 Thank you very much Sergio for the introduction and not commenting on the affiliation and saying that Princeton not for long. Well, I'd like to thank the organisers very much for the invitation to the conference and to speak and also to stay here. I had a very good time actually in Bure, although people discouraged it. What about the wizard? It was good. I think it was better than Princeton. That's for sure. I think it was okay. But all right, so I apologize to those of you who have seen, if not me, one of the collaborators talk. Okay. But somehow, what I'll try to do, since this is also a diverse audience, in the first part I'm going to give some sort of more general introduction to the problem. So that could be boring for the experts. And then in the second part I'll talk more in detail about our most recent result. This is joined with Udeng, who's a postdoc at Courant, Benoit, posto there. And I didn't get the names in the right order, but that's fine. And Alexionescu. Okay, so this is the main. So in the second part I'll talk about our most recent result on gravity capillary waves. But before, let me give you a sort of more general introduction, talk about the models and the results that have been proved recently. So the general goal is to understand the fluid motion for a fluid that occupies a region that is moving, evolving with time. So a standard example is waves on the surface of the ocean. And this is a question posed by Laplace some time ago, where basically he posed the Cauchy problem for water waves asking, given any initial disturbance, which is also localized, say, of the liquid surface, what will happen next, right? So and here you see two pictures of some two different phenomena where you say perturbed maybe a drop on the surface of the water and you have created little ripples that will interact. That's an example of sort of a nonlinear interaction of waves. And here you see some sort of phenomena which maybe looks more two-dimensional, right? Okay, now if you want to be more ambitious, which I won't be in the rest of the talk, but the models that we study and the equations that I want to do also describe, I mean, are meant to sort of make us understand more complicated phenomena than just what Laplace asked, such as, for example, how can large wave form in apparently calm water be reported? And so this would be some sort of questions about some type of growth, right? Maybe connected to singularity formation somehow, or for example, you can think of models where you have two fluids, right? And here the solution might be more unstable. One type of instability that one sees is, for example, if you have two fluids that shear past each other, creating some sort of formation like that. So these are very poorly understood phenomena, although, of course, observed and there's lots of experiments. And they can be modeled by the water waves equation which I will introduce, at least in some approximation, all right? So this could be a nice future research. But so for now, let's concentrate on, so what is the model, okay? So we're interested in a fluid that is inviscid. So I want a viscosity and it's incompressible. And the fluid will occupy a region that changes with time. So I call it omega t of Rn and n will be 2 or 3. And I call v, which is a function of space and time, the velocity. So this is only defined for x in omega t. p is the scalar function, the pressure. And rho is the density, which is constant for me in this more basic model. And the equations of motions will be inside the region of the fluid, will be Euler's equation, okay? So just Newton's law here. And on the right, you have the internal force given by the pressure, right? Excited by each particle, on each particle by the surrounding ones. And you might assume, and for most cases one does, the presence of gravity, okay? So acting E n will be the last coordinate of space Rn. And the fluid is incompressible, expressed by the divergence free condition. And I will look at most of the Cauchy problem, so specifying an initial data, okay? So the data to be specified is v, okay? Which has to be incompressible. And one also has to specify the initial domain, okay? Now the system is not closed yet, but somehow, so these equations are, because p is recovered from v, at least inside the domain by the divergence free condition. Now what one needs to specify is how does the domain evolve? Okay, so let's say, for example, how does the boundary move? And this is done as follows. So as the fluid moves, the domain is moving, and this is happening consistently. So in the sense that particles that are on the boundary will always save this is the fluid region, and I have a particle on the boundary. At initial time, at later time, this particle will still be on the boundary, and the fluid will just evolve consistently with this boundary. And moreover, the way that this will move is in the direction given by something that's proportional to the normal component of the velocity. So if the velocity is tangential, there will be no motion, otherwise the velocity of the particles inside will push the fluid to move, okay? So this is just expressed by this one, v is tangent to the union of the spacetime boundaries. And there's one more, so these are, say, n conditions. Then you need one more condition of the boundary about the pressure. So the pressure inside is determined, you need to know what is it on the boundary. And now what you can do is either say that the pressure is constant or say you can normalize it to zero, anyways, to match the atmospheric pressure. Or if you assume that you take into account the surface tension's effect on the surface of the liquid, one would say that this, the pressure is proportional to the mean curvature at, say, at the point x, okay? So this is, so either you could have or not surface tension in the sense that you might want to include it in your model, depending on also the characteristics of the liquid