 So, first I'd like to thank the organizers for the invitation to speak here. And also thank them for putting together this very nice program in here. I have enjoyed very much the time spent at IHES and benefited from it. The topic of my lectures is two-dimensional water waves. And for those of you who are not familiar with the subject, water waves are the waves that you see forming on the surface of the water and, of course, are the fluids, but most importantly, it's water that we have in mind here. Now, when you think about water waves, you will most naturally think about water waves in our physical space in three-space dimensions. The two-dimensional water wave equations would be, you could think of those as, for instance, three-dimensional water waves which do not depend on one of the variables. So there are special cases of three-dimensional water waves. Maybe I owe you a bit of an explanation why am I talking about two-dimensional water waves and not about also higher-dimensional water waves. And fortunately, some of the motivation for that was already given in Nader Masmoudi's talks. There are significant differences between two and three-dimensions when you look at fluids. And particularly, as far as the problems that I will try to tell you about are concerned, two-dimensional water waves and three-dimensional water waves are very different both in the way you set up your equations and in the kind of issues that you are facing in the analysis. What is common, though, between two-dimensional water waves and three-dimensional water waves is the large set of problems that you want to look at. And later today, I'll spend some time trying to give you an overview of maybe what are, in my view, the interesting problems. Those are the same for two-dimensional and three-dimensional water waves. But other than that, the setup of the equations is different and the technical issues that you face are different. Now, the work that I'll tell you about is collaborative work. And my collaborators for this stuff are, first of all, Mihail I. Frim. For all of the stuff that I will tell you about and then for some portions of this work, two other collaborators, John Hunter and then Benjamin Harrow Griffiths. There's an S here at the end. Sometimes I forget to write it, but this time it just doesn't fit. And just for your curiosity, John was Mihail as PhD advisor and Ben was my graduate student. All right. So I'm not going to assume you know about water waves and I'll try to give you some idea of where the equations come from. But I will not be able to do complete computations for time restrictions. And so if you will want to see more details, I'll tell you where to look. And maybe just to start with, for a general introduction to the subject, there are two recent books that I would recommend. So one book which is devoted to water wave equations to their sort of general analysis is the book of David Lan. And another book, but both books have appeared within the last two or three years. And another book which is devoted to more to a topic that I will not touch very much, but which is very hot if you want in connection to water waves, the topic of traveling waves of solitons if you want. And that's a book by Adrian Constantin. Of course, there are lots of other books on the subject, but these are two of the most recent and they somehow tie into what I'll have to tell you. And if you know nothing about water waves, reading maybe the first chapter in each book, we'll give you some idea about what the problems are. We can now have to figure out the boards. So there are two on the side, right? And three in the middle. All right, it doesn't go any higher in any case. So before I tell you about water waves, I have to start with the incompressible Euler equations. So the incompressible Euler equations will model the evolution of an incompressible fluid. My main variable will be capital U, which is going to be the velocity. Here, for now, the dimension doesn't matter very much. And then the equations for U will be U t plus U gradient of U is equal to minus gradient of P minus, and here I'll introduce the small letter G for gravity, J, where J is the unit vector point in the vertical direction. Maybe that's a specific notation for two dimensions. And the picture that we'll have in mind is, so this would be, and of course, we have the divergence of recondition, which is related to the incompressibility. And so if you are looking at Euler equations in all of R2 or all of Rn, these are the equations that you have to consider. But now the picture that we'll look at will be instead the picture where the water does not cover all of R2 or all of Rn, but instead what you have is some free surface, which is the interface between water and air. So this is air and this is water. And maybe here you'll also have a bottom, right? And so now the function U that describes the velocity of the fluid is defined in this region between the water-air interface and the bottom. Let's call this region omega of t. So this region will depend on t because the interface between air and water will move. And let me call this interface gamma of t. And so now U is a function from omega of t into, and let me stick with the two dimensions, which is what I want to talk about in two dimensions, the velocity field. And so now once you, when you try to set up your problem in a domain like this, you're going to have boundary conditions. And you're going to have boundary conditions on the top and boundary conditions on the bottom. And the interesting boundary conditions are on the top. But just to get this out of the way, let me say one word about the bottom. So here we're simply assuming that the fluid