Thank you, and also I'd like to thank the organizers for the kind invitation to this conference.

It's a pleasure to be in Paris for a month, a month of June. It's really nice. And the conference is also really nice. Now, I'm going to talk about waterways. Most of my talk is going to be about singularities and some formation of singularities in this model.

Also, I'm going to start with some more general description of the problem in order to understand better what the problem is. Also, Fabio Pusateri, I'm going to repeat a few things that Fabio discussed two days ago.

So the model that you're looking at, at the simplest level, the model is a model that has... So it's simple to draw a picture. So you have a fluid that presumably lives under an interface.

And the fluid is described by the interface. The interface is a moving surface, z of alpha and t. And it's described by velocity, so v. And we have equations.

And inside the fluid, so looking at the free bound incompressible Euler, inside the fluid we have the Euler equations. The material derivative of v is minus gradient of the pressure. And then we also have the gravity term minus gen, so it's a gravity term pointing down.

Now, this is inside the fluid, this is the equation for the vorticity. However, there's also an equation for the moving interface. The interface itself is moving. And the interface is moving with the fluid, which says that dt of z, so the particle on this interface. If the velocity of the particle points in some direction, then the interface wants to move in the same direction.

And one can write this simply as saying that dt z minus the velocity v would have to be tangent to the graph of the interface. Now, in order to make this into a system and to close the system, we need to prescribe something.

And one can prescribe the pressure on the interface. And the pressure on the interface would be prescribed, the simplest way is to prescribe it proportional to the mean curvature of the interface. So p of x and t sigma times kappa of x and t, where kappa is the mean curvature, as I said.

And sigma is a positive parameter. If one just looks at the system this way, even just making sense of what's the equation, what evolves, there are several things that evolve. So making sense of the system, it takes a little bit of, so one has to think for a minute. The way to look naively is to take the divergence and then the first equation would say that one has an equation for delta of the pressure.

So if you know the velocity at some point, at some time, then we get an equation for delta of the pressure at that time. And then the pressure is also prescribed on the bond. So in principle, we get the pressure at that time out of this information. And then once we have the pressure and the velocity and the interface at one time, then we get the infinitesimal increment.

So we get from the equation vt, from the Euler equation, we get what the increment is. There's a slight imprecision in the equation for z, because one can think that the parameter is to do with how you parameterize the surface. There are many ways to parameterize the same surface and there's a slight imprecision there.

So this is the system now. One can raise, like for any, it turns out that one can make sense of it as a well-posed evolution system. And one can raise at least three basic questions, like for any system of this type, one would have the local regularity question,

which is, can we construct solutions locally in time? If one starts from nice initial data, can one construct solutions locally in time? One can also ask the question of global regularity. So can one construct solutions that extend for a long time or even long-term regularity that goes beyond the local existence time?

And one can raise the question of dynamical formation of singularities, meaning starting with data that's nice at time zero, and at some point beyond the local regularity time, one would form something that will look like a singularity in the flow. And I'm going to quickly describe, most of my talk is going to be about the last point, the issue of dynamical formation of singularities.

But I'm going to describe quickly the other pieces as well. And Fabio also described them two days ago. Now, the local regularity is well understood, and it's taken a long time.

But the general picture that has emerged is that one has local regularities, one has a well-posed system. If the surface tension is positive, or when it's equal to zero, one has to make a certain condition, the radial condition is satisfied. And the time of existence depends on the natural, two natural features of the system.

One of them is the smoothness of the parameters, so we want to have z to be smooth and v to be smooth, let's say in some norm. And the other one is what's called the r-cord constant of the interface, meaning that this picture is nice, but you could start from a picture of this type at the initial time.

And then what we expect, no matter how smooth things are, the time of existence has to record the fact that if it advanced for too long, it could have created a self-intersection. So the time of existence somehow has to know that one cannot expect to advance, so it depends on the fact what's called r-cord constant.