itself, right? So, and water is actually one of the liquids with the highest surface tension, okay? And this is, so water surface tension, you can think of it as a restoring force because if the liquid tends to bend, right? This will push up the mean curvature, which will push up the value of the pressure at the boundary. And so create a larger gradient, which will then decelerate the fluid, right? So we'll tend, we'll make the fluid tend to relax. So in some sense, this could be thought of a restoring force, at least on certain scales. And for example, you see that, right? So at the interface of air and water, the surface tension essentially results from the greater attraction of this molecule as with the ones inside the fluid rather than the ones of air, which are less dense, okay? And now as the fluid moves, right? So you can have, okay, so first of all, these are the equations, right? So this is the system. We have equations inside and the boundary conditions. Now you can have several possible scenarios, depending on what type of physical phenomena you want to model, okay? So you can have your domain omega t to be anything. So basically I didn't draw anything, right? So just through some sort of curve, and the fluid is on one side. Now the other, now you can have the model with gravity and or surface tension, and usually we call gravity waves the one with gravity and no surface tension. We call capillary waves the one with surface tension, no gravity, okay? Now the fluid, in addition, you might or may not want to assume that as curl zero, in which case we call it irrotational, or curl not zero, okay? And now as I said, you can also consider more general models where you have two fluids, okay? And that will create all sorts of possible scenarios. You can have a fluid that is lighter than the other, right? So they can be one on top of each other, they might shear, and so on. But this will be essentially if you have two fluids, you'll just have the same set of equations on the other side essentially. And then there's just some condition about that the boundary needs to move consistently with the two, okay? So there's many possible things one can consider. But so what are the main questions for this evolution equation, right? So three main questions we want to, one would like to study are, first of all, the log over posedness. So given an initial velocity field which may be nice, divergence free, and an initial domain, which is also smooth or sufficiently regular, can you construct just a solution locally in time? This is a preferable question. And we'll see that it's already non-trivial for this equation. Now, are there some configurations that after you start and you can solve maybe for a short time lead to some breakdown, okay? And what does breakdown mean also, okay? Some sort of, so how do we think of blow up in this model, say? And then another question is, can you, so once you know that you can construct solutions for a short time, can you avoid blow up? So can you put some conditions maybe on the initial data so that solutions will live for very long times or maybe forever and see what is the asymptotic behavior, okay? And of course there are many more questions you can ask, okay? But so these are the three basic ones and the ones that have been studied about the Cauchy problem mostly so far, okay? So I start, this is a suggestion of Rowan to start with saying that things can go wrong. Okay, so to try to understand why the model is sort of complicated to study, we have examples when things actually can go wrong. So the first one is by Castro-Cordoba-Ferrero-Mangand said that Lois Fernandes, which is the following. So if you think naively just of the situation where you have a wave like on the surface of the ocean, right? So maybe something like this and this is a graph of a function. Then you could have solutions where if your description of the fluid is a graph, okay, that might fail to be so at later times. So you might have a graph that is very steep and maybe at a later time it will overturn, right? So in some sense this description will not hold. But usually nothing wrong is happening with the fluid and the equations also will make sense, just a description as a graph would not. But even more importantly, the result about splash and spot of Castro-Cordoba-Ferrero-Mangand said of Dom Serrano and Cunans Scholar, which where they constructed solutions where what you can have is, well what is in the picture, I don't need to do the picture. What is in the picture? So here you have a fluid inside here where the density is one and the vacuum outside and you can have, so this is a very smooth interface between the fluid and air and nothing is wrong. You start with such a data and after a short time, if you prepare of course the velocity say in a certain way, this will self-intersect, okay? So now your description of the fluid as being inside the region bounded by a smooth curve, okay? That sort of stops making sense, right? And you cannot continue the solution at least in a classical sense. You see that the velocity changes and this happens all in the smooth category. So there's a touching point here, but the interface as a parameter as curve remains smooth, okay? This is an example that has been constructed. And so you see that if something is large, right here for example what is large is the inverse of the distance, right? But nothing else is wrong. Then actually the model, the equation can stop making sense. But for example, if you had another fluid inside, okay? Such a touching in the smooth category of parameterized curves cannot happen, okay? And this was proved by Fairfield-Magnon-Escoli and Kudansk-Koller, okay? So this is example of some type of breakdown of the equation. Don't you get instability if you have the fluid whose higher density above the one is lower density? No, so if you have surface tension it's fine. So right, so if you have two fluids, even just for local existence, you