 OK, well, first of all, I'd like to thank the organizers for this beautiful conference. And I'm going to talk about turbulence and the Navier-Stokes equations. So this is some work I've been doing in collaboration with Bertrand de la Motte and Nicolas Schwebo, who are here, and also Vincent Rossetto and Guillaume Bellarac from Grenoble. So in the first part of my talk, I will show you how one can set up a non-perturbative or functional RG formalism to study Navier-Stokes equations. And I will show you the fixed point you can get from a reasonable approximation. And in the second part, I will show you that we can do much better than that. And what is very unusual in the NP-RG context, we can get, in fact, exact equation for the two-point function in a certain limit which is the limit of large wave number. And in fact, I will then show you that even more unusual in the NP-RG context, we can get analytical solution of these equations. So let me start with a very brief introduction. So we're interested in fully developed turbulence, so really stationary state of homogeneous and isotopic turbulence. And the standard picture in 3D is that of an energy cascade. So energy is a pointer, yes. Energy is injected at some microscopic scale which is called the integral scale. And then it is transferred, conserved through what is called an inertial range of scale until it is dissipated at a microscopic scale which is a Kolmogorov scale by molecular viscosity. And in fact, this inertial range extends with the Reynolds number. And through this inertial range, there is a constant flux of energy. And what's typically observed is scale invariance in this inertial range. So what is typically measured are the velocity structure functions. So moments of velocity differences between two points separated by a distance L and they're observed to behave as parallel with some exponent psi p. And for instance, here is the second order structure function and the exponent psi 2 is 2 third. So which is used in experiments. And what is also measured is the energy spectrum. So two point velocity-velocity correlation at equal time in Fourier space. And it decays with the very famous minus 5 third slope which is very robust. So it's observed in many Earths or industrial flows and it's something which is typically observed over many, many decades. What is much less known is what happened after here in what is called the dissipative range. And I will talk to you about that. And also note that I've told you that the energy spectrum decays like K to the power minus 4 third here. And here in this picture, it's plotted as a fraction of frequency and it shows the same slope. And I will also come back on that later on. So the first statistical theory for turbulence have been proposed by Kolmogorov in his very famous K-41 theory. So really in a nutshell, assuming that symmetry are restored in the statistical sense that is the flow is locally isotropic and homogeneous away from the boundary and forcing mechanism. And assuming that there is a finite dissipation rate per unit mass in the inviscid limit that is when viscosity goes to zero, then it derived the energy flux constantian relation and it yields an exact result for the third order structure function which is exactly given by minus 4 fifth epsilon L. So from there, if you assume universality and standard scaling variance, then you deduce that velocity scales like L to the power one third and then dimensional analysis leads to scaling prediction that the structure function just scales like L to the power P over three and it also gives you the Kolmogorov spectrum so they decay with the K minus four, five third. But of course this is not the end of the story. In fact, in many experiments and numerical simulations, systematic deviation from Kolmogorov theory are observed and emblematic example is how the structure function where this exponent in fact deviates strongly from the linear prediction Kolmogorov prediction P over three. So this is an example of data collected from numerical simulation and experiments. This is the exponent as a function of the order and you see that it deviates the more, the larger the order. This is Kolmogorov prediction. So it means that simple scaling variance is violated and it's usually referred to as multi-scaling. So with a non-trivial spectrum of exponents and also what is characteristic is that the probability of both very large and very small situation are enhanced and this gives very bursty shape of typical velocity signals and this is typically generally referred to as intermittency. And understanding K 41 theory and even more intermittency form first principle that is from Navier-Stokes equation is really an open challenge. So if you want to go from a microscopic theory to large scale scaling variance, then the ideal tool is RG. And in fact, RG study of Navier-Stokes equation has started back in the 70s and it's a whole field of research in itself. It's very extremely complicated and it has not really allowed to fulfill this program. So some attempts have already been made to use non-perturbative or functional RG to describe that and I'm going to show you that we can do that without truncations in the certain limits that we can achieve an exact closure of this equation based on symmetry. So the microscopic theory is Navier-Stokes equation so you all know it. We consider incompressible fluids so with this condition here and since we are interested in the stationary regime, we consider Navier-Stokes equation in presence of a forcing. Then because of universality is expected with respect to forcing, it is convenient to choose a stochastic forcing as long as the correlator is picked at the relevant integral scale where forcing has to be applied. And then under this form, it's formally a Langevin equation so you can apply the usual Martin-Singer-Rose Johnson-Denome's formalism to turn it into a field theory and you obtain the standard structure so you have the deterministic parts of the equation are encoded in linear terms in the response field so incompressibility constraints, deterministic Navier-Stokes and the characteristic of the noise correlator are encoded in the quadratic part in the response field and we are going to study this action using your vector equation. So more precisely, M is to compute two-point functions so velocity-velocity correlation