The r-cord constant is simply the fraction between the cord between these two points and the arc. And when that goes to zero, that will be a parameter that's small. Now one can also state all these problems, so these problems are stated, I just drew the picture in

two dimensions, it makes no difference in three dimensions, one can write exactly the same equations in three dimensions. One can also put periodic conditions, now periodic, there are various ways in which you can put periodic conditions, the simplest way, the closest to this picture is to assume a periodic relative to saying that the world is a cylinder.

So to be periodic in the x-axis, but infinite in the vertical axis, so infinite in this direction, and periodic in the horizontal direction. One can also put a finite bottom here. And one can also consider a model that I'm going to consider a little later, which is the two-fluid model, in

which one extends, it's not just one fluid and vacuum, but one would have one fluid and the second fluid on top. And it turns out that there is a reason, it's not just an extension, there is a reason to consider the two-fluid model at the same time as the one-fluid model. And now the local Poisson, as I said, is well understood and it goes back to the earlier work of Nalimov, Yoshihara, Craig.

Then the local Poisson is in this shape that I wrote here with Sobolev norms and what we like to think is the natural local Poisson in Sobolev spaces goes back to the work of CGO from the late 90s. Now there are lots of models and there's been a lot of work proving this local Poisson's theory in all

of these models and I wrote some names, probably not all of them, and I apologize if I miss people.

So there's been a lot of work on these models and I stopped at 2011, there's more work after 2011. In any case, the picture is well understood.

Most of the work after 2011 has to do with reducing the regularities. Instead of thinking of the objects being in H10, they would be, let's say, in a lower Sobolev norm. And one could try to reduce the regularity relative to how low one can go. On the other hand, this is a quasi-linear problem. So once you think that this is a quasi-linear problem, in none of these problems, it's unlikely one can get to the critical regularity by, it's a quasi-linear problem.

So it's probably not possible to prove well Poisson, I said, the critical regularity. Now, the global regularity results, they are much fewer and they're much more restricted, not only much fewer but much more restricted.

The only time when we know how to prove global regularity is if we have small data. It's the same kind of picture that one has in quasi-linear problems. You need to know that you are small or close to some solution that you know. And in this case, you also need the data to be irrotational, so you cannot have any vorticity. And they also need to be in the entire space. We cannot be in a periodic case.

So all of the global regularity results, they require these features, smallness, irrotationality, and the domain itself has to be the entire space. Because that's the only one that would allow, so the way the mechanism would be is through dispersion. And one can only have dispersion if one is in the full space.

And it goes, it's also pretty recent. So the first result of this type was an almost global result of CGU from 2009, in which he proved global, almost global regularity for 2D gravity water waves. Gravity means that gravity, the gravity coefficient is positive, but the surface station coefficient is zero.

And this was followed by German mass, Moody and Chata in three dimensions, the same problem in three dimensions. One can look at the opposite problem, which one has the capillary water waves, when G is equal to zero and sigma is positive. This is also work of German mass, Moody and Chata. Then afterwards Fabio Pusateri and I, at about the same time, Al-Azhar and Delore, we looked at the 2D problem, 2D gravity problem.

And we proved global regularity, so passing from almost global to global. There are new proofs of this and also of CGU's result by Hans and Yefim and Tataro, and Yefim and Tataro, global regularity. My student Wang, he revisited the problem of global regularity and he showed that one can do global regularity in an infinite energy class.

And in that class, he can remove one, infinite energy, but still small in another norm. In that class, he can remove one momentum condition. Now, the opposite problem, when the capillary, so this work with Yefim and Tataro assuming one momentum condition in the Hamiltonian variables.

And also Fabio Pusateri and myself without that condition. And there's this last result that Fabio talked about a couple of days ago, in which we looked at the full problem, the G, the full problem with both gravity and surface station in three dimensions.

So this is all by now, I think this is all very well understood. And I only want to make one point about this global regularity work, because I think it's useful for other problems. So if you don't care about water waves, there's still one point that I think is not about water waves, but it's something that we understood very well in the context of water waves.

Okay, so all of the results in three dimensions, the mechanism is to prove control of the highest order energy and the same time prove dk. And it turns out that in this case, one can prove 1 over T dk.