need to have surface tension, okay? Otherwise it's unstable. Otherwise it's unstable, that's correct. And then for short times you can solve, it doesn't really matter, I think, right? So you could have the model with two fluids except that this cannot happen, right? So you cannot squeeze the fluid if the other one is also incompressible, right? If the- Kind of a comment about that, which is with density zero on the other fluid so to speak, there's no way that one component of the fluid can apply any force to the other component of the fluid. But when there are two fluids then it can. Yeah, so you mean of this fluid, right? This doesn't, I mean there's no position. Look at the second where the interfaces are touching, right? It's at that point where one interface can start applying force to another interface. Before it touches, there's no way where one component can apply force to another component. But when there's two fluids it can. Okay, well that's what the theorem said, yeah? That if it does apply, I mean it does apply such a force that it prevents the, at least it prevents it in the smooth category. Nothing prevents it from touching, in this case. You're right. Okay, so this is what can go wrong. Now, on the other hand, the equations for nice and smooth initial data can be solved locally. And there's been many works on this, so early works from the 70s and 80s. I'm not gonna go through the whole list, but at first people started solving the equations in the analytic category, then smooth, then for small data. There's been lots of very nice contributions by Walter Craig, of course. And Schneider and Wayne and more. And there's also relating the local solutions constructed to other models such as KDV, Boussinesque, maybe trying to understand a bit more the behavior beyond the local well-possessed, but this is for small data. Now the real breakthrough came at the end of the 90s, when C.J. Wu showed that any sub-level data, by this I mean, say, a sub-level initial velocity in the domain that can be parameterized locally by a curve, which is of sub-level regularity, can be solved for short times in two and three dimensions. So this was, and if you think that for most of the equations that we know, usually the local opposing is a reality, or it's something simple, at least in the smooth category, while it's not for this problem. And this is why there were many more works that try to improve on this and also give us a better understanding and cover many more models. So for example, this case, the case that we treated was irrotational, Christian Nurell-Nidbad had the vorticity, and then there were many works. David Lan, of course, had the very general results where you can have the bottom with the very general conditions. And then more works, Shatal Zen, Kudan Scholar. And then people started thinking of, okay, what is, how low can we go in terms of regularity, right, on the surface? And so Stryker sets are introduced, and these are in the works of Christian Nurell-Nurell and Stafilani, and Arzab Burkensuli have a series of really great works where they treat lots of models, so gravity, surface tension, and have the best results in terms of regularity. And, okay, so then there's also works on two fluids, but, I mean, beyond the list of people that probably are missing someone, worked on this. I would say that the local opposing is very well understood, but not that well in the sense that there's no unified theory, I think, that really can treat models in very large generality and give some kind of simple proof that can be read easily or satisfactory. I mean, all of, right, okay. So you have to make some effort somehow also to go through the local theory, okay? Now, the third question was about, and it's the one that I'm most interested in is the question of long time or global well-poseness, okay, so what is the setup? So before I start writing down a bit more about the equation, so the setup is the following. First of all, you want to assume that the fluid is irrotational. So you have Euler's equations here and you will add that the curl of V is zero, okay? So if you think of this, so in three dimensions, it's obvious that you want to do it. Otherwise, you'll be facing Euler's equations, okay? And that's, so this will be harder than Euler in 3D and you don't want to do that. Now, in two dimensions, vorticity is transported, right? So in principle, it won't be something that grows, okay? But it will be definitely a component that doesn't decay, okay? And the idea here will be that if you want to control the solution for long times, things have to decay and having a component that does not could be a big problem, but so far this has been assumed. But maybe in the future, right? So, especially, so in two dimensions, probably something can be done, right? But now if you have, okay, so the second thing, this is one thing. Second thing is you want your data to be small. Mostly because of the examples that we saw before, that if something is large, right, then there could be breakdown, okay? And it's also, right, the very reasonable assumptions for what would see as a linear equation. So, you want your solution to be small and stay close possibly to one of the equilibriums, and the most simple one will be the trivial equilibrium, which is a flat interface, okay? Which has infinite extent. And for example, you could have infinite depth. That would be a model for waves in the middle of the deep ocean. But you could also potentially have a bottom, and there's been some work on this, so I'll talk about this later. So now, in this case, so you will have all these equations inside, okay? You think that you have a graph here of your, of your waves of T and X, let's say. This is X will be in R2 or R. So depending if you're talking about the two dimensional case or the three dimensional case. So your interface will be either a curve or a surface. And you'll be summing all these equations. And now the boundary