 Okay, so welcome everyone to this afternoon's session. It's a pleasure for me to introduce Professor Irene Gamba from UT Austin. So Irene will talk about global LP solutions of the Boltzmann equation with an angle potential concentrated collision kernel and the convergence to a land-out solution. Thank you, Irene. Organization. As I've been invited to contribute to the scientific committee, so I can see it's an extraordinary event for young people and I hope many of the things are keep on happening as much as here. I think the United States, and I said that because in the United States we do not have as much activity as kinetic theory as they have in Europe in kinetic theory and for many years, and maybe still today, I have to travel to this continent to be understood more or less what I'm interested in talking about while finally starting to pick up there. So it's a great pleasure to see more young people getting engaged into these issues worldwide. It's an interesting problem. You have heard a lot about the Boltzmann equation so far and the Landau equation. So my introduction is going to be very simple, very, you know, I'm not going to, there are some slides that repeat what you have heard over and over and over again. And I will try to stress on the new aspects or in the different things we want to address. It's clearly related. What I'm going to describe today is to the problem that somehow around the billet has been discussing since the beginning of the lectures and by now he's fully embedded in the aspects of the Landau equation. But he said suddenly yesterday that the way you get to the Landau equation is from Boltzmann. That's the way Landau, you know, Landau is a Russian physicist, perhaps a Soviet, we would say at the time, or, you know, at the time he was a Soviet physicist, was obviously a very strong inclination in theoretical physics or mathematical physics. And he wanted to try to address the problem of how to understand collisional plasmas as was well articulated by Loran. And, you know, the natural way to thought about this problem is to use the Boltzmann formulation. I mean, as a historical remark, I'm not sure if Loran has addressed it that way. I didn't see the first lecture. You know, when Boltzmann did his work in 1880, 1880 plus between the 1880 and 1890s, there was no concept of statistical mechanics. People didn't understand that. The world was describing continuum mechanics and there were probability. And probability was very good to do game theory. That's exactly how probability was born and raised in France. As, you know, how do you get an edge when you gamble? But there was no concept of putting that into physics. And it's Boltzmann, the first one that articulated after following some work of, I mean, I don't remember all the names, the particular one I want to say. I forgot it as well. So, it was quite interesting that while he proposed this model and the Billiard model of Boltzmann, which is cumbersome, which we still keep on studying and we still try to squeeze and dissect and understand every piece of it, it's in a sense a toy model in physics, in my view. I mean, it's in a sense, you know, the Billiard model is too idealistic to have, to be able to describe all physical phenomena. So you need to really move along, but it's very hard to move along if you don't even understand how to do the basic problem, whether how we do mixtures or mixtures in plasmas, electronials in plasmas, how would you do it if you don't even understand the basic formulation? So, in a sense, it's in the very much spirit to see how can you really complete the program that was initiated perhaps by Landau without saying it very specifically. The paper of Landau is very scant, it's written in a very physical way. He quotes only one individual saying, you know, this reference is wrong and doesn't quote anyone else. It doesn't quote Rutherford, who is a British physicist, who actually made the first measurements and quantification of how Coulomb interactions, by measuring the scattering cross sections were particles interact until under active potentials. He doesn't quote that, but yet it's encoded and it's used in that derivation. He could not have done it without it. So it's very interesting. He says, well, how can you come up with these connections in a way that is, you can say, something more rigorous from the mathematics point of view? Historically, I'll browse by everybody, but let me start by the slides that perhaps are the things that you have seen and heard, because it would be a common framework to work out and see what can we go from there. So, from the Wolfsman classical point of view and billiard model, it's exactly what you see at the billiard table, with the exception that there is no friction with the table, neither there will be a loss of energy after the balls interact. And it's very important that this concept of many atomic gases is identical particles that are indistinguishable, and so you can do this molecular chaos, independently distributed and indistinguishable. If you don't have that answer, you really cannot even start with the reduction of the model. And so, in the setting, this is very interesting, because the only thing you do in this model from a statistical point of view, a statistical mechanics point of view, is just to say, what is the collision law? The rest is more or less going to set up by saying the interactions are binary, because the probability of seeing three bad interactions are so low that you disregard it. And it's a reasonable answer if you have, in between the interactions, a lot of free flow, or as it's well phrased by Wolfsman, and all people have written about Wolfsman, and I followed a lot the writings of Giorginiani in this area many years ago. You have to have a lot of intercity aspects. So if you have a box, there is a scaling of how big the particles can be in the size of the box in order to get that this billiard is in an eternal motion that could be going back and forward because it's a complete irreversible motion, and yet you are successful to make sense of this particular model. So having said that, the interaction is going to be elastic at the billiard level, exactly what you have heard in previous lectures. And this is what usually would be called, so when the