and then the next step in the renormalization group theory is just to find the solutions to this equation. It actually turns out that this is possible and then the result is just the following that if we consider five parameters sigma 1, sigma 2, h1, 1, 2, h2, 1 and h3, 0, 1 these are coefficients in the expansion of unknown function with respect to Hermit polynomials then the solutions are written in the form which is presented here and this series converges well enough so the sum has a corresponding meaning. So we shall consider some very special solutions where c1, c2 are equal to 1 all other three coefficients are 0 and then in this case the solution h of y takes a very simple form which is written here minus 2 y1 and minus 2 y2 and so what will be our next problem is to consider the linearized operator corresponding to the quadratic operator in the fixed point and to study its spectrum this also can be done in complete detail and it turns out that the spectrum has four-dimensional subspace corresponding to four unstable eigenvectors unstable means that this eigenvector has absolute value greater than 1 then six-dimensional subspace generated by six neutral eigenvalues and then remaining subspace of infinite codimension where the linear operator is stable so now just it leads to the following picture which just explains the structure of solution and it says the following that we denote by a tau of u and tau of s unstable and just one second u and zero unstable and neutral subspaces and then the remaining subspace which is tau s and then suppose that we have just ten-dimensional ten-dimensional submanifolds of initial condition in this theory we cannot have considered just a single solution we must consider solution which has enough parameters and then from general theory of renormalization group theory it follows that there is a set of parameters for which the solution converges to g and if you analyze the consequence of this assumption then it exactly means that the energy of solution which we consider tends to infinity and this is exactly blow up so a remarkable thing of solutions which appear in this way is the statement about the growth of the energy namely if we just denote by e of t the energy of solution at time t then at the point of at the critical point t critical it grows like constant over t critical minus t to the power five this power five is universal in this theory in other words it's the same power for all systems of arbitrary dimension and even just for other system the Navier-Stokes system and just in the numerical studies which we have done by Carlo Bondargini and his colleagues this power five appear in excellent way much better than all other numerical results so this power in a sense is universal the reason why it does not depend on the system is just connected with the structure of renormalization group theory which we have here namely if we just look at our results from the point of view of Fourier mode Fourier coefficient then it turns out that we begin with initial conditions concentrated in a small bowl and due to non-linearity the solutions I'm sorry the modes corresponding to solutions spread along one dimensional line and this one dimensional line appears everywhere and the fact that it is this line is connected with the asymptotic of the energy which is presented here it is certainly much stronger just growth than all just phenomenological theory of turbulence but this kind of singularity was confirmed very well so now I just want to show you the results which were obtained by Carlo Bondargini and his colleagues and they considered the simplest case namely two dimensional Burgers equation without the external force where it is here is the equation and so here is the solution which they constructed you see that time just goes down and initially the solution was more or less just normal then at some place there appear just bump and then in another moment of time there appear much bigger bump and then there are all reasons to believe that if we follow along this solution a little bit more in time then this bump will become infinite so this is for solutions of Burgers equation which we discussed before and I just gave you the results which show the real valued solutions or gradient like solutions do not have any just singularities but if we consider complex valued solutions then the singularity appear very well and the growth of the energy in the solution satisfies to the inequality which I just explained before then also just Carlo Bondargini and his colleagues just presented the solutions of the three-dimensional Navier-Stokes system which are also complex valued solutions this is the picture which corresponds to just to Hurricane Sandy which just made so many bad things in the United States recently and then you can see that there are just some vortices which resemble very much vortices which corresponds to solutions which were just given in the previous picture I shall say about this a little bit more then just I'm sorry then we come to another part of this talk namely to the properties of bifurcations I shall show you two pictures here just if I know how to switch this on here it is one I'm sorry just okay not that, no I'm not done so here just we consider the following question which in a sense is model question for many other cases and suppose that we have just say linear partial differential equations and suppose that initial conditions are such that at some point say at point zero this initial condition has maximum which is degenerate meaning that if we write down the Taylor expansion of this function at t equal to zero then this expansion begins from terms of the fourth degree then the question is what will happen with solutions so this picture corresponds to the case of heat equation and there is a theorem which was proven by a student of Princeton University named Ben Shaffer in which he just showed that for the heat equation for any positive t the point of maximum will be non-degenerate so just heat equation is such that it just somehow destroys all the generacy of solution this equation I'm sorry this picture corresponds to corresponds to bi-harmonic equations so the derivative du over dt equals to minus delta u square and then you see that from one maximum there appear for just maximum which exists for some time