conditions, if you have such a graph, can be expressed more simply by saying that the DTH is V dot essentially the normal times the surface element, okay? So these are the equations. Now, if, so besides the reason I gave for the curve being zero to be something fundamental. There's also some advantage in the description of the equations. So if the flow is irrotational, then V will be given by a potential, which is harmonic. And so you can reduce everything to starting the trace of this potential. So you take the potential from inside restricted to the boundary, which is given by this curve. So which I'm describing by X, H of X. And you can write down the equations. Okay, so the equation, so the previous equation, this one, since V now is the gradient of an harmonic function, becomes what we call the Dirichlet-Tunnan-Mann operator. And this is Euler's equation. Once you express it for the potential, it's essentially Bernoulli's law restricted to the surface. Okay, so this is a short computation. It's already, not here, but also these are the equations. And posed on the, just on the boundary. So X here is just R or R2. And this is a system of two equations for the evolution of the interface and the potential. Okay, so now, that's not a thing too much about the form of this. Okay, one thing you see here that the gravity and the surface tension are the only, so contribute linear terms. So this is GH, and this will be sigma, and the linear term will be Laplacian of H here. Okay, so you already see that having or not having them has already a big impact already on the linear, on the linear solutions. Now this is a time-reversible quasi-linear system. With quadratic nonlinearities, you see it everywhere. Okay, since it's quasi-linear, you cannot use fixed-point arguments to construct local solutions because of the loss of derivatives. And one thing is that if you write it this way, but also any other way you write it, somehow the equation is nonlinear and non-loga, just because you're solving Laplace equation from the interior. This is already from Laplace and Cushy. Now, one can deal with the issue of the domain moving in a different way than what I did here, which is by fixing, say, a graph. Of course, if your domain is not a graph, you can do something else. But you can use Lagrangian coordinates, Morphic coordinates. So you can treat this problem in various ways, but these difficulties will still stay. So constructing local in-time solutions is already challenging, as we said before. And now if you want to go beyond the local existence time and maybe construct global solutions, you have to. So the basic mechanism would be to use dispersion to control the nonlinear interactions. And now in a frame or so where you have a quasi-linear equation and with quadratic nonlinearities, which potentially decay very slowly. So in this setup, you need to rethink somehow and combine and extend the classical methods such as energy methods, the vector fields method of Sergio Kleinerman, normal forms and all the free methods that have been used in semi-linear equations. You have to rethink of this in this different context. Now, let me make a few other basic observations about the system. So first of all, it has an Hamiltonian structure, okay? Has a point out by Zakharov in the 70s. By the way, this is usually called the Zakharov, these are Eulerian coordinates, the x and h of h and phi. And this is called usually the Zakharov of Greg Shansoulin formulation. And Zakharov observed that this has an Hamiltonian structure, okay? So where this is conserved energy, it's also the Hamiltonian. And so you see here that you have gravity and total mass. And this would be the potential energy of surface tension here. And this is the kinetic energy, okay? Which corresponds to half the derivative of the potential because G is a self-adjoint first order operator. Now h is the only known coercive conserved quantity, okay? So you don't get much control on your solution just by the conservation. And now, since we're interested in global regularity, I said I want to stay close to a flat interface. So if you linearize around zero, what you get is dT of h, the linearization. So for the half-space, the shaded phenomena is just absolute value of grad. And here you will get minus gh and sigma Laplacian h, right? So this will be the linearization close to the flat solution. And so the linear dispersion relation, this I'll write for later, is given by lambda g sigma is root of g c sigma c cubed, okay? And also, so all solutions of the real problem of the form are oscillating waves of the form e to the plus or minus i t lambda of the initial data. So now, as I go towards explaining our main result and stating it, let me first say what has been done on the question of global regularity. And I condensed everything in one slide. So the first result was Cichirou's result. And she showed that if you are in two dimensions with gravity waves, no surface tension, then you can construct solutions which live up to time e to the constant over epsilon, where epsilon is the size of the data, okay? Then later, at the same time, Germain-Masouli-Chadda and Cichirou proved that you can actually, if you're in the higher dimension, you have global solutions. And this makes sense because in more dimensions, you are conserving the mass always, but you have more directions to disperse. And so your solution is supposed to decay faster. This gives you more control on the nonlinearity, okay? But also, sigma is here. Right, so this is for just exactly. So all of this light is for either g or sigma zero. And this part is for sigma zero or g positive. I mean, the same result is also true in three dimensions, also Germain-Masouli-Chadda that comes later. Then in two dimensions, Alex and I proved that one, so one could improve this result and actually pull solutions to infinity. And the reason why this couldn't be done before is that there's something nonlinear that