particles touch, this is what we call a heart sphere, is when the intramolecular potentials become infinite. But there is the other range, which is when the particles are going to be subject to long-range interactions, which may be kind of forces that far away are attracted, but they get very close like this dipoles and then tend to repel to each other. And these are the case of what would correspond to Coulomb interactions. And so in particular, that constitutive relation is not going to be changing from the fact that the particles are inelastic. It's going to be changing in the way you model the scattering cross-section. The scattering cross-section, as it was introduced in many of the lectures, and I'm thinking more on what you've been systematically, Lorand de Villet has been describing, is you can call it the collision cross-section, but instead it's a transition probability rate of change, because what the Boltzmann equation has been modeling for us is actually a rate of change in time of a probability distribution function, which is predicting what is the probability of finding a particle at a position X with velocity V and a time T. So in a sense, you can think that whatever it comes on the right-hand side, aside of those integrals and so on, they have to be reflecting this rate of change in time. It's like a balance equation in conservation laws. So in fact, this is something that you have, as I said, seen several times. The way, let me just adhere a couple of comments because I don't want you to get lost on notation very quickly. What we call here Q of F is an operator, which denoted, I mean, sin in binary form. We are going to be assuming that this is actually sort of, you know, I have to say, Loran, I don't like to call it a quadratic, a nonlinear operator. I think it's the equation itself. I mean, the problem, the nonlinearity gets into game with X into an action, but when you look just at velocity space, it's really a mixing operator. It's a very special bilinear form that is going to allow a lot of the magic to happen. Otherwise, you know, you have a quadratic term, it's very hard to get good estimates out of it, okay? So in the bilinear setting, which we are going to understand, I mean, bilinear, symmetric form, and so from now on is what I'm going to be using. I'm going to take this object is symmetrized in this form, nothing spectacular, but perhaps what is more important that the object itself, right, written in bilinear form, is going to be an operator that will have some symmetry, and I will come back to that, and that's what we call the game, which is something very, very important for the stability of the equation, and then it becomes a local form in F, multiply by, sorry, let me call it a term, that is no local in B, and all of these have names, come from physics, so what this is going to be is the integral of G, being proportional to the transition cross-section that the physics is bringing to you, okay? That's something that it needs to be given to us, in some way, so it's going to be data, the constitutive form of what happens here, which is usually modeled as a function of the relative position or the relative speed associated to the particles, and the reason is that potentials between particles depend on that quantity as much as on the angle of interaction, okay? And so if I do that, then I don't have the picture here, but you'll see it very soon, so that means I have my beam, my beam star, and then you can actually do these drawings in a very systematic way, given two pairs of velocity, right? So you draw the difference between the two of them, that's the relative velocity, the speed is the magnitude of that vector, it could be very small or very large, and then you draw a circle of that diameter, sorry for not being so accurate geometrically for that circle, and then what is going to be random as much as you fix V, you write on V, I mean think of yourself doing a Lagrangian, writing in a Lagrangian flow, what does it mean? It's just sit on the particle V and move along and see what happens, and someone comes and bang with you, and that banging is going to come in any direction, any possible direction, which is V star, right? And it could be on, sorry, any velocity V star, and it could be hitting me on any point of my sphere of influence would I be a sphere, okay? And that means that it would be two quantities that are going to be rather random, average picked up, you know, aleatory, and that direction, you fix it, and that direction is going to preset the direction of the post-collitional relative velocity, and then you do this picture here, and you immediately pick up the collision law, and that is going to be elastic, because here you get V prime and V prime star. If you assume that this angle, you constrain it to be between zero and pi over two, otherwise, you know, you're going to be assuming symmetry with respect to these rotations, okay? With respect to the upper and lower semis here. So essentially what this subject is going to be doing, if you see this interesting, here you get the V star, then here is V minus V star, and then here is, this is a renormalization, U minus, sorry, V minus V star over the relative velocity, C dot sigma, but this sigma, as I said, right, is in the same direction, is the unit direction with respect to, so I'm rewriting it exactly as Loran wrote it yesterday, this is in the same direction of V minus V prime star, because I designed it to be there, once you have all this law, you can play with it and replace them and use them as you need them. There are a lot of combinations of which you can handle this, you know, equations or manipulation of this precise structure that is rising here, and then the absolute value of V prime minus V prime star actually is the same as the one without the prime, because it's in the same diameter as here, and that's exactly what actually makes it to be an elastic collision. Inelasticity would break that, okay? Although there is a lot of interesting stuff that you can do when you're doing elastic interactions, mixture of particles that have different masses would not have that property, and then you are forced to use, to model them, you know, systems of Boltzmann equation and they would give you something that is, is going to be different, you