In the cases when g is equal to 0 or sigma is equal to 0, one can prove 1 over T dk for the linearized flow. And the hardest part is to prove 1 over T dk for the non-linear solutions. And that's the argument that works in the three dimensional problems. All of them except for the last one, the one that Fabio talked about a couple of days ago.

Now in the two dimensional problems, there's one idea that developed very well and which got very well understood, which appears in all of the results, starting with CGO's result. Which is the so-called quartic energy inequality.

Which is an inequality that one would like to prove as if the equation was cubic. You'd like to prove that you have an energy inequality, an increment as if the equation was cubic. Because in two dimensions, it becomes a 1D interface and the best you could hope for is 1 over square root of T dk. And as we know, 1 over square root of T dk cannot get very far. But if one could pretend that the equation was cubic, then that would have

led to, then in principle, could have at least gotten the almost global result. And so the mechanism that I think it's kind of a, it's well understood in the context of waterways. And I think it's kind of a general mechanism by now, is to prove this quartic energy inequality, which the energy is controlled by an integral of four terms.

And the point is not to lose the derivative, so there are two points in this. One of them is not to lose the derivative, so to have the same number of, the highest number that is not to be different in the right hand side and in the left hand side. And the other one is to get the four terms.

Formally, one can think about this as a normal form. But then also is to do it carefully in the process to make sure not to lose the derivative. And this is, as I said, this inequality or a form of this inequality is present in all the global results in two dimensions. It goes back to the first work of C.J. Wu.

And the way one would construct, one would solve the problem afterwards is to prove an inequality like that. Have 1 over square root of T point twice dk that would almost give, that would give the almost global results if it's. Okay, so I'd like to thank the organizers for the kind invitation to, to bear.

Okay, so basically what I was saying is that, so what I was saying is that if one proves the disquatic energy inequality.

And if one can couple this with proving 1 over square root of T dk, that will almost lead to, that will lead to an almost global solution just out of this, just out of this piece. In order to get to the global solution, there's some more Hamiltonian, there's some more structure in the system that comes from the Hamiltonian structure. And that's what Fabio Pusateri and I found, for example, for the gravity waterways.

There's been a lot of improvements, as I said, this kind of inequality goes back to the work of C.J. Wu. Where she actually had a logarithmic loss in time in this inequality. Did not affect what she was proving, which was almost global existence, but she did have a logarithmic loss. Fabio and I, we went through her proof and it was a removable logarithmic loss, it was not a significant logarithmic loss.

Like I said, there were several improvements throughout this, the reason why this got well understood. Starting with the power differential energy estimates, and we learned this from the work of Alazar and Delore.

So they had a certain way, a very nice way to decouple the two issues. Which is one of them that you want to have, you want to pretend that the equation is cubic, at the same time you want to make sure you don't lose the derivative. And one can decouple these two issues to address them, to address them. Then what Fabio and I did was to use what we call the compatible vector field

structure, which means that we construct a certain, not all vector fields work the same way. It turns out that one cannot quite implement the Kleinman vector field method, with all the vector fields counting the same way. There was the modified energy method of Ifrim and Taro, that clarified, what I really like about that is that,

it says that if you do a normal form for the purpose of doing an energy estimate, then it's a very good idea to do them together. Because the normal form is basically a division, the energy estimates, it's a symmetrization. And if one does them together, they work very well. They would even simplify a denominator, for example, which is the real concern here.

And then what we did, so we took the point of view that energy estimate is always better to them in the Fourier space than in the physical space. If it's more flexible in the Fourier space, so we put everything in the Fourier space, in the spirit of the high method coming from the semilinear theory. And in any case, the reason why I wanted to bring this up is because this is the kind of thing that I think is relevant in any problem.

So we made this work well for waterways, but I kind of feel that this is a general, it's basically a general picture, that this quadric energy estimate is very robust and I think the only thing that's needed there is not to have small divisors.

So in principle, if you have any problem that formally expands in a way that there are no small divisors, then one should expect to be able to do a normal form while not losing the derivative, so to prove this quadric energy inequality. Of course it will fail, the quadric energy inequality will fail if there are small divisors, it's not possible to prove the inequality of this type.