happens a little past this time at time e to the one over epsilon squared. And once you capture that asymptotic behavior, it's one of the main things that makes you understand the solutions and close the argument for time going to infinity. And then independently at the same time, Alazare and the Lore gave a different proof and if Riemann-Tatarou had a nice proof of, reproof of C. Jebou's result, almost global, and then complemented it to a proof of the global result. And then Sushant Wang, Cecil of Alex, what he did is he reproofed this result by improving the spaces in a significant sense, which is he could consider data that have infinite Hamiltonian, so infinite energy. Okay, so in particular, removing a momentum condition that it's implicit in two dimensions if you assume that the kinetic energy is finite. But you have to have one of them different from C, right? So all of these so far is just G, yeah, in general, yes, yes, of course. So all of this is for G positive and Sigma zero, all of this here, right? And now I do the opposite. So now the results with G equals zero and Sigma positive. So in 2012, remember we showed up proof that in 3D, indeed. So this is a harder problem. So in 3D, there's something geometric that is different. And so it's a really different problem. So but they proved that you can have global solutions. Now in 2D, it's also a harder problem because the reason is that, so you see if either you have C to the one-half for gravity waves, or you have C to the three-halves, okay? And so this is the dispersion relation, which means that waves with small frequency travel much slower here. And so they tend to hang around. And this creates lots of problems. So the zero frequency is a big issue here. And nevertheless, we were able to solve the problem. And also, if Riemann-Tataru gave a proof of global solutions with slightly different assumptions, okay? So this is all for global regularity. Now, so what I want to talk about is the problem where you have both gravity and surface tension. So and try to explain why this is really a different problem, okay? So first of all, so let's take the picture. And we said that we can describe the solution by the profile of the wave, and the potential restricted to the interface phi. And now if you, so let's just call you this combination, which is essentially the variables that are in L2 by the Hamiltonian, okay? So now if you give a sufficiently nice initial data, so think of SVAR data, now these are the more precise assumptions. So U has to be in HN0 for some N0 large. It has to have sufficiently angular regularity. So omega here is the rotation vector field, two dimensions. And it has to decay to zero at infinity. This is how you quantify that it's also close asymptotically to the flat solution. Now if this is true, then you can have a global solution for the problem where you have both gravity and surface tension in three dimensions. So in the case of two dimensional interfaces, okay? So now what I'll try to do in the next, I don't know. At least for the next 10, 15 minutes is to say why this problem is different than the others and what are the main difficulties. And then if I have time I'll give some ideas of the proof, at least of some parts of the proof, okay? So there are two major difficulties. One is that lambda, the dispersion relation has an inflection point. If you have both G and sigma and this gives you less decay even for the linear solutions. And the other difficulty is that if you look at the quadratic mononarities and the way they interact, they interact much more strongly than in the other cases, okay? So I'll try to explain this. Now if you don't want to think of the waterways equation in the local theory, a good model to think about is just you can take just the linear equation. And then add a quadratic nonlinearity, okay? So now here, also the way we explain it, we added it in such a way that the L2 norm is preserved. But okay, if I have time I will discuss why this is a bit more relevant. But if you tell me that you can solve this problem, then definitely you can solve the whole problem, okay? So and here you can think of this model and you can also restrict say V to be a smooth function just by restricting its frequencies, of course it has to be real, so that you don't lose the derivatives. And so this model has the same linear part, quadratic quasi-linear nonlinearity. And you disregard all other interactions that are not so relevant, okay, and cubic terms and so on. So this is a good model to work with. It conserves the mass, like it should, because of the Hamiltonian. And now if you want to reduce to this, you have to work a bit with the waterways equations, you have to use the parallelization, in particular in the way that Al-Azhar Berkens really used it with the y quantization. That's something that is different, and you have to diagonalize the equation, use the good unknown of Aynak, but so I'm gonna skip all of this, okay? But somehow, so you can concentrate on such an equation. So this is a linear part, just like a Schrodinger equation which is with a different dispersion relation and some quadratic nonlinearity, okay? Of the form du squared, okay? So now what is the basic strategy in general? Basic strategy is the following for all quasi-linear problems, I guess. You, the local theory will tell you that the solution can be continued as long as it has finite h and norm, right? And the basic way you try to push the solution further is by constructing an energy functional, which is like your h and norm. Plus, you can put other two norms maybe based on vector fields, which if you have in variances, okay? But for example, in our case, we only have rotational in variance, but not scaling, okay? We should instead have for these other two equations, right? And then you try to prove some kind of granular inequality like Cecile showed, right? And typically, so if your solution decays fast enough, right? Or you're