know, something that is going to be more like an elastic type of configuration. It's interesting that, that when the particles tend to go to zero, right? Even if you do, and I'm not going to get into the theory of inelastic collisions, when the, sorry, when the angle gets to zero, what is what we call the gracing collision angle, or when the particles are assumed to be under strong potentials, right? Because they are coming from far away, it is going to be, it remains an elastic interaction, what it really changes is that when they are getting close, instead of touching each other like the billiard table, right, they're going to start repelling, but that repellent, the replacement have to come with some little time that we somehow disregard in the model that is going to do a deflection of the particle. And then at the time of the closest range in between two of them, which they're not going to touch, they are basically moving in parallel motion, and then they depart again, okay? And that's why you saw the picture in the previous slide. So going back to this object, this is called the collision frequency. This is called the collision kernel, or transition probability. I like that word a lot. Because, you know, if you do this type of model, in the probability setting for multi-agent dynamics for, you know, describing, you know, wealth distribution, whatever, they're not such a thing as collisions, but they are transition probabilities. And what things, you know, how much you quantify what happened before and after the interaction, that's exactly the meaning. And then you decide what does the dependence on. And this is called the game. So what is interesting is that when you write the equation, this equation, I'm not sure if this has been said or not, but I think it's very important. Maybe it was not said in the same way. It's that, in my view, this is a typical model, or is associated to a master's equation that corresponds to what people like to go into probability, Chapman-Komogorov flows. Because what makes it to be very special is that they have, it's exactly the difference between two positive terms. The game term is how much probability you are going to be taking in the direction of V, but due to interactions, you will see it, maybe you saw it before, or so it's the, you know, is to look at the grain of probability rate of my, I mean, all average impossible directions in V and in the sphere of particles that would take the direction V because of pre-particles that have interacted and go into V, while the loss is the particles that abandon the direction V. And so you can view this as a city like a birth, death rate equation that appears all over in modeling of probability. And what, it has the ability that by doing that trick is the only hope you have that if you are going to have some flow of multilinear nature, it can stabilize into something that is probability distribution function via sound, good sound state in the domain of definition of the equation, of which you can solve it in the name of space of solution. So the equation, as I said, have all of these properties. I will not repeat them, but this is exactly what Babylet has been describing in the last four lectures, okay? And essentially, conservation laws are encoded here because you test by one V and V square and then you give you zero. I should say that there is an extraordinary weak formulation. Did you introduce the weak formulation? Yeah, okay, maybe. And on which this become a trivial statement, okay? And that's very important. This is not so trivial. This is not so trivial. This inequality actually is due to the fact that the log is a monotonic function. And it's very interesting that perhaps once you write the equation, molecular chaos has been already settled. And if you read carefully, how do you do the factorization of endpoint probability distribution functions into a single product or products of single point distribution functions is that it needs to be done before the interaction occurs. It's like particles are recorrelated before they interact. And once they interact, they need time to forget about what happened before and so they can recorrelate again and interact with another person. That's why it's very important that you have a lot of intercity space. And intercity space is a scaling issue and that is a little bit complicated to go through details. But, you know, okay. So, in fact, this inequality actually encodes and clearly sees the fact that the sign here is not arbitrary, that what you get here is a local function in V, while here this is not local in V. This is going to be a double integral, all right? Would I have changed the sign that reverses this inequality and then physicists have different ways to interpret plus or minus entropy, but they are equivalent because what we like to call the entropy, they call it the minus and so on. But it has to be the sign that produces the stabilization. And that, I guess, Loranda explained very well. So, what I'm going to be discussing is a little bit the... Sorry, I'm not going to be discussing L1K bounds, but the description of the Boltzmann and Landau equation, the Grayson collision limit, the different class of Boltzmann equation. And in fact, more than... I mean, the Boltzmann equation is going to remain the same. What is going to be different is this, right? Because it's quite interesting. And for me, it was a big surprise. And I will show you some calculations that you can... There is a sum of transition probabilities that are going to take you to the Landau equation. And within a range. And I will explain that in a minute. Okay? And then once you realize of that, and I said, well, you know, we are all debating how to approximate the Landau equation, but it looks like you can do it from many, many, you know, ways of assuming how the particles are going to pass from one state to another. And so that is actually something that was very telling for the work we are going to be focusing, which is, I try very hard to do it with the existing cross-sections. Can I actually show rigorously that the solution of Boltzmann, and this is in the Space and Millennium setting, converges to a solution of Landau? That question was addressed yesterday. It was proven by Vilani and Alexander in the sense of