And it's exactly the issue that Fabio discussed in this last model that Fabio discussed two days ago, that's exactly the central issue, that there is a full set of resonance, so one cannot prove the quadric energy inequality in that case. And also one cannot prove enough decay, the decay.

So what Fabio described is some kind of a partial normal form that we did that depends on a non-degeneracy condition. Okay, so I'm going to talk now a little bit about the main topic on which I'm going to discuss some proofs, which is formation of singularities.

Now if we think of the local Poisson's theory, the local Poisson's theory says that we have a time of existence that depends on the smoothness of the objects, and on the r-cord parameter, which has to do with how far the surface is from self-intersecting. So there are two possible scenarios in which you could say that you are going to create a singularity.

One of them is to find something that leads to loss of regularity, and the other one is to find something that creates a self-intersection. Now, in the loss of regularity scenario, there are also several things we could think. Losing regularity in the middle of the fluid, so inside the fluid, appears to be very hard,

because one has less control on this problem than on the Euler equations. There's also the moving interface, so that appears to be a very hard problem. But one could also try to understand the loss of regularity at the level of the interface.

So the first, what I mean to say is that the loss of regularity problem can be decoupled into something inside the fluid, or something on the interface. I guess like in 2D, you don't expect to lose it inside. No, you don't expect to lose it inside. Well, you can say that inside also,

you can say that there's no vorticity. You could do other. In 2D, in the Euler equation, you lose it because of vorticity. You'd lose it because of vorticity. Now, the point is comparable. I'm not saying that you can really solve this inside, and you should wonder why can't you do Euler.

In any case, so the two theorems that are, there's only one. So basically, the only mechanism that's known that loses regularity, and it's proved, and for which there's a mathematical proof, is this mechanism of display singularity, which is drawn here, which says that if we start,

it's possible to start with an interface that's very close to being a self-intersecting, and a smooth. Everything else is smooth except for the fact that it has, it's panda pinching at the point like that. It's possible to create data that's like that and which advances for a short period of time in a way that it creates the self-intersection.

This is a very stable phenomenon. This is a singularity discovered a few years ago by Kostro, Cordoba, Fefferman, Gansed, and Gomez Serrano, and there was a new proof of Koutan and Schkoller. It's a very robust phenomenon in certain ways, in the sense that the original proof,

for example, was in the gravity problem, but one can put surface tension without changing the conclusion, or one can put vorticity without changing the conclusion, or I can go from 2D to 3D without changing the conclusion. So it's robust in that sense. The only thing that prevents this mechanism is what we found a few years later,

in joint work with Charlie Fefferman and Victor Lee, which is that if one puts a fluid in the middle, so if one has this picture with one fluid outside and the second fluid, no matter how light the fluid in the middle is,

it's not possible to create a splash while preserving things smooth. So these are the two themes that are now relative to singularities. I'm going to describe why is that I think that's relevant. So the second theme is relevant to the question of producing a singularity through loss of regularity of the interface.

But let me introduce first the model to make it an exact model. The model, as I said, for two-fluid interface is very similar, looks very similar to the model for one fluid, which is what one has two fluids with two different densities, and they live in separate, they have to be separate, so there's an interface.

There's still an interface. The fluids are separated and they both evolve according to Euler equations in their respective domains. So one has these equations, the material derivatives of UJ, so they refer to the fact that there are two U's, there is U1 and U2, and they both evolve according to the Euler equation in their respective domains.

The interface itself has two slides, so there's some compatibility condition. The two velocities, they are not independent of each other, because the interface itself has to move relative to both fluids simultaneously. The condition on the interface is that dTz minus each velocity

has to be tangent to the interface. The condition here is written in dimension one, in which it's a dot product with the derivative of d alpha z. Also, the condition of the pressure on the interface

becomes the difference of the two pressures, and the difference of the two pressures would have to be proportional to the curvature of the interface. So this is the system. It's a little bit harder to understand that this is a well-posed system. It takes a little bit more effort.

For this particular model, the local opusness theory was proved by David Lang, to prove that indeed this model is well-posed in the sense that if one starts with initial data, that's nice. One can extend the solution on a short period of time. So this is the model. Now I'm going to describe first.