does not penetrate the bottom. And what this means about the velocity vector field is that when you are on the bottom, the velocity vector field has to be tangent to the bottom. So this means that U is tangent to the bottom. And then the second, now, a set of boundary conditions, the interesting ones happen on the top, all right? And on the top you have two boundary conditions, and they have names. One will be what is called the kinematic boundary condition. And this kinematic boundary condition has to do with how the free interface between the water and air moves. So this moves with respect to time. But now what happens with the particles which are on the surface at some given time? Will they go inside the fluid or will they stay on top? And the kinematic boundary condition tells you that particles which are initially on the top, they stay on the interface. So in other words, the interface, if you want, moves with velocity. And later on, we'll also write down a formula for this. But for now, let's just keep this very simple interpretation in mind. And then there is the second boundary condition. That's what we call a dynamic boundary condition. And this dynamic boundary condition, if you want, is nothing but the balance of forces on the surface of the fluid. Precisely, the air will have a certain pressure, atmospheric pressure. And then the water will have a different pressure. And you'd expect these two pressures to be equal, right? And that's usually the case, unless you also assume that there is a surface tension at the surface of the water. Now what does the surface tension do? The surface tension tries to keep the surface of the water flat, okay? So if the surface of the water is very curved, then the surface tension will try to flatten it. And so what will matter in terms of the surface tension is the curvature of the water surface. And so the relation, let me make sure I get the correct sign. The relation is that the P, P will be the pressure of the fluid, is equal to P0, P0 will be the atmospheric pressure. And if you have surface tension, then here, you have to add the effect of the surface tension, sigma H. Sigma plays a role of a surface tension coefficient. So let me say, and since I'm making, let me also write down G, which is the other important parameter in our analysis. This is graph, okay? And then H will denote the mean curvature of gamma t. All right, so we have the evolution inside given by the incompressible equations together with the three boundary conditions, one boundary condition on the bottom, and two boundary conditions on the top. We're not yet at water waves. This is if you want full fluid equations. And when you look at this evolution, you have to track two things really. You have to track the evolution of the velocity field inside the domain. But you also have to track the evolution of the domain, the evolution of this free boundary. This is not an easy problem. And there are people in the audience who have studied this problem, or postedness of this problem, maybe Hans Limblat, Limblat and Christodulu, then Kutard and Scholar, David Land, whose book I mentioned. So there's been work on this, but this will not be what I want to tell you about. Instead, what we'll do to get to water waves, we're going to specialize to one class of flows. And maybe I can put something more on this board. So one object that plays an important role in the analysis of two-dimensional flows, and also three-dimensionals, but even more in two-dimensional flows, is the vorticity. So if omega is the vorticity, this curl of U, then the vorticity will solve a transport equation. This is a scalar function. So omega t plus U dot gradient of omega is equal to 0. So omega is transported along the fluid flow. And so this leads us to an important class of fluids, and those are the irrotational fluids. Omega is equal to 0. That's what we call irrotational. And these are the class of flows that I want to really tell you about. So all right. So when you look at the irrotational flow, you have two relations which are satisfied by the velocity vector field U. One is that the divergence of U is equal to 0. And the other one is that curl of U is equal to 0. And so clearly the two components of U are not independent. And what one usually does in a situation like this is you try to characterize the two components of U, which are not independent, by a single scalar function. And that's what's called the velocity potential. And so this is a scalar function phi. If you want phi in our setting there will be defined from omega of t with values in r. You can define this at every time. And this is defined so that the function U is equal to gradient of phi. Now there's another function that one, another scalar function that one can associate to a flow even if it's not irrotational. And this will also play a role in my discussion. So let me introduce it here. That's what's called the stream function. And we're going to denote that by capital theta. And so this is defined so that U is the gradient perpendicular of theta. This is in two dimensions. And so once your velocity vector field is represented in terms of the velocity potential, then it's natural that you can try to write down your equations in terms of the velocity potential. Now if you look at the definition of the velocity potential and you remember that divergence of U is equal to 0, then for the function phi, for the velocity potential, you're going to get a Laplace equation. So phi will be Laplacian of phi is equal to 0 in omega of t. And if the fluid happens to have a bottom, and some of the stuff that I'll tell you about the fluid will have a bottom, and in some stuff I'll tell you the fluid