renormalized solutions. But renormalized solutions is an analytical tool. I don't think that has been explained in this course, right? No. It was an analytical tool proposed in the late 80s, 70s, 90s, which actually allowed to construct the form of solution, which was good enough to satisfy conservation of mass and momentum. But it's not a classical weak solution. And part of that is that the conservation of energy as much as conservation of entropy and entropy decay as much as entropy bounds remain open for that type of solution. So, you know, the natural thought is, if you do, if you do, and that is a true statement for the space in homogeneous setting when things depend on X, as much as when the solution only depends on velocity space, which basically says, well, all particles are the same. I should say it by now, when you write the Boltzmann equation, particles are not any longer spheres. They are point masses. And the radius of the sphere has been scaled, of the particle itself, has been scaled with respect to the number of particles in order to obtain a constant that appears, you know, in front of the subject. Okay? So the space homogeneous setting would be something like that, perhaps with a constant, and that constant of order of one has a ratio that depends that given the size of a box, how many particles of certain size can you have in a box in order to get the correlation? Okay? And that is the, usually is, is, was calculated as what is assumed in the Boltzmann grad limit. When you do this calculation, the number that results here is the, called the Nuchsen number, or associated the main free path, is how much you can fly in between one another interaction average. So, so, existent theory, I will revisit it, and I will propose, I bring to your attention, not propose, I'll bring to your attention a new way to get existence of solutions for the Boltzmann equation. And, you know, we have the existence of Archery and then, which actually get raised to a lot of words that came up after it. And it's, in a sense, if you read, I mean, if I, I have to say when I read Archery for the first time was extraordinary. I was like, how can anyone do that? Very hard object to read. And, but it had something that was interesting. He's trying to build up a Picard iteration. And for that he has to work a lot in how to make this Picard iteration and he was trying to use the concept, I mean, he needed the concept of entropy. By now we realize you can actually remove that condition that you need bounded entropy. And, but what is very interesting, it was pointed out in 2006 by Bressin in a, actually, in a summer school here in CISA. And he presented the, how to solve the Boltzmann equation for hard spheres. And he writes a statement, I mean, he gathers all the estimate, writes a statement on the theorem and the, the statement that he writes, you'll see it later on. He checks some of the points, some of the points is very straightforward. There is one point that was very subtle. He says, this is a trivial statement. It is not trivial. It needs to be an extra work. And, and so because of that, I communicated with Bressin. He came up to my attention that everything was encoded in martin's theorems for ODE's. And he said, you know, and he says in his notes, anyone probably sees I should have them. I mean, they are posted on the website and I have them and I called them and you can find them. They are basically the Pozman equation. You need to solve them in a setting when they send ODE in a Banach space. This is not the way you do classical PDE's in divergence form where things are in Hilbert space and you have an extraordinary theory for that. Banach spaces, as you, we all know, are much subtle to work with. And that's exactly the, the point that I'm going to do. So in this particular setting, you'll see it's very simple because we get the existence now too. And then we need, I want to describe the existence in LP. The existence in L1K, we still have some issues because of the natural of the cross section and the, and the, you know, and the particular cross section I'm going to be choosing for the Landau equation. But if you take the problem classical for hard potentials, when you split, right, when you split this object into a function of u times the function of u hat dot sigma, and for instance, you assume, by now you can assume integrability or not, but that is, is going to, to, to thing. I mean, this could be integrable or not. So let me put it plus this or this. If it is incident, you need more conditions. I will not touch that today. Okay. So let me just say what, what it means in the classical setting. And on the file of u, you can actually put here, imagine, put in a constant times u to the gamma, and here a constant u, a capital C u to the gamma, gamma between 0 and 1. What you get is that the L1K estimates are gained or are created, which means you can start with, with set L1, 2, and the solution, you know, the Boltzmann operator actually will bang you, you know, solutions that are going to be in L1K for any K, which is quite a stroke deny event and we know much more on that and, and if B, and if gamma is equals to 0, and remember that, I will somehow come back to this, then L1K estimates only propagate. So this is a, you know, it's an interesting result. I like to always pause at this point and, and there were a series of work very nicely developed by Lorande Villet and Bember in the, in the early 90s, when they made an observation for the, yeah, you were a baby. You were the age of the student here. Where the observation is that, you know, as we all get estimates in time, you always have to find some inequality in the norm, right, as a function of time for which you can get bounce. And what they observe at different times, Lorande, I think you did first the propagation for all Ks and then, and then Bember realized that just tweaking a little bit the calculation that, that Lorande did, you can actually go into, into generating them. It's like, like everybody likes to go about the Georgie theory. Maybe some of you have heard about it, that you start with very little and you get a lot out of it. Okay, so the Boltzmann equation has that property instead of saying, you know, from L2 you pass to, to bounded, here from, from L2 you pass even to