clever enough, then you can prove something like this, okay? Now, it turns out that, so to do such a thing, you assume a priori that your solution decays, okay? So you wanna control the energy by assuming that the solution is already dispersing. And the way you do it typically is by some estimate of the unlimited norm or some other weighted norm, okay? I'll get to this a bit later. And the second part of the argument, you try to prove that assuming that you have control on the energy, you will prove the decay that you have assumed to control the energy, okay? So it's a two-part Boussrap argument, okay? So this is the basic strategy. Now, what happens in the water waste cases, okay? So this is the linear equation. Now, if you have one of the two sigma or g equals zero, then the dispersion relation is non-degenerate, that's non-vanishing action, and you get the full decay. So d here is the dimension of the physical space I'm in, but d minus one is the dimension of the x, right? You have different dispersion relations between g equals zero or sigma zero? No, so it's the same. The rate in time is the same, but the behavior of, say, it's more frequencies will be different. But say, if frequency is one, the time rate, so that's very important if you want, right? But the time rate is still full, right? So, still full decay, safe for the Schwartz data, right? Now, on the other hand, if one of them is, if both of them are not zero, right? Then you have an inflection point, and let's call this sphere of inflection point gamma zero, okay? And you see that, so this is the dispersion relation, the red curve, the blue curve is this, the group velocity, which has a minimum at gamma zero, which is this value here, okay? And so, and therefore you will lose one over six over the decay, okay? This is like KDV, say, compared to Schrodinger. Okay, now here you see one important frequency then is gamma zero. There is another one, which is gamma one, which I'll get to hopefully at the end, we should don't notice anything particular on the linear solution, right? So it's something that will appear later nonlinearly, but nothing is special here. Now there are two major issues with our problem, which is all the previous results on water waves, and other, there are no quasi-linear dispersive equations, had one of the two following features. So either the linear decay was ready to do the minus one, allowing you to close the Groner estimate more easily. And this is the case when you have one of them being zero in three dimensions. You have the full decay. Or, if you didn't have the full decay, you could remove the part of the nonlinearity that decays slowly, okay? So there's a procedure that can be done called a normal form that removes the slowly decay in terms. And this is the case of the 2D models, both with gravity, no surface tension and surface tension, no gravity. Okay, so either one of the two things was necessary, absolutely important. So here, as we said, A fails, because we have decay five over six, and also B fails. And B fails, this is a brief explanation. So if you want to remove quadratic terms from an equation, you should look at how two waves interact. So let's say you take away with frequency eta, and away with a different frequency, now I call it c minus eta, so that the sum is c. And this is the dispersion relation. Now if two of them never combine to give you the dispersion relation of the sum of the frequencies, then, so if this expression is not zero, then at least at the formal level, you are able to remove the quadratic terms, okay? Now, so if you have gravity waves or two decapillary waves, then you have some solutions, but these are the trivial ones. So say, for example, eta equals zero, c equals zero with the right signs, okay? So that could be treated, although in some models it's a little harder. Now if you have both gravity and capillary waves, you have lots of solutions of this, okay? So there's no way that you can remove the quadratic terms. And you see this is just a picture. So here, c is fixed to be 100 zero, just a vector, and this is the trace in the eta plane of all the solutions, okay? And there's nothing special about this, it's some sort of ellipse, okay? But so this expression, this has lots of solutions, okay? So there's no way to do that, and so the mechanism has to be different, okay? To propagate, say, the energy, okay? And I'd love to make a comment about that, which is the frequencies that go into such resonance relations are forcibly bounded. So if I cut those out, it's actually a compact operator. Yeah, but you, I mean, how do you, you can't remove them? I mean, even if you remove it from the data, they're not going to be. I'm not telling you, not saying to remove them, but they result in a compact operator. I'll let you, well, we'll talk about it later. Okay. But yeah, yeah, but okay, no, yeah, yeah, okay. If you, okay, so if you remove just the neighborhood of that, then that's fine. But you still have to prove some bounds on that, right? So, and they have to decay as fast as if it was, right? So that's, okay, that's the main thing that I'm going to explain, right? So you start with an energy that sort of looks like the nth derivative of your solution in L2. You take the evolution and try to do the standard energy estimates. Now if you really want to see how this behaves, you want to look at, it's convenient at least to look at it in free space. So you just write the products out and this becomes convolution in free space. And so you have two functions with the maximum number of derivatives and then one more function. Now, as we said, if you're trying to do an energy estimate, these two things are in L2 and this will not decay at least for general frequency, will not decay like 1 over T. So you cannot just put the absolute values inside and estimate. But okay, if you're not at the frequency gamma naught, you should be fine. So let me restrict when you are at gamma