Oh, there's one more. I have one more slide to describe. There's one remaining slight imprecision in these equations, which is the fact that it's the choice of the coordinates. So the exact condition is that dTz minus some velocity,

it has to be tangent to the interface. But exactly what tangent means, it would mean one can specify, one can make a more precise condition by saying that dTz is equal to u, plus a constant times d alpha z. So this would pick a vector in the tangent space.

And now this constant can be made to depend on everything. It's a parameterization constant of the interface. And one can take it, one can also not specify it, in the sense that any smooth function would work the same way. But the two typical coordinates, the Eulerian coordinates,

which we don't use here, or the Lagrangian coordinates should be more useful. Each one would say that dTz is equal to u. It's exactly equal to u. So the constant would be taken equal to zero. So I'm going to quickly describe the construction of the splash singularity of Castro, Cordoba, and Fefferman,

Gansed, and Komeserano, just to contrast the case of, the point is to contrast the case of one fluid versus a two fluid problem, and try to understand what is it in the two fluid model that prevents this. And the exact definition, so the definition of f of z,

this is the exact definition of the r-cord constant. It's the picture that I draw here, that one looks at. It's the worst value that one gets by looking at two points, alpha and beta, and taking the cord and dividing by the length of the arc. And the worst value one gets relative to the points is called r-cord constant.

And so the exact theorem of these people that I mentioned, is the following, that if one can, there is a solution. So there is a periodic solution. It's worked out in the periodic case, but it makes no difference.

There is a smooth periodic solution in the gravity problem. So there's a smooth periodic solution. What I have here are the equations written all together. So the first equation is the Euler equation. The second equation is the equation for the interface. The pressure in the case when there is no surface tension is said to be zero,

and then the incompressibility and irrotationality conditions in the domain. And there is a solution that either starts out from, so the way it's written, starts out from the r-cord condition being equal to zero. So it starts from a contact point and it develops in time to a curve,

which doesn't have any more contact points. One can also go the other way. So the more natural way to go the other way, the equations are time reversible anyway. So one starts with a picture that's separated and one creates a contact.

I'm going to give a very quick idea about the proof. The proof is basically local well-posed. One can reduce the problem to local well-posedness result. The way to do that is to think of how to express the velocity in terms of,

how to express the velocity in a way that accounts for all the conditions. There are several ways to express it. One can think of it as the gradient of the velocity potential, one can think of it as a gradient perpendicular of a stream function. But the way to do it in this problem is to think of the velocity as being the Birkhoff-Roth operator applied to a function omega.

And the function omega is thought to be leaving. So the function omega leaves on the interface, the interface vorticity. And the Birkhoff-Roth operator looks like, it's basically an inter, it's like a variable coefficient Hilbert transform.

In which one it's looking, the fraction has in some sense the singularity of a Hilbert transform. We are in one dimension, this is all written in one dimension. And has the singularity of a Hilbert transform, it's integrated against omega and there's the formula for u. The system can be rewritten in terms of z and omega.

So these two variables z and omega they will capture everything. However, it turns out that the only way you can have a splash, the only way they construct a splash is the function omega going to infinity at the time of the splash. And the idea that goes in this paper is to make a change of variables. It turns out that a possible change of variables is tangent of z over two that splits the plane.

So at the end of it, after this change of variables, one has the graph basically, the graph of the change of variables it will look,

there will not be no splash, it will be like a smooth graph. And so they're able to make this change of variables and then think about the problem as if it was a local world poisonous problem. So they start to data that is of this type that would be consistent to the splash, rewrite the equations in terms of this tilde variables,

which are in this change of coordinates, and then run the system, so show that the system is well posed for a short period of time. And the short period of time being such a way that this species separate from the line such that after one goes back, one can get the separation.

So one can think about the problem in this way. One can still solve it. The point is one can still solve it by energy estimates after this change of coordinates. Now, I'm going to describe now the problem

that has to do the two-fluid problem. So what's the difference? So what's the two-fluid problem? In the two-fluid problem, I want to show exactly that this mechanism doesn't work. The proof itself obviously doesn't work because if there was some fluid in the middle, then making this change of variables shouldn't be very smooth. So it'll create problems. The change of variable itself would not be...