said, not only that you have L1K but you add all the moments and that is also going to be true. So you have, you know, exponential tails being said as much as you would have point-wise bounce in an infinite. But you can do that if this is a split it, okay, and if it is positive. When you are on zero things are going to be propagated and in fact this is going to be crucial for the fact of the L1K bounce but I'm not going to talk about it because we are still tweaking it. I tell you what is the difficulty when we get there. It's going to be very interesting. I know. Okay, so again you've seen this picture. This is my drawing. This is what we call, so here the, the, the V and V primes are below but you have all this diagram and it's very important that if you're ever going to work with Boltzmann, right, if you don't learn it in the first two, three weeks my advice is just work on that that comes from the scheme because it's really needed. It's really needed. You need to always refer to how the interchange of the interactions are. So, so, so in particular the collision cross, the, the, the interaction law or the collision law can be written in the classical way which is, which is the center of mass as much as the relative velocity. The fact that this, this interaction here says that the two radius cross in the middle and that vector that appears in the middle is center of mass and it's the same for the pair V V star as much as for the pair V, V prime star. You read it from the picture and the other statement says that in fact the four, the four magnitudes square of the velocity vectors, right, V, V star, V prime, V prime star is a lot of a stiffness from the geometrical point of view. Okay. So it's a, this is not an arbitrary kind of interaction. So, so the other interesting thing and this becomes crucial to understand what we'll be doing is that this interaction law can be actually written in a different way which is another thing you really want to get under your skin because all the estimates actually operate at this level and it's to say that the post-collitional velocity okay, it's going to be equals to the pre plus an object, right, which is proportional to the difference in between, you know, this, so the vector, this vector in these two directions multiplied by the magnitudes view and here you have it with plus and here you have it with with minus, all right and essentially what this is telling you that you can actually rewrite this interaction by modifying a little bit the two vectors that are going to be orthogonal and I'll go back to that. So, so, so now I'm writing what all I said so I'm going to move forward. This you have heard a lot about it. Soft potential is this range not very soft potentials. These sometimes are called not so soft potentials. The first time I heard that word was from Laurent. I have never heard it before. Right, right, right. You have been around my kinetic life since more than 20 years ago. And but then it comes another one which are the very soft potentials and the very soft potentials when you do, you know, all of these quantities and numbers really depend on the dimension. So, so if we're in three dimension, a very soft potential means the dimension, so the potential has to be in between minus, the dimension minus one and minus the dimension. And when you hit the dimension at that point, you are lost. Why? Because you go back to this point, if the cross section is split, you put here u to the minus three, right? Absolute value of u to the minus three. You are calculating then here, right? A convolution because suppose that the cross section even is constant or integrable. So the cross section here you would have that a convolution of g with u to the minus, with v minus v star, the g of v star, v minus v star, u to the minus three and that is not integrable. And that is what motivated, I mean, lambda to, well or at least I don't know, people probably figured it out earlier, but they didn't know what to do. So I said, there is only one way out and it's to do, to use the collision cross section and in fact, I can actually do it on this drawing now. So when you see it, you know what is coming, right? Basically, what you can say is, you know, I have a bad singularity with respect to this term. Okay? So what can I do? I mean, after all, I have my collision diagram. I have my birth and death rate. There is nothing else. There is a binary operator. And so what I do know from here is that if I look at the distance from any point, from say B, B prime to B, to B, as much as B prime to B star prime, these are going to be identical, this is a little rectangle that you saw before. This is going to be exactly the, this is basic trigonometry, high school, this sign of theta over two. And so there is one thing that can save you. And it's the fact that if this is going to be very large, right? Well, you can make this very small, right? I mean, actually, sorry, if this is going to be going to zero, because it's, I mean, it's u to the minus three is going to be very, very large. Then this is going to be to zero or this is going to be to zero. And then you somehow need to use that these two things need to be very close. Okay? All right. So all of the things are coming up. And so, and so the, the, so the way to do the calculations is exactly what we started to study and this everybody has to study this, which is essentially to look at the sphere and start to focus how I'm going to be absorbing something that is extremely singular here. All right? Well, the first observation is, okay, let's see what you can do by using this object here. Well, if you, let me right now what is this object here, right? This is going to be f of b prime g on b prime star and when you want to look at the, and I like to put them in the back because they are prey and I remember they are prey. Okay? They will be paused in the week formulation and that is going to be also proportional because you do answers of symmetry for getting here always the same fall. All right? And so the only hope that you have is that you're going to be comparing these two terms. All right? And by looking at how the solution may be differing from b minus b prime as much as from b star minus b star prime. And so you are forced to integrate the flow into a whole single