naught, okay? So I can assume that this frequency is gamma naught and these two are very large, I'm doing an energy estimate. So this would be like large, large frequencies. Now I want to study the oscillations because if I just put the absolute values inside, this is not gonna be integrable. So there's, I cannot apply tolerance in the quality. Now, so let's try to, in first approximation, describe you as an inner solution. So let's, let F be the profile. So once I pull back the linear flow and substitute for UF and get out exactly this expression from before, which sums up all the oscillations, okay? And now, so what do you do? Now, you can look at two cases. So when this function is away from zero, okay? Then, well, we'll see what to do. So first of all, let's look at the two cases. So either it's away from zero or say close to zero, okay? For some parameter, let's, let's think about what that is exactly, okay? Now what do you do? If you are away from the resonances, you can average because this will oscillate very fast and in particular, you do it in practice. You can integrate by parts in time, okay? Using this, you will divide by something that is vanishing. But hopefully, you gain more by integrating by parts in time. So for example, if this was a linear solution, DS of it will just be, if it was a linear solution, this is just an initial data, so DS will be zero, okay? So you are dividing by something. But supposedly gaining because DS of f is a quadratic expression, okay? So there's many issues associated to this, but this can be done. So you're dividing by something that it's possibly small. But if it's not smaller, say, than t to the minus five over six, the extra function you gain, right? To at least make you think that this is, this is okay, all right? Now, if you instead are close to the resonance set, what can you do? Right, so let's think about it this way. If you were close one over t, right? And this was a function with gradient one, then this is the sub-level set of, and the measure will be one over t, right? So you will be able to integrate that. Now, of course, you don't want to be that close, otherwise you wouldn't gain on the other side, okay? So there has to be something special, some extra oscillation that you have to be able to extract. And this comes from the following lemma, okay? For this type of reintegral operator. So this is exactly the same expression that I have here. So I think of this guy, I'll take it in L2, and then I have an operator acting on L2 function with such a kernel, where I have an oscillating factor like this, t goes to infinity, and I restrict the oscillation, the oscillating function phi to being less than some parameter. And moreover, you can restrict to being close to gamma naught for that, for this function, right? Now, the statement is the following. If, and this is a simplified version of the statement, if you are close t to the minus two-thirds, say, to the zero set, to the resonances, then this operator decays like one over t, which is what you want to close the runway estimate, okay? And as I said before, let's think again, if this is just a regular smooth function, and you're looking at the sub-level set of, so here is the, so here is your resonant set, which is just some sort of ellipse, right? And then you want to cut close to it, right? So this is with two to the p. Then, if you just have a smooth function, right, which says non-generate with gradient non-zero, then the measure of this set will just be two to the p, right? But this is saying that actually what you get is more than that, okay? So this is the main thing. Now, the proof is based on a TT star argument, sort of similar to how you prove the standard of two bound for non-generate free interloperators, which have non-generate ashen, okay? And, but it cannot be done that way because this is sort of a rough function, okay? So it's a cut off that cuts very narrow, okay? And what one finds is that one has to study not just the ashen, but the ashen of this phaso. This ashen is the same as the ashen of phi, restricted to some specific directions. While you want to restrict to this direction, you see that if you want to integrate by parts, you don't want to integrate in the direction, so you don't want to integrate in the direction that it's gonna, where the function, where this function is not smooth, okay? So you want to integrate in the direction that it's tangential, which is the direction of the gradient perp of phi, right? And this was already some, an earlier version appeared already in the work on to the other Maxwell by Dengonis Composite there. Now I'm gonna skip the proof. That's why I want, very nice. I hoped I would use this. Okay, three minutes, I can do six slides. No, I'm fine, no, I mean, this I skip also. I skip also, so I'll go to here. So in six minutes I'm gonna, so okay, so first of all, with that, and many other things, you can close the energy estimates. Okay, so that was one of the main ideas. There's other structural things that are very important about the equation, which I skipped here, but if I left time, we'll see. So, but okay, so you can close the energy estimate. Now this is the first part of the argument. Second part will be, now you have to re-prove everything that you assumed about the function decaying and being localized in space and so on, right? And you don't have enough vector fields to say that the energy of the vector fields is enough to give you the decay, right? Okay, so in particular, you want to prove that the solution decays and you want to control some weighted norm that guarantee that. Now, the very general basic approach is the following. Again, you look at an equation of this form. You write it, you write Duano's formula. Now you can do it because you don't care about the loss of derivatives and you write it in free space. So this is exactly the same as I did before. You have the product, it's a convolution and you extract the