One wouldn't be able to translate this into a smooth problem by making this kind of change of variable. But the question is, is there something intrinsic or is it just that the proof doesn't work? And what we showed is there is something intrinsic. And at the beginning, we didn't know.

So we started out either way. We didn't know whether it was going to be possible to have a splash or not. So I'm going to stay now the precise theorem that we prove in order to explain why is this related to, possibly related to formation of singularities. So the exact theorem says the following. If we are looking at the solution, so assume that we have a solution of these two-fluid system

and I wrote it here, so it's all written in these brackets here. So we have these two fluids with the Euler equations, the equation of motion of the interface, the difference of the pressures, and the divergence, the incompressibility conditions. So assume that we have a solution of this.

So I need to describe the precise assumption. So we assume that z is smooth. We assume that this interface stays smooth throughout the evolution. And we assume that the velocity on one side stays smooth, just as the problem, just as the theorem for the splash.

And then we assume that the time t is equal to zero. The picture is better. So the time t is equal to zero, there is no contact in the interface. And we assume that the time t is equal to zero. The other fluid is also smooth.

But we don't assume anything about the other fluid at any other time, except for time t is equal to zero. Then the conclusion is that as long as the equations make sense, which means that as long as we don't have the function f, as long as there is no self-intersection, we have a lower bound for the r-cord constant,

which means that the interface will not be able to touch, which means that basically in terms of a continuity argument, this means that the interface will always have to stay bounded away from self-intersection. So in other words, it's not possible to develop splash singularities

while keeping the interface smooth and one velocity smooth, in the case of the two-fluid interface. Now, in order to be able to say that it's not possible to develop singularity of this type,

we don't say that it's not possible to create a splash, because it's actually likely that if you start fluids to develop singularities. The other part that's left is that, so the reason why this mechanism of creating a splash while keeping this smooth is not possible is because the fluid in the middle doesn't have time to get out of the way.

As long as these two curves are smooth, they will be approaching in a way that's quadratic, because there are two curves that would be approaching, and if they are smooth, then the distance between them is too small, and the fluid in the middle doesn't have time to get out of the way.

A very likely scenario that I think is very possible is that as we approach, so if we start with a picture this way, but at a later time as we approach, it will start creating a corner. You can picture if it starts creating a corner, then it leaves more time for the fluid in the middle to get out of the way.

I think that's quite possible if we prescribe data that would want to push the fluid in the direction towards self-touching, then the only way that can keep advancing is if it gets sharper and sharper in order to allow for the fluid in the middle to get out of the way,

and that could potentially create a curvature singularity. And now I'm going to describe quickly what's involved in the proof. So as I said in the beginning, we didn't know which way to go, because this splash mechanism seems to be robust to almost any other change,

and it's actually very close. It's a log, it's close by a log. So I'm going to describe the two main ideas that we have in the proof of this theorem, and there are really two ideas. One of them is dynamical, the other one is at one constant time,

so analysis at one constant time. Dynamical part has to do with this boundary vorticity. So if we look at the equations in terms of the boundary vorticity, we can use the Birkhoff-Roth operator and can express the two vorticities on the two sides in terms of the Birkhoff-Roth operator of this function omega, and we have these formulas, v1 is Birkhoff-Roth of omega plus omega over two,

and v2 is Birkhoff-Roth of omega minus omega over two. Then if we write the two equations, so the two Euler equations get two equations for the velocities, and we can take the difference of the two equations.

The main difference between the two models is on this slide. We can take the difference of the two equations, then we get the Burgers-like equation, and the Burgers equation is something of the form dt omega is equal to one half d alpha omega squared plus some kind of multiples of d alpha f omega and some simpler terms.