configuration. Okay? So one observation that I should set, you know, sigma is a vector on the sphere, which means you need to decompose this into two directions. So I draw it on the flat, you know, on the plane but this is an object that is actually evolving on the sphere. And so what happens is this angle here is theta, which is the theta here and so that basically says that sigma is going from now on. When you start to write all the sins you basically said well sigma is going to be a vector, a unitary vector that has to have the direction of u with respect to the cosine of theta. Right? Because I'm going to be defining that which I haven't said it yet. u hat dot sigma you define it to be the cosine of theta and that starts to frame, you know, how you put all of the things together. And but then you have another direction which is my azimuthal direction with respect to the polar. So this is, you may think that this is the polar angle with respect to u and this is then the azimuthal direction with respect to u and that means that u is orthogonal to omega. And so, you know, this is a local configuration because you have the interaction, you look what happens with you at that point with the letter u, right? And then you have to think of that frame and whatever you do, you're going to be doing a calculation and an integration and I should say that when I finish this calculation now I can, you know, somehow dissect it, I'm going to be integrating in r3 times s2 and here is a minus and then now I can really talk about what here there was an integration I forgot my f of v and then here I have dv star d sigma for this big hole form. Let me put a bracket and a bracket, okay? So in fact, what is going to be extremely important to derive the Landau equation is not what happens with the polar angle, it's what happens with the azimuthal angle because if you remember what Loran has been showing you the Landau equation doesn't have any angle anymore. It only remembers u in a very specific form. In fact, it is the u in the projection with respect to the azimuthal direction. That's exactly what has to be coming out from this calculation. Whatever you do it is really the fact that if you properly do an expansion here with a single purpose to absorb this, you know, columbic interaction the single purpose to absorb something, to be able to say something about or reduce the singularity when this object is u to the minus 3, okay? The only way to say that is because this object depends on the cosine of theta is to perform very careful azimuthal integrations. Maybe I'm saying something that is silly or, I don't know that you may have her or maybe this is knowing from day one but it's something that one has to really keep in mind very much. Okay, so again keep this picture in mind. So the Coulomb potential was actually not even addressed by Boltzmann. Yet in 1911 it's a British physicist that did a lot of calculations and in particular he measured the cross section or interaction rates for Coulomb potentials. His name was Rutherford and he said something, okay? I'm trying to put it in pieces. 1911, so Boltzmann is 1888, right? I mean about that time it's where he does the equation maybe earlier, 1828. That's a lot of work. And he died in 1906 actually very close to here, as you may know. He died in Duino. So Rutherford says, you know the cross section the angular part of the cross section. Measure as a function of the scattering angle in terms of the polar angle with respect to U should be a quantity that is of this form to the fourth and then, you know, if you think that this is a differential it's a rate of change with respect to the variation of the angle, this is something that in three dimensions the transformation the Jacobian transformation would give you an extra this area. It doesn't say much. I mean, even when you look at the pictures, yes. This was from experiments and measurements. And actually, and that's my understanding. One, okay. I tried to read Rutherford. I'm not a physicist and it's hard to tell. And it's very hard to distinguish at the time physics from mathematics. It's the same bifurcation that you have these days. Yes. The interaction is not based on... Yeah. Well, okay. That perhaps would take me to think about how to answer at first and then probably much more time what I want to spend. But they do this with measurements and what they can see is to look at the deflection angle that they observe at the time of interaction. Okay. I'm going to put a caveat here. Okay. All right. So I need to speed up. I'm actually going very slow. But what's interesting is that the measurements and he fits something like that and it goes into the... into the... I mean, or puts up a law that maybe nowadays we translate it into something like that. I'm actually going very slow. But what is... It's a very singular decay. But what is clear that what Rutherford said and in any experiment even today there is a frame or an horizon on which have no information in longer. You cannot resolve zero angle. Okay. So you have to negotiate with some fort of cutoff so... So in fact, what then Landau said the equivalent is somehow... I mean, it's not quite Landau and I need to speed up. But was after perhaps the well articulated work on the gong and looking that if you take that the cross-section is going to be actually now I'm taking the whole B. All right. And you assume a parameter. And that parameter is going to aid a little bit or quite a lot what you have. So let me put here the cosine theta or I can write it as a function of the sine of theta over 2 so you can see after the transformation that takes you from cosine theta to sine of theta over 2 it has to be an object and they write exactly this and here you get 1 over sine that way this theta and then in this and there's one other thing that needs to be said you need to actually cut off this is sine of theta you need here to cut off the angle so you can say I know nothing or I assume that there will be no contribution for the transition probability if I actually get into angles which are smaller than 2 because I have no information but it has to be compensated and this is what Landau says and I think it's an extraordinary observation. He said this alone would not make it but there is something else and he