oscillations. Now, if it is not zero, that would be okay, but we said that there's many solutions. On the other hand here, you also can resort since you're not at the level of the highest number of derivatives, you could resort to integration by parts in eta, okay? So, exploring the fact that the F hat, for example, you can think of them being Schwarz functions or being smooth so you can take derivatives of them, okay? With respect to the frequency variable. Okay, so the real abstraction to not being able to prove that your solution disperses is if you have no oscillation, so if phi is equal to zero and the gradient at the same time is equal to zero. And this is the basic approach that was put forward by Germain Masoudin-Chata and very successful in water waste problem before. Now, in this problem, unlike in all the other water waste problem, there's a following issue that if you look at this phase, he does the large set of zeros and if you intersect it with the gradient, you'll also find that this is a co-dimensional one set. So it will be of the form eta has to be c over two for the gradient to be zero and there's also another solution which gets discarded, but if you also want the phase to be zero, c has to be of length gamma one, okay? And this was the other special frequency that I sort of hinted at before. Now, what happens? So let's say that we look at this expression and we want to understand how it behaves for very large times, okay? And then from there, get some intuition on what is the space that we're actually allowed to put the solution in and see if we can actually prove the dispersion that we assumed before. So let's look at this expression and let's call two to the m the time parameter, just as a parameter, okay? And then how does this behave? So if either the phase of the gradient are not zero, you can supposedly get decay, okay? Assume that, for example, you're f hat as far as functions, okay? So you can restrict by this stationary phase heuristic that give you the oscillations to a set where phi is quite small, like say one over the time parameter and where the gradient is more like one over the root of the time parameter. And if you calculate just formally, what this gives you, it gives you that the output of such an expression will be a smooth function that is localized on an annulus around exactly this resonance set. Now I'm just gonna use the same picture, okay? Of with one over t. Or two to the minus m here, one over s. Okay, so if you input first functions and do this stationary phase heuristic, you get such a function. So whatever space you wanna put your solution in, it has to contain this, right? So you want all sorts of functions to be in your space. You want functions that are free to transform that is supporting on this fin annulus. And you also want your norm to be strong enough to give you the decay, right? So typically that would be, for example, that one norm would work, but of course it doesn't work with this, okay? So typically you want a weighted norm, something like an L2 norm with some weight on top of it, but a weight for your function means a derivative for this guy and this guy is quite rough, okay? But if you take a derivative, you see that it actually behaves, right? And take the L2 norm, it behaves like the square root of the distance from this sphere of resonance. And so if you take such a function, it will be in such a space, modulo plus or minus small losses. And so another choice where to put your function and prove your bootstrap will be to have dixie of f hat, which is like x on the real side, and degenerates as the root of the distance to the, to this frequency that appears non-linearly, non-linearly as the output of all the resonances, okay? And this was, so one way to deal with these norms, I mean one way to actually implement this, a good way to do it is to decompose in atoms that are localized in frequency and in space. And here in this case you would also have to decompose depending on the distance from this sphere. And then to control this norm, it's a quite involved argument because you have by linear terms and for those you can maybe use for some of them you can use the heuristics from the stationary phase. And then when you are in non-resonant situations you will have lots of linear interactions to take care of and you have bad frequencies at gamma naught where you have not enough decay at gamma one where you have not enough localizations with degenerates. And so controlling all of this is quite involved but so it can be done and this will close the bootstrap of the way the norm which together with energy gives you the global existence result. Now, I think I can stop. I don't know what that's here. Yeah, I can stop here. So if you look at the scattering data at infinity for F it should have the singularity at X equals gamma one. All right. Can you say what this singularity looks like? So I think maybe this is the next slide I had but so you're thinking that, right. So basically what happens for example is that if you look at F hat or U hat then this will be, this will be. So for some solutions, right? Unless you restrict something about the support in frequency space, you will have that this goes like log T, right? Because so say, well, okay. Maybe I don't know if this is what you meant, right? But say this, right? So you, if you look at the stationary phase say in eta, you will get something like one over T. These are the two things that interact at C over two and C over two. But if you put T equals infinity then F hat at infinity is gonna have a logarithmic sequence. Right, if you're close to gamma one, at gamma one, right? Yeah. And then you can also quantify that so depending on, so exactly in such a neighborhood of gamma one of width one over T, right? So you can quantify, yeah, exactly. So this will be the scale. And also for the, you see also delta norm, right? So also delta norm that generates. Yeah, so I guess this was, this one.