Now we can think about this Burgers equation, and for the Burgers equation, we can preserve the strongest one we can preserve for omega is the infinity norm. So everything is to be thought of as local, but what we can preserve because we have this Burgers equation, we assume that these functions f1 and f2 are smooth as part of the assumptions for the theorem, then we get that this boundary vorticity would have to stay bounded

over the time of the evolution. So this is the piece that makes a difference relative to the case when there is only one fluid. And now, once we know this information

that the boundary vorticity stays bounded, what we want to say is that if we have a situation in which we have two points, so let me draw it like that. If we have two points in which, so this would be the points alpha one and alpha two.

So if we have two points in which the distance between them is small, then we want to be able to say that the difference between the normal velocities of those points would have to be small and small with the same constant. So if the distance is smaller than epsilon, then the difference between the velocities

have to be smaller than epsilon. Turns out that we lose a log of one over epsilon that has to do with the fact that all of these operators are essentially Hilbert transform. At best they are Hilbert transform, in fact some kind of variable coefficient Hilbert transforms and we have to work on L infinity because there's the best information that we have about the boundary vorticity.

So that leads to a logarithmic loss of log of one over epsilon. The proof goes in the following way. So we are looking at, so what is the information in the problem? So we are looking at these operators, but maybe I should start from here. So the information in the problem is the smoothness

and the smoothness, we can draw these pictures, so let me draw these pictures separately. So we have two curves described by the functions f and g that are supposedly smooth. And we have these operators t1 f of omega plus.

Omega plus has to do with one side. So this would be, the things with plus leave on the side on top and the things with minus leave on the side on the bottom, let's say. So on the top we have this omega plus and we have the function f and we have the velocity.

So f plus plays the role of the velocity so you obtain the velocity out of the, so this Hilbert transforms, this operators t1 and t3, they are written on this slide and they are essentially the normalized versions of the Birkhov-Roth operator I had earlier.

So the Birkhov-Roth operator, if we write it in this picture in which we let, in which we, so we expand the picture, so we have these functions f and g that correspond to these two sides of the curve and also have to localize them properly. So if the distance here is epsilon, then it turns out we have to take out

a distance in the horizontal direction of size square root of epsilon. So we take out the essential parts of the Birkhov-Roth operators and we write them, so we write these operators that are the parts of the Birkhov-Roth operators after these normalizations. And the system that, so the information that we get

is contained in the first two lines. So f plus is one velocity and f plus is a formula in terms of omega plus in terms of omega minus and f minus also has a similar formula and the function g is what measures the, is what measures the difference between the two velocities

that we want to show this smaller than epsilon log of one over epsilon. So the harmonic analysis problem if you want becomes, so after these reductions, it becomes that we have, we know that the functions omega plus and omega minus, they are bounded. This is coming from the dynamical part of the problem. Now we fix the time t.

We just have to prove it's a fixed time t, but we have this dynamical information that omega plus and omega minus are bounded and we know that the functions f and g are smooth, suitably smooth. And we know that functions, what's more important is to know that functions f plus and f minus, they are smooth as well. And out of these, we want to derive the fact that

this function g is small, g of zero is small. And it's like, we like to, the problem, the difficulty is the fact that the information we got about omegas is coming from the Burgers equation. When you do the Burgers equation, you cannot get information about the derivatives. You are going to get information

about the function, but it's not possible to get information about the derivative. So you don't have good information about the derivatives. We only know the L infinity norm about these verticities. So we have to run an argument according to this L infinity norms. And we use, so to prove this proposition, we use what we call the Z-norm method,

in which we define a norm. We use this also in other problems that have to do with global regularity, but we have to define a norm that's consistent to the problem. And we have to analyze this system in a bootstrap way in terms of this norm. In our case, the norm that we define, it's basically a Hilbert transform.

So these are essentially Hilbert transforms that we are looking at. But they're not exactly, they are variable coefficient Hilbert transform coefficients that depend on this other functions f and g. So the function that we have here is a certain type of Hilbert transform.

And they are looking, so we measure a function by testing against, by essentially calculating its Hilbert transform and then measuring it in a proper space. It's written here in a duality way. And then it turns out,

it's very important to get the bound on g to be epsilon log of one over epsilon. You cannot lose more than log of one over epsilon. If it was like epsilon to the one half, then they do not have prevented the splash. It has to be epsilon. And the most we can lose is log of one over epsilon. It turns out it's quite,

we are quite surprised that in fact, it turns out we get it exactly. So we get this exactly. So we can close the argument to prove this. And I only have five more minutes. I wanted to say a few words about why is it that we are looking at this model.