will have an infinite bottom. So there's no such boundary condition. But if there is one such, then what is the corresponding boundary condition for phi? And since all these hypotheses are invariant with respect to rotations, it's not hard to see that the reflection of our first boundary condition on the bottom, that U is tangent to the bottom, is simply the fact that d phi dn, the normal derivative of phi, is equal to 0 on the bottom. So we have a harmonic function in the domain. Satisfies Neumann boundary condition on the bottom. And so this means by standard deliptic theory that the function phi is actually determined by its values on the top. So if I know phi restricted to gamma of t, that will tell me what is phi restricted to the entire domain omega of t. All right? And so once you have this piece of information, it's very natural to think of the evolution really as the evolution of the top. So if you look at water waves, you don't need to look at the entire fluid body, but instead you want to look just at the interface between fluid and water. And here you're going to have two variables if you want. One variable, which is the interface gamma of t between the fluid and the water. And the other variable is the velocity potential phi restricted to gamma of t. And if you have these two pieces of information at any given time, then you should be able to compute the velocity at that given time. And you should be able to see how in turn these two quantities vary in time. So my first goal for you would be to show you that this actually happens and how we can write down the equations for gamma of t and phi. And for the moment, let me stick to what is called the Eulerian formulation of the problem. And I will assume that the interface gamma of t is a graph and that this graph can be written as a function. So y, which is the height, is eta of x. And then let me call the restriction of phi to the free boundary to be called psi. So c will be now a function of x because the free boundary is parametrized by x. So now we have the set of two functions, eta and psi. And we want to see what is the evolution equation for these two guys. All right. And so the evolution equation for eta and psi will come from the two boundary conditions that I have shown you before on the top. You have the kinematic boundary condition, the dynamic boundary condition. And we want to see what is the system of equations for eta and psi. And now if you look at the relation, and I'm not going to do the full computation here. I'm just planning to wave my hands a little bit. But I'll tell you why. So why you get the equations that I'll write in a moment. So if you look at the definition of u at the velocity vector field in terms of the velocity potential, you see that it requires the full gradient of the velocity potential. Now if the initial data that I give you is just the restriction of the velocity potential to the free boundary, that by itself does not give you the full gradient. If you want to get the full gradient of phi, you also need to know what is the normal derivative of phi restricted to the free boundary. How do you get the normal derivative of phi? Well, you solve the Laplace equation with this boundary condition on top and the Neumann boundary condition on the bottom. And that will give you the normal derivative of phi. And this operator that takes you from phi to the normal derivative of phi is a very important object in this story. It's called the Dirichlet to Neumann map associated to the domain omega of t. So let me make sure I have consistent notations. So I'll use the letter g for this. So g takes the function phi restricted to gamma of t, which would be noted by psi, into g of psi, which is the normal derivative of phi restricted to gamma of t. All right? What kind of operator is this operator g? And it's clear that it's not going to be such a simple operator to deal with, because the domain is not a simple domain. You have one boundary that you don't know. So g, in particular, depends on what the domain is, of course. So it's g of eta, really. So it depends on one of your unknown functions. And furthermore, g is not a differential operator. It's a non-local operator. And if you want to think of this in micro-local terms, it'll be a pseudo-differential operator of order 1. And in a suitable setting, you can think of g as being like absolute value of t. But this, of course, will depend very much on how you parametrize your free boundary. So let me not use the equal sign here. But that's a good starting point if you want for this. You have to keep in mind that g depends on the function eta. All right. So once we have this Dirichlet to Neumann map, you can write down what those kinematic and the dynamic boundary conditions mean. And let me start with the dynamic boundary conditions. And that's for a very simple reason. In terms of the dynamic boundary conditions, actually, before you get to the dynamic boundary conditions, you look at the expressions in the incompressible Euler equations. And you see that in this context of irrotational flows, you can write down each of these expressions in here as a gradient. So this will be the gradient of phi. This will be the gradient, essentially, of phi square. This will be the gradient is the gradient of p. This will be the gradient of the y function. So if all of these expressions in here are gradients, this means that you can integrate this equation. And so if you integrate the Euler equation, then what you're going to get is what's called the Bernoulli equation. And the Bernoulli equation, and let me make sure I get all the signs right, is an equation for, essentially, for phi t plus 1 half gradient phi square