and this is some kind of bifurcations which is just quite interesting and it would be just an interesting problem to investigate the number of singularities which appear from a given initial condition so what is important in this case and why this picture is different from the usual theory of bifurcations so as you know bifurcations first were studied in the works of Poincare on celestial mechanics and he just used various just expansion, various theories in order to explain what happens with solutions of some kind and there appeared the notion which is called Verso deformations and Verso deformation are just the families depending on just depending on some parameter of system which are in a sense universal so the changes in solution which appear in Verso deformation just appear more or less in all other cases in the problem which we consider this theory of bifurcations is changing namely instead of Verso deformations we have solutions of our equations our partial differential equation so we just start with some initial conditions and we just switch on the dynamics determined by the equation and we want to know what is happening with solutions it turns out that the pictures which appear here can be just absolutely different and have just other properties so here are just two examples which I mentioned but then I just want to explain the other set of problems where we have these kind of bifurcations now suppose that we write down the equation now it's not here fluid so Navier-Stokes equation and suppose that we write down this equation did I break something? no okay good no I just wanted yeah I need this here okay and how about this one? ah yeah thank you very much yes so we just write down the equation of fluid dynamics for the so-called stream functions stream functions in many cases are just more useful than the usual just components of velocity u and they are connected with the velocity u is just perpendicular gradient of the stream functions then just if we consider our system equations on the two-dimensional torus then it is a periodic function and as every periodic function it has just maximum and minimum if we want to reconstruct fluid dynamics trajectory then near maximum and minimum the trajectory of fluids are just directed along level sets of stream functions therefore near this near the extreme of points the trajectory are in some approximation closed curve we call these sets as viscous vertices viscous because they appear from the equation of the Navier-Stokes which has a viscosity and they are vertices because near these points the trajectory really just show behaves like vertices and then I just showed you some time ago the picture of Hurricane Catherine and in the center you see the vortex which was just appeared as a result of some bifurcations and let me formulate some number of theorems which show what are the bifurcations what kind of bifurcations can appear in these problems these are not all the bifurcations which we know at the moment but these are just the most natural ones so the theorem one which means the splitting of vertices says the following suppose that initially our just stream function which is just more or less normal Morse function on the torus has one maximum then we have Navier-Stokes system which shows what is happening with this maximum under the action of dynamics then it is possible to show that the initial conditions in the left can be chosen so that at some moment of time this maximum becomes degenerate and then after it became degenerate then there appear instead of one vertices appear three vertices two corresponds to maximum one corresponds to minimum we call this just bifurcations of young camel by the natural reason and the theorem which is here says the following says the following that there exists an open set in the space of stream functions such that for each stream function in this open set there is an open neighborhood in the time space and two moments of time t1 and t2 so that for all just time t between 0 and t1 our stream function has only one just maximum which corresponds to one vertex and then for t equals to t1 the stream function has two critical points and this corresponds also just to vertices but some of them one of them corresponds to degeneracy and for t greater than t1 the stream function has three critical vertices in U so this is creation of vertices think about this as dynamics of atmosphere you can see that there can be evolution such that initially there was only one vertex and then the dynamics was such that there appear several vertices another theorem of this type just gives the merging of the vertices so when merging of the vertices considering the merging of the vertices we have initial picture when we have three critical points then at some moment of time these vertices merge and they appear one maximum or one vertex which is just degenerate and when time grows this vertex becomes non-degenerate so there is just a big picture of vertices which can appear here and certainly they have some meaning for applications of fluid dynamics and in particular we think that they can be also used for creation of tornado and changes in the climate and all that I think just I spoke 45 minutes therefore I can stop, thank you very much Our speaker has taken exactly 45 minutes that leaves us plenty of time for questions No, this is generic by four occasions because just initially we have just one non-degenerate Normally you would have just this Yes, they are certainly on a different level I don't say that they are on the same level one can be higher than the other one Yes No, in this case cubic does not appear because evolution just follows from Navier-Stokes system it's not the biological deformation So the Nierenberg which says that singularities have zero measurement for dimension This is just different set of questions it is just they have so in the paper by just Kaffarelli, Nierenberg and others there was some estimate from above of just how the dimension of the set of singularity Here we just describe the set of initial conditions or the set in the functional space where we can claim that locally the set of singularity consists of finitely many points So it's not global results but only local Any more questions? Does anyone take advantage of it? If not, let's thank Professor Stein I for his talk