 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. So I'm going to start with some more general description of the problem in order to understand better what the problem is. Also Fabio Poussateri, so I'm going to repeat a few things that Fabio discussed two days ago. So the model you're looking at at the simplest level, the model is a model that it's 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 is described by a velocity, so v. And we have equations. And inside the fluid, so it's 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, inside the fluid 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 the 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 is 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 body, 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 increments, 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 parameterized, it has 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, cannot 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 would 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 these is the smoothness of the parameters, so one would 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-chord 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 you'd expect, no matter how smooth things are, the time of existence would have 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-chord constant. The R-chord constant is simply the fraction between the chord 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, I 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. There's 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's a reason, it's not just an extension, there's a reason to consider the two-fluid model at the same time as the one-fluid model. The local Poznan, as I said, is well understood, and it goes back to the earlier work of Nalimov, you'll see Hara Craig. Then the local Poznan is in this shape that I wrote here with Sobolev norms, and what we like to think as a natural local Poznan in Sobolev spaces goes back to the work of CGU from the late 90s. Now, there are lots of models, and there's been a lot of work proving this local Poznan's theory in all of these models, and I wrote some names, probably not all of them, and I apologize if I miss people, Bayer Günther, Christodulu Limbled, Ambrose, Ambrose Masmoudi, Lan, Limbled Kutanshkolaer, Cien Kutanshkolaer, Christensen Khurstafilani, Al-Azhar Borjuli, Shata Zeng. 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 I should 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-posed, and I said the critical regularity. Now, the global regularity results, they are much fewer, and they are 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 a quasi-linear problem. So 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 you 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... The way the mechanism would be is through dispersion, and one can only have dispersion if one is in the full space. And 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 almost global regularity for 2D gravity water waves. Gravity means that the gravity coefficient is positive, but the service station coefficient is zero. And this was followed by Germain-Masmoudian-Chateau in three dimensions, the same problem in three dimensions. One can look at the opposite problem, which one has the capillary water waves, where g is equal to zero and sigma is positive. This also works as Germain-Masmoudian-Chateau. Then afterwards, Fabio Pousateri and I, at about the same time, Al-Azhar and the Lore, we looked at the 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 Hunter-Infimantataro and Infimantataro 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, the capillary... so this worked with Infimantataro, assuming one momentum condition on the Hamiltonian variables, and also Fabio Pousateri and myself without that condition. There's this last result that Fabio talked about a couple of days ago, in which we look at the full problem, the g, the full problem with both gravity and surface tension in three-dimension that's worked with Deng, Pousater and Pousateri. 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 word 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 three... all of the results in three-dimensions, they go... the mechanism is to prove control of the highest order energy and the same time prove decay. And it turns out that in this case, one can prove 1 over Tdk. In the cases when g is equal to 0 or sigma is equal to 0, one can prove 1 over Tdk for the linearized flow and the hardest part is to prove 1 over Tdk for the nonlinear 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 is... which appears in all of the results, starting with CJU's result, which is the so-called... 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 and increment as if the equation was cubic because in two-dimensions becomes a 1d interface and what we could hope for is 1 over square root of Tdk. And as we know, the 1 over square root of Tdk cannot get very far, but if one could pretend that the equation was cubic, then that would have led to... then, in principle, it 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. 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 also is to do it carefully in the equation in the process to make sure not to lose the derivative. And this is, as I said, this inequality or a form of inequality is present in all the global results in two dimensions. It goes back to the first work of Cj1 and the way one would construct the... one would solve the problem afterwards is to prove an inequality like that, have one over square root of t-point to SDK that would almost give... that would give the almost global results if it's... I'd like to thank the organizers for the kind invitation to... to beer. Okay, so... so basically, what I was saying is that... so what I was saying is that if one proves the aquatic energy inequality and if one can couple this with proving one over square root of t-dk, that will almost lead to... 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 Bussateri 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 Cj1 where she actually had a logarithmic loss in time in this inequality. It did not affect what she was proving because it 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 and starting with the... the biodeferential energy estimates and we learned this from the work of Al-Azhar and Elor. So they had a certain way... a very nice way to decouple the two issues. It is one of them that you want to have... you want to pretend 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. 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. So we're starting the same way. There was the modified energy method of Ifrim and Tataro 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 estimate is a symmetrization and if one does them together, they work very well. Which is the real concern here. And then what we did, so it took the point of view that energy estimate is always better to do them in the Fourier space than in the physical space. It's more flexible in the Fourier space so we put everything in the Fourier space in the spirit of the i-method coming from the semi-linear theory. And in any case, the reason I wanted to bring this up is because this is the kind of thing that I think is relevant in any problem. Maybe this works well for waterways but I kind of feel that this is a general... it's basically a general picture that the squatic 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 losing the derivative, so to prove this squatic energy inequality. Of course, it will fail. The squatic energy inequality will fail if there are small divisors not possible to prove 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 one cannot prove the squatic energy inequality in that case. And also one cannot prove enough decay, the decay. But Fabio described it as some kind of a