divides and he divides by an object which is the logarithm of the sine of epsilon over 2 which is not quite like that but you can read you can put the pistol together so I'm going to explain you why that happened and why it's successful and why we make it with a different one in the limit what they all compute is the Landau equation as Loran explained it alright so different collision cross sections well the classical is exactly what I wrote for you here and in particular I'll show you very fast the slides now I mean you can keep them later but so that is the Rutherford right but it turns out that physicists when they want to compute Landau and they want to use their Monte Carlo codes they said that computing Landau with Rutherford potential it takes forever whatever that meant right I mean you know with people with numerical analysis sometimes they talk in a jargon well that forever is this object because this is going to be the rate of decay and that's exactly what Leukean and De Gaulle showed heuristically that the rate of decay of solutions from actually the decay of the collision operator to the Landau operator is this one over the logarithm of something that goes to zero and it's very very slow and then physicists said well actually you can do better than that if you increase the singularity right and if you increase the singularity you converge faster provided that you actually change here this rate so what this rate here is nothing else it has to be now a function of delta that is going to depend on the sign of epsilon over 2 that has to go to zero right as epsilon goes to zero but what is very interesting is that you increase the delta here and then what I personally discovered maybe you know and then it was very easy to cook up much more functions what you have here is nothing else that is primitive of an exact integration of the collision cross section in the sign of theta over 2 so you really have to work a little bit like physicists and do the calculation and then once you have this primitive what you are dividing is by the value of the primitive evaluated at the sign of the cutoff value so for instance if you put here delta and delta actually can be zero but cannot be two this is rather four and this is no way to control the tail and so that was actually very interesting for the first time in my life I realized wow you can get there as many cross sections as you want at least a single family one parameter family on this delta parameter and then it made me think that then I'm actually looking at the problem maybe like people are trying to prove that Navier Stokes converges to Euler right I mean there are many ways that you can get even from the burgers equation that you can regularize the viscous burgers equation and you always recover the inviscid burgers equation are we in the same sort of a scenario who knows you know later or earlier I mean this is something we did a few years ago but certainly in between after looking or approximately about the same time and then it was work out but he they actually indeed realize perhaps you Laurent knew this try a completely different cross section put a different answer scale it properly and you pick up and out the formal calculation and you pick up and out and in fact we actually did I mean and let me show you what is the difference in between the David lead versus the rotor for potential or cross section the rather force says look if you have no information what happens for small angles I assume that there are no particles colliding with that that is you if I assume and David lead says if I assume what happens below a certain angle I'm going to say that they collide more and more and more and more and you put infinite diffusion they are bang and you put concentrate a lot of interactions okay and I should say that there was a nice paper of he in 14 where he actually shows that you can pass to the limit and he construct Schwartz solutions in the Schwartz class that converts to the Landau equation I'm not sure if the convergence is a fully rigorous calculation it's a lot of estimates I can never get the guts to get into that but I I expect so but it's a local in time and says can you do any different so let me say this is the history and so on this is what I want to show you that why we actually computed all of this objects and it was by using the Fourier transform and this is where I discovered that you have a very beautiful problem that perhaps would remain open now someday maybe close we actually did a numerical calculation on how to come to approximate landau and we use it by a Fourier method and if you do it by a Fourier method okay I know there have no internet and I'm not sure why the machine keeps on saying the same okay so what happened is the following so you do this approximation and understanding how you're going to manipulate the integration on the sphere but if you do in Fourier space something quite rather spectacular happened the Landau equation the Fourier transform the Fourier transform of the Boltzmann equation is a convolution as Fourier space but it's also the Landau equation and the difference of both of them in weight form is exactly the same object actually acting on the weight of the Landau equation versus the weight of the Boltzmann equation whatever comes from the cross section in this case you have an angular contribution in this one you don't okay and so basically we were able to come with this equation doing all this class of possible cross sections we were able to set the following I mean this is really a chip theorem okay it says if I assume that my solutions of Boltzmann are strong enough to say that the Fourier transform of f times is shift its own shift all right it's bounded by a quantity that essentially comes up to say that the solution of Boltzmann would have at least you know two derivatives or three derivatives and decay with cubic tens then you can actually get a decay rate to equilibrium from Boltzmann to Landau of the operators not from the solution okay just the operators we try like hell to work into that and I put so not to work into that and we go nowhere five minutes two minutes okay so in any case let me just crop up because of the fine work so can I have three more minutes so basically what we got here is that so for instance