plus y. y is the vertical variable plus p is equal to 0. And you'll say, why am I setting this equal to 0? And one thing that you can be a little bit fast and loose when you talk about these equations is the constants. And the reason for that is that, first of all, when you're writing incompressible Euler equations, the constant in p is irrelevant, right? When you write, therefore, the dynamic boundary condition in there, the constant in p cannot have an intrinsic meaning. The constant in p0 cannot have an intrinsic meaning, so the constants you can essentially discard. And so here, for a suitable choice of the pressure, you can set this equal to 0 if you want. And now the next thing you do is you combine this Bernoulli equation with a dynamic boundary condition, and you get one of the two equations that govern the motion of the interface and the velocity potential restricted to the interface. And did you decide that gravity is 1 plus 4? No, actually not. Thank you. And I will not decide that gravity is 1. So if I do this again, please correct me again. All right. So now I'm going to write for you the water wave system. And as I said, this is a system for the evolution of phi and eta. And let me make sure, again, I get the signs right. OK, so this is the evolution of eta. This is the equation that gives you the kinematic boundary condition. It tells you that the interface moves with the speed that's somehow given by the normal derivative. So this is the normal derivative of the phi of the velocity potential. And then the second equation in here is the equation that you derive by combining the Bernoulli equations with the dynamic boundary conditions. But again, when you look at the Bernoulli equations, here you have the gradient of phi. And so if you want to express this in terms of fear restricted to the boundary, this will also involve the Dirichlet-Tunai-Mann map. You need the normal derivative of phi. And so the full expression that you get will be like this. So this is psi t minus sigma h of eta plus g eta. So for now, I have incorporated the effect of the surface tension, the effect of gravity. And now the effect of, if you want the gradient of phi square combined with the pressure, that gives plus 1 half grad phi square. And I'm worried that I'll run again out of space. So minus 1 half. And the expression that you get in here is grad phi grad eta. This is a dot product plus g of eta psi. Sorry, everything is psi in here. Psi square. And this gets divided by 1 plus grad eta square, OK? So this is what is called, and of course, to finish this, this is equal to 0, all right? Well, this is what is called the Eulerian formulation of the water wave equation. So this is the equations I want to tell you about. Except I'm not going to use this formulation of the equations to look at water waves. And I'll tell you in a moment why. All right. So before I go any further, I want to show you some features of this equation. And really, if you want to do just one computation, the simplest computation that you could do, what you should look at is the linearization of these equations around 0. So 0 means that your interface is flat, and the velocity is 0. And then you want to compute this linearization. And this computation also helped me with something else, namely that hopefully you all know that the incompressible Euler equations are not dispersive. But this is a program about dispersive equations. So when we compute this linearization, you're going to see that the linearized equations at least are dispersive. So then the nonlinear equations are also dispersive, which is very, very fortunate. So what is the linearization of the first equation? Well, when eta is flat, then the Dirichlet-Neumann map is exactly the absolute value of d. So the first equation becomes, and I'll use, hope you'll forgive me for using the same variables, eta t minus d c is equal to 0. So nicer operator, but still non-local. And from the second equation, psi t minus sigma. And now what is this mean curvature of the free boundary? Of course, we're in two dimensions. So in two dimensions, the free curvature is a curve. We're looking at the curvature of a curve. So that's just it, the curvature. And that will be essentially at the linear level the second derivative of eta. So minus sigma d2 of eta. And then plus g eta, the effect of gravity. And all the other terms that you see in here are at least quadratic. So we wouldn't care about them. So this is equal to 0. All right. It's not so immediate if you want to see what is the character of this equation. But one thing you can do is you can write down a second order for eta. So I can write down the equation eta tt by simply differentiating this equation once more with respect to time. And this will be, well, let me make sure I get my signs right. So it will be minus sigma. It will be d cubed plus g d of eta. And so if you look at the principal symbol of this associated to this equation in here, let's use the letters tau and psi for the Fourier variables. Tau will be for the time Fourier variable. Xi will be for the space Fourier variables. What you're going to see in here is that the principal symbol for this will be tau square minus sigma psi cube minus g psi. So this is the principal symbol. And so the important thing in here is whether this vanishes or not and whether the roots of this as a polynomial in tau are real. And it's a very simple analysis. If the roots are real, then your equation will be essentially a dispersive type equation because this symbol is not a linear symbol. And if the roots are not real, then in principle, this equation is going to be ill-posed. So to be more precise, if you're on the leading role in