partial normal form that we did that depends on a non-degeneracy condition. So I'm going to talk now a little bit about the main topic in which I'm going to discuss some proofs which is formation of singularities. Now, if we think of the local well-posed hysteria, the local posted hysteria says that we have a time of existence that depends on the smoothness of the objects and on the R-quart parameter which has to do with how far the surface is from self-intersecting. So there are two possible scenarios which 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. Now I guess like in 2D you don't expect to lose it inside? No, you don't expect to lose it inside but well, you can say that inside also there is no, you can say that there is no vorticity. You could do other in 2D in the Euler equations you lose it because of vorticity. You lose it because of vorticity. Now the point is comparable. I'm not saying that you can really solve this inside and you should want to work into the Euler. In any case 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 the Spley 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 kind of pinching at the point like that and 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 the self-intersection and this is a very stable phenomenon. This is a singularity two years ago by Kastro, Kordoba, Pfeffermann, Ganceda and Gomez Serrano and there was a new proof of Kutane-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 one can go from 2D to 3D without changing the conclusion robust in that sense the only thing that prevents this mechanism is what we found a few years later in joint work with Charlie Pfeffermann-Victoria 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 no matter how 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 these are the two themes that are not relative to singularities and 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 so the model as I said for two fluid interfaces 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 and the fluids are separated and they both evolve according to Euler equations in their respective domains so one has one has these equations this the material derivatives of U, Uj so they refer to the fact that they are two U's these U1 and U2 and both evolve according to the Euler equation in the respective domains the interface itself has two slides so there's some compatibility condition the two velocities are not independent of each other because the interface itself has to move relative to both fluids simultaneously and the condition on the interface is that dTz minus each velocity has to be tangent to the has to be tangent to the interface and dT is written in dimension one in which it's a dot product with the derivative of dA and 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 to be proportional to the to the curvature of the interface so this is the system it's a little bit harder to understand this is a well-posed system it takes a little bit more effort for this particular model the local oposeness theory was proved by David Lahn 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 and so this is the model now I'm going to describe first oh there's one more I have one more slide to describe what's there's one remaining slight imprecision in these equations which is the fact that the it's the choice of the coordinate so the exact condition is that dTz minus the sum velocity it has to be tangent to 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 that dTz is equal to u plus a constant times the 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 you know 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 the tangent coordinates should be more useful and 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 Pfeffermann, Ganset and Comet Serrano just to contrast the case of the point is to contrast the case of one fluid in the 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 arc 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 that's called the arc cord constant so the exact theorem of 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 in the gravity problem so there is 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 the 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 arc 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 so it can also go the other way so the more natural would be good to go the other way the equations are time reversible anyway so it starts with a picture that's separated and one creates a contact and I'm going to give a very quick idea about the proof the proof is the proof is using it's basically a local well point one can reduce the problem to a local well poseness result and 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 I can think of it as a gradient of a particular stream function but the way to do it in this problem is to think of the velocity as being the Birchhoff-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 of vorticity and the Birchhoff-Roth operator it looks like it's basically an it's kind of like a variable coefficient Hilbert transform in which when it's looking the fraction has in some sense the singularity of a Hilbert transform we are in one dimension this is already in one dimension and has the singularity of a Hilbert transform it's integrated against omega and there's the formula for u and now 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 the possible change of variables is tangent of 0 over 2 that splits the plane so at the end of it after this change of variables one has the graph basically after the change of variables there will not be no splash it will be it will be like a smooth graph and so they are able to make the 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 the steal the variables which are in this change of coordinates run the system so show that the system is well-posed for a short period of time in such a way that this piece is separate from the line such that after one goes back one can get one can get a separation so one can think about the problem in this way 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 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 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 a change of variables wouldn't be very smooth would not be one wouldn't be able to translate this into a smooth problem by making this kind of change of variables but the question is is there something intrinsic or is just that the proof doesn't work and what we showed is there's something intrinsic and at the beginning we didn't know so it 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 state now the precise theorem that we prove in order to explain why is this related to possibly related to formation of singlite so the exact terms as the following if we are looking at a solution so assume that we have a solution of this two-fluid system and I wrote it here so it's all written in these brackets here so you have these two-fluids with the Euler equations the equation of motion of the interface the difference of the pressures and the divergence of singlite conditions so assume that we have a solution of this and so I need to describe the precise assumptions 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 0 the picture is better so the time t is equal to 0 there is no there is no contact in the interface and we assume that the time t is equal to 0 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 0 then the conclusion is that as long as the equations make sense which means that as long as we don't have 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-chord constant which means that 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 split 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 singularities 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 but the other part that's left is that the reason why this mechanism of creating a splash while keeping things 