So my hope is that, now we don't have a theorem, but we have tried a few things. The hope would be to try to create, like I said, a splash of this type. So I'm sorry, the hope would be to try to look at an evolution that creates both a splash

and a singularity at the same time. So what I would like to see dynamically is that as the interface starts approaching towards splashing, it would also have to create a corner. And that seems to be pretty tough, so we have some work in progress.

One feature of this model that I like, which is better than other model, one could ask the same question for other models. You can ask the same question for SQG, for example. Like why you can put any equations here and ask this question,

can you form a singularity in the interface? One feature of this model is that, in some sense, there is no place it can go. So if it advances, what we showed with this theorem is that if the interface wants to, it's also possible to create a symmetric picture. So one can have everything symmetric relative to an axis.

And if we can get it to advance, then the only way the evolution will finish is by creating the curvature singularity, because otherwise we would violate this theorem. Otherwise, it would create a splash without forming the singularity. Now, of course, the difficulty,

what's hard and haven't been able to do is to control the flow. One needs to be able to say that this interface advances for long enough and it gets start to control the flow as you lose smoothness. So the more smoothness you lose, it's a quasi linear problem. It's hard to control the flow.

One has to use energies, but at the same time, control the flow. So in any case, we don't have a very clear, we don't have a theorem of this type, but I think this would be something that would be very interesting for this. And it's the kind of thing that one can do it for this model, one might not be able to do it

for the model in which one has only one fluid, we know that the splash can be created without losing smoothness. So it's not possible to have this mechanism there. Okay, so I'm going to stop here. Thank you.

The question. So is it possible to try and solve backwards from the corner situation? Yeah, if we could find a good answer to...

It's possible. It's not so easy to do it, but it's possible. I mean, of course, that's how the theorem for the splash was proved by solving backwards. Now there are more proofs, but the original proof was by solving backwards. It's pretty important how you make the answer. I think it's pretty delicate. My thing is pretty delicate to do it this way.

But yeah, it's certainly possible. I think that's a very nice idea. And your picture is, I would like to modify to make it a self-similar corner and then solve backwards. So what would happen in self-similar variables?

Make a very symmetric corner touching and then solve separately? Yeah, if I could... I'd love to be able to tell you. It's the kind of... There are several things. Ultimately, my feeling is that the only way you can really do this is through some sort of monotonicity because if you start losing...

If you try to solve the equation as if it was a well-posed problem, I don't think that's... I'm not sure this can be made to work. At the end of the day, this way you can say that the problem is well-posed in some coordinates. But what sometimes works better is for some sort of monotonicity which we are working on understanding.

You have another question, if you permit me. Sure. Which is, I know it looks very different, but it does nonetheless remind me of Cordova-Delayave-Feferman squirt singularities.

That is, these interfaces are approaching each other and they are smooth, you assume, so it's pushing the interior fluid rapidly. And so the squirt singularities avoid that situation by the time integral of the l-infinity normal velocity.

You're studying something else, but is that geometric configuration forcing, say, the vorticity to infinity? It's almost a geometrical picture. It's making the interior fluid, the lighter fluid as you've drawn it, move fast near the boundary walls

and so omega is getting larger. It's not the only scenario that it can happen. So this is my hope because I like to prove lots of regularity of the interface. This is not the only scenario. In fact, Charlie Feferman and I, we had lots of discussions. He feels that things will happen differently, maybe according to what you say.

My feeling is that if you set up the velocity to be high enough, you can set up the data, the velocity to be high enough, there is no time for it to develop something so complicated. It just wants to form a corner for the thing in the middle to get out of the way. Now, that's my feeling. Of course, this is not the proof, so there are no proofs in this formation

of curvature singularities. But I think Charlie's... I had lots of discussions with Charlie Feferman about this and we could never really see exactly one reason why one is more likely than the other. Thank you.