function and response function. So we have to start from the vector equation for the two-point functions. As you all know, of course this is a very difficult task because what you have to solve is an infinite hierarchy of flow equation. So your usual way to deal with it is to divide some approximation scheme that amount to truncation or approximation of higher order vertices to close this equation. And in fact, if you do that, you already have a reliable result and I will show you in two slides what it gives and then I will show you what else we can do. So first thing, from the symmetries, you can deduce the general form. So first approximation, you can write an and, that's for Gamakar for the effective average action. And from the symmetry, you can write down the general structure of this object. So there are a bunch of terms which are not renormalized plus something which satisfy very constrained symmetries. And on that, a really reasonable approximation is to truncate this object at quadratic order in the field so that you cut out higher order vertices. And in fact, but still, you keep a functional dependence in momentum in the renormalization function. And in fact, this is a kind of approximation that works very well in a related context which is Burgers equation or Karda-Parek-Zezang equation if you, in the language of interface growth. And this is the example of the scaling function associated with the two-point correlation that you get with such in D equal one where there are exact results using such approximations. Okay, so if you do that for Navier-Stokes, so you write down the equation and you integrate them numerically, and what you get is a fixed point. So the already, the nice thing is that within the same unified framework, you can get the fixed point both in D equal two and D equal three. So you can describe both three-dimensional and three-dimensional turbulence in the same framework. So this is the typical fixed point function shape. And from that, if you compute the kinetic energy spectrum, you recover what is expected. That is, first in the energy containing range, the spectrum grows like K to the power D minus one which is expected from quasi-equilibrium equipartition arguments. And then you recover the column over of decay minus five-third. And in 2D, you recover the equivalent theory which is the Christian and Bachelor theory which predicts the steeper decay with K to the minus three because part of the energy flow in the inverse cascade. So this is already nice. But in fact, we can do much better. So the two ingredients to do that are first the symmetry of the Navier-Stokes field theory. So the Navier-Stokes equation are symmetric under Galilean transformation. So you can shift all the velocity by a constant. But in fact, you can, sorry, you can do a general, it is also symmetric under a general form of this Galilean transformation which is a time-dependent form. So if you consider a time-dependent velocities. And the advantage is that it yields water and entities that are local in time so that are more powerful. But in fact, this was well known in the context of RG for Navier-Stokes since the 90s. What was not known is that there is also another symmetry which is also time-dependent which amounts to a shift in the response field sectors and which also yields many water and entities that will be the key to do our program here. So from these two symmetries, you have a functional water and entities that is local in time and you should take derivative of them with respect to field. You can generate an infinite set of exact water and entities for all vertices as long as one momentum is zero. So this is an example of the water and entities you get for Galilean from Galilean invariance. So you see that three point function with one momentum to zero is related exactly to two point function and four point with two zero is also related to two point function through these identities. And the second ingredient is a BMW-like approach. So if you look at this equation, a key property is the presence of the derivative of the regulator. So this is something which effectively suppress all internal momentum Q, which is larger than the RG scale K. So now if you consider the regime of large wave vector, so P much larger than the RG scale K, then or equivalently if you consider K to zero, then since the internal momentum is cut out to value less or further K, it means that the internal momentum is negligible compared to P and in the limit of large P, this means that you can set Q equals zero in all vertices. So this is what is done here, you put zero in all vertices and then you can use the water and entities to re-express this in term of two point function and then you close the equation exactly with water and entities in this regime. And this is what we get for the two point gamma function. In fact, if you express this now for the connected correlation function that you want to compute, so correlation function and response, and if you also consider the sector of large frequency, so large with respect to KZ, then the equation takes a very simple form, which is written here. You see that it's linear in C and all the non-linear part is hidden in this quantity which does not depend any more on the external frequency and momentum. It just depends on the scale K and you have a similar equation for the response function. Then what you can do is, well, focus on the fixed point, five minutes. Oh, focus on the fixed point. So at the fixed point, this is just a number and in fact, depending on its size, it defines two solutions which are relevant to describe the inertial range and the dissipative range which I will show you now. So first, the analytical solution in the inertial range, you can derive it and it has this really simple form here. So first of all, you can compute the kinetic energy spectrum, so which is integral over omega and it displays the expected K to the minus 5 third decay. So of course, the equation is linear. So the complex solution is an infinite, linear combination of such solutions and if you consider a certain distribution of parameters and bounded distribution, then you can generate correction to this leading behavior. That is intermittent C-effects. But this, you cannot compute them just looking at the fixed point. You need to integrate all the flow to compute what is the precise