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 would be approaching in a way that's quadratic because there are two curves that approach that would be approaching and if they are smooth then the distance between them is too small and it 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 the picture this way but at a later time as we approach it will start creating a corner you can picture that if it starts creating a corner then it leaves more time for the for the fluid in the middle to get out of the way and I think that's quite possible if we prescribe data that would want to push the fluid in the direction towards self-touching then 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 hopefully will that could potentially create a singularity curvature singularity and now I'm going to describe quickly what's involved in the proof 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 closed by a log so I'm going to describe the two main ideas that we have in the proof of this theorem and really two ideas one of them is dynamical the other one is at one constant time so analysis at one constant time the 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 Birchhoff-Rott operator and can express the two vorticities on the two sides in terms of the Birchhoff-Rott operator of this function omega and we have this formula as v1 is Birchhoff-Rott of omega plus omega over 2 and v2 is Birchhoff-Rott of omega minus omega over 2 and then if we write the two 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 and we get a Burgers-like equation and the Burgers equation is something of the form dT omega is equal to 1 half d alpha omega squared plus some kind of multiples of d alpha f omega in some simpler terms and now we can think about this as a Burgers equation and for the Burgers equation we can preserve the strongest one we can preserve for omega is the infinity norm 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 1 and alpha 2 so if we have two points in which the distance between them is small then we want to be able to say that the distance the difference between the normal velocities of those points would have to be small and small by with the same constant so if the distance is smaller than epsilon then the difference between the velocities have to be smaller than epsilon and because of that we lose a log of 1 over epsilon that has to do with the fact that all of these operas are essentially they are essentially Hilbert transform at best they are Hilbert transforms in fact some kind of variable coefficient Hilbert transforms and we have to work on an infinity because this is the best information that we have about the boundary vorticity so that leads to a logarithmic a loss of log of 1 over epsilon and 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 t1f of omega plus omega plus has to do with one side and the things with plus leave on the upside 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 so that and we have the the velocity so this so f plus plays the role of the velocity so you obtain the velocity out of the so this Hilbert transforms this operas t1 and t3 they are written on this slide and they are essentially the normalized versions of the Birchhoff-Roth operator I had earlier so the Birchhoff-Roth operator if we write it in this picture 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 so we take out the essential parts of the of the Birchhoff-Roth operators and we write them so we write these operators that are the parts of the Birchhoff-Roth operators after these normalizations and the system that so the information that we get is containing 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 the two velocities that you want to show they are 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 it fix the time p we just have to produce a fixed time p 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 we know that functions f plus and f minus they are smooth as well and out of this 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 omega 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 functions but it's not possible to get information about the derivatives so we don't have good information about the derivatives we only know the infinity norm about this vorticity so we have to run an argument according to this 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 with 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 are also they are not exactly they are available coefficient Hilbert transforms coefficients that depend on these 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 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 as in a duality way and then it turns out it's very important to get the to get the bound on g to be epsilon log of one over epsilon we cannot lose more than log of one over epsilon if it was like epsilon to the one half then to not have prevented the splash it has to be epsilon and the most we can lose this 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 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 one would like to see dynamically is that as the interface starts approaching towards splashing it would also have to create a corner and that's now that seems to be pretty tough so we have some work in progress one feature of this model that that I like which is better than other one could ask the same question for other models 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 show to 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 only way the evolution will finish is by creating a singularity because otherwise you would otherwise you would violate this theorem otherwise you would create a splash without forming the singularity so that's now of course the difficulty what's hard and haven't been able to do is to control the flow one needs to control one needs to be able to say that this interface advances for long enough and it gets hard to control the flow as you lose smoothness so the more smoothness you lose in a problem so it's hard to control the flow one has to use 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 for the model in which one has only one fluid we know that the splash can be created by without without losing smoothness so it's not possible to have this mechanism there okay so I'm going to stop here thank you Is it possible to try and solve backwards from the corner situation? Yeah if we could find the good answers to it's possible it's not so easy to do it but it's I mean of course that's how the theorem for the splash was proved by solving backwards now one can now there are more pros 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 to separate I'd love to be able to tell you it's the kind of there are several things ultimate my feeling is that the only way you can really do this is to some sort of monotonicity because if you start 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 it's at the end of the day this way you can say that the problem is well posed in some coordinates but what sometimes works but it's some sort of monotonicity which you are working on understanding you have another question if you permit me which is I know it looks very different but it does nonetheless remind me of Delayave-Pfeffermann's 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 norm of the velocity you're studying something else but is that a geometric configuration forcing say the vorticity to infinity it's just it's almost a geometrical picture it's making the interior fluid the lighter fluid as you draw it move fast near the boundary walls and so omega is getting larger this is not the only scenario that 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 Feffermann 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 the curvature singularities but I think Charlie is so I had a lot of discussions with Charlie Feffermann about this and we could never really see exactly one reason why one is more likely than the other