if you compute the Landau and you put the rather for potential this is what you get but if you put the Quintic potential here so you take delta equals to one then you get something much closer to Landau so the first question is what is Landau the physics is rudder for the physics flow maybe rudder for has the physics and so should be that but the difference between one and the other and as you browse in changing cross sections is that you change the rate of decay of Boltzmann to Landau both of them are going to convert to the you know to the statistical equilibrium that are going to be the convergence doesn't have a spectral gap is a lot of understanding about that but actually I take one minus so to be matching the quantity which is the medium between one and epsilon gamma over theta so this is almost like a master model because because this is a measure if I average to that it forgets completely about the potential I put gamma here minus three and that is gone what I get is the average of that is eight over epsilon it's just a scaling so by that is very important that is you do not succeed getting Landau without that scale right well the choice of your potential could be very different but something has to push it to compensate in the expansion so essentially what you get then is to deal with an equation that behaves like Maxwell molecules but it's not because in the Q plus operator you still have this big this big enchilada as they call it it's a mess to look at that what is that I mean does it look like rather for like the devilate interaction where it looks like this I mean just look at this part of the picture I mean I just took it from another slide okay I didn't have time but basically what this one says if I don't know what happened for a small angle I assume that I keep my last observation okay and that is actually going to be moving as you know as you get epsilon to zero with that minimum because you have the one over epsilon this starts to be raised and then you pick up in the limit the u to the minus cube alright so it does the job of approximating Landau so we did the calculation and in fact this is something that we were able to get it's quite interesting and I give a lot of credit to Sona to have read every single line of the first part of the paper of Laurent which he discussed today and come to the understanding on how to manipulate LP estimate but I just want to focus on this particular well the Cauchy problem is the Martin theorem I'd like to also credit to Bressin for realizing that it would work here although he didn't use it itself but it worked we have used it for quantum Boltzmann equation we are using it for wave turbulence for stratified flows which are integral equations in web theory and it works like a charm as people would like to say the other observation this theorem would not need to use entropy and then the other observation is the following one and this follows that's part of the things you need for the proofs and the sub tangency condition and and so let me go into this slide just a word that's for Michael I hope you like it Michael this is something that I was actually thinking how do I explain this in two slides so with Alonso and Carnero in 2010 what we did is we actually started to revisit it and try to get young inequalities for the collision operator that was done by Gustemson and many other people and Villani and Mujo but what we did at the time was extremely careful estimate that first of all take a complex combination of exponents a la brush can leave a la loss can leave and then regroup them accordingly in a way that you can actually get a good hold of it and so what you observe is that these parts here are crucial because they are encoding your cross-section now my cross-section now my mu is going to be the steric delta when I do the integration with respect to the sphere this mu gets replaced by the minimum right? actually 1 minus mu gets replaced by the minimum between 1 and this Coulombic interaction rescaled by epsilon so 1 over epsilon will come out you see but this one is going to be very nasty and this one is going to be very good and then you have to work like hell to get this actually come up to chance so in the classical splitted case you recover the young scene equality what was new in our work with Carnera and also is that C was exact and that was what got me to be able to do the calculation and in the search of that cross-section I realized and I will not go to the detail here I have to finish that I need to actually to and the trick was and I finish with this the trick is that you start with any LP function it's going to be the initial data that you want to put on and out okay could be infinite, could be now but could be something you have to be in LP it actually works for an infinite solution we are finishing all the details and it's going to come very soon I need to assume because of that nasty cross-section that in an epsilon interval which depends only on gamma on the Coulomb interaction I have to do a correction of this function so I'm going to take in this it's a subspace in LP because if this function is in LP this is also in LP then I actually say trust me that that works and I get then the equivalent to the young senior quality but with a negative weight and it's actually interesting I'm trying to see if it is removable it is needed I actually get better than minus 3 and that worries me a lot but I cannot get you know if it is an issue I cannot find it but there is a weight and that is because of this negativity and so with that then we can show gain of integrability by recent work we did with Maya Tascofi and Ricardo Alonso which allow us to come up with a maximum principle and get a full control which the data is going to depend on the initial data but in the limit you are going to lose the initial data and then the last statement is the convergence and I will not say more than in fact we can actually show that if I have an initial data in L12 and L1P, L log L then I can actually construct a solution in LP epsilon minus 2P and that in the epsilon limit gives me a weak solution of the Landau equation in the classical sense and that conserves mass momentum, energy and makes entropy decay and with all of that I actually thank you very much for your attention