distribution of integration constants if you want. So this is what we progress but anyway for the two-point correlation function, intermittent C-effects are expected to be very small. So I will neglect them here and I will show you that already the leading behavior yield very interesting prediction. So the first thing that you can look at is that the kinetic energy spectrum, not in wave vector but in frequency that is integrating over K. And what you find is that in frequency it has precisely the same exponent. And this is not trivial at all because if you use standard scaling theory which predict a decay like omega to the minus two because for Navier-Stokes, the dynamic exponent is Z equal to two-third which means that the relevant scaling variable should be omega over K to the Z that is omega over K to the power two-third and if you do that it would yield Z is decay which is in fact observed for Lagrangian velocity but not for Lagrangian ones where the proper decay is this one. And this is really non-trivial it means that standard scaling theory is violated and physically we know why in fact this is attributed to what is called the sweeping effect that is the random advection or small velocity by large structure, by large eddies and for those who know it's related to what is called random Taylor hypothesis and this is really something non-trivial which is captured by a solution. But of course we have more than that because we have the full time dependence and if you look, if you inverse for your transform and if you look at the function of time and wave vector you get a really simple form which is a Gaussian in the variable KT. Okay, so is it meaningful? So what we did is we compared it with numerical data. So we use numerical data from simulations that we run using a standard integration code from Navier-Stokes equation developed by Guillaume Balarac in his team. And also we use the huge database which is called John Hopkins turbulent database where turbulence data are made available and you can fetch some data and compute things. So what you obtain is so we have different Reynolds lambda here and you saw the typical spectrum. So first energy contingent range, Kolmogorov decay and then dissipative range. So let us look now at the time dependence. So I remind you the analytical prediction is an exponential minus K2T2. So if you plot that in log scale as a function of KT2 square you should observe straight line with the same coefficient. So straight parallel line and this is exactly what we find in the numerical data. So more precisely if you represent this alpha the value of alpha that you get for each K it's a perfect constant as soon as you have a high enough Reynolds number. So it's a perfect constant. And another way to see that is you see that if you divide this quantity by C of zero and K to remove this parallel you should have just a Gaussian. So you should observe a collapse of all the data and this is precisely what is found for all our data for there are 200 values of K here in this curve. So this is already very nice. So we have a prediction in the initial range that is confirmed by numerical simulation but now there is another solution that describes the dissipative range. So again you can write it's analytical form it's given by this. And if you compute the energy spectrum it gives a very interesting behavior which is a crossover from the number of decay to a stretch-trait exponential. So in fact there have been many empirical proposition for the behavior in the dissipative range more or less under the form of stretch exponential with values exponents but essentially the common wisdom is that it approximately decays exponentially in this regime. And here we have a precise prediction with this exponent two-third. So if you look at the shape of this so you find that and qualitatively compares quite well with the experimental data I have shown you earlier. But now of course we can be more quantitative and we've analyzed our data in this initial range. So if you plot, so you multiply by the column of decay so that in the initial range just appears as a plateau and you plot it as a function of this K to the two-third and you should obtain straight lines and this is precisely measured in the numerical simulation for all the Reynolds number. So this is a clear indication that this behavior is a stretch exponential with the power two-third. So this is already very nice because this is also confirmed in the initial range but what is even nicer is that now we have confirmation in experiments. So I won't show you the plot right now but it's in preparation. In fact, the experiment is from von Karman flow. So you take big cylinders with counter-retating plates and in that you take cylinder of many different sizes. You put many different fluids from helium to water to glycerin and you use also different forcing mechanism and you collect all this data and it beautifully shows the stretch exponential that we predict in the dissipative regime. So this is very nice. So this is my conclusion. So what is not so frequent in the NPRG context and even less in the context of Navier-Stokes we have an analytical solution for the two-point correlation function. So full frequency momentum correlation function that is confirmed by numerical simulation and experiments at least for the equal time part. So now there is a nice work to do. So the first one is to compute the full numerical solution of the flow equation. So this has two purpose. The first is to compute the inter-mensancy effect. So I've told you about. And the other is to really pinpoint the interplay between this, so this change of sign of the constant IK that we can get from the numerical solution. Another direction is to do the same work for b-dimensional turbulence. So we already have the exact equation. We already have the analytical solution in the direct cascade. Now we need to work on the inverse cascade, which is less easy. And of course now the beautiful things to do would be to do the same program for higher order structure function. So we are quite confident that we can also close the equations for higher order structure function exactly in the same regime using wild identities. And this would lead to the first determination of the inter-mensancy exponents for higher structure function which has never been done. Thank you for your attention.