this analysis, it's played by sigma because it's psi cube is larger than psi. And so the first thing that you can say is that if you want to have well-posedness, and this is well-posedness in any sobola space if you want, the first thing you need to know is that sigma is greater than or equal to 0. And then if you are in the special case when sigma is equal to 0, then g becomes important. And then you want g to be greater than or equal to 0. You also see in this computation that the case when sigma and g are both equal to 0, when you have no gravity and no surface tension, is a very degenerate case. So you can think of that as a case when these equations are indeed very weakly coupled. And that's a case where your equations are more like the incompressible Euler equations where you're not really going to have much to say in terms of dispersion. So for the purpose of the stock and the purpose of this program, therefore, I want to stick to the case when either sigma is positive or sigma is 0 and g is positive. All right. So this is one thing. The other important object that plays a role in this story is it has to do with the balance of forces on the surface of the water. And that's really given by the balance of pressures. And then the object that plays a role there is the normal derivative of the pressure. And what role does this play? Well, you can think of, so I'm going to give you two motivations of why this is important and what role it plays in our story. So one motivation is the following, that if the normal derivative of the pressure has the wrong sign, then the water particles that are on top of the water are pushed somehow into the air. And you get what you might call turbulence. And the water surface will not stay coherent. So that will tell you that from physical considerations that you need the proper sign somehow for the normal derivative of pressure so that the particles on top of the fluid are pressed into the fluid. They will not go out into the air. There is a second interpretation for this, which has to do with the way I wrote the linearized equations. So if you look at this second term in here, in the case that the interface is flat, the pressure in the fluid is given exactly by g y. So this g plays the role of the normal derivative of the pressure. So if the interface is flat, if the velocity is 0, then g is exactly the normal derivative of the pressure. And I'll show you a little bit later that when you compute the linearized water wave equations, you're not going to get something like g in here in the principal symbol. But instead, you're going to get something like dp dm, the normal derivative of the pressure. So everything that I said about g before really is not something that applies to g in general, but it's something that applies to the normal derivative of the pressure. So especially if you have no surface tension, you care a lot about this sign condition about the normal derivative of the pressure having the correct sign. So let me make a choice in here. And of course, the choice that you make depends on what you define your normal. And I'm sure that if I start caring about signs at some point, I'm going to mess them up. So let me be a little bit faster and lose with signs. And so this is called the Taylor sign condition. And so what Taylor observed, and I think also some other people, but I don't remember their names, is the following that at least if you look at the linearized equations, this is a necessary condition for well-posedness. So not only that you, so these equations, in other words, these equations will not be in general well-posed unless you have something like this Taylor sign condition. All right, so let's see the next issue that I would like to talk about a little bit. And I still want to keep the Euler equations for now. So what it has to do with the question of coordinates. So one thing that you already saw in Nader's talks is that it's important when you look at fluid problems to work in a proper frame of coordinates, because some issues that may arise might just be a matter of having a poor choice of coordinates. Similar issues arise in lots of other problems, for instance, in general activity, in really any problem that has some gauge invariance where you need to fix the gauge. And so it also plays an important role here. And so the question is the following. We have this system of equations which modeled the evolution of an interface and of some function restricted to this interface. And the question is how do I parametrize the interface? What is the best way to parametrize the interface? So how to? And you already saw here implicitly one choice of such a parametrization. And that's the Eulerian parametrization. And in the Eulerian parametrization, at each moment in time and each position in space, you look at the values of phi and eta, psi and eta at that position. And you parametrize the surface eta, as I wrote it in there, in terms of the Euclidean coordinate y at any given time. There I should have written really y is eta of x and t. And everything should also be a function of x and t. So you might say the Eulerian parametrization is the most natural one. But we also know from other problems that not always the most natural parametrization is the best one for the problem. And another classical parametrization is the Lagrangian. And in Lagrangian coordinates, what you do is you fix a frame at the initial time. So you're going to have some initial configuration of the fluid. And then for each particle of the fluid, you move your frame according to the way the particle moves. So one big difference between the Eulerian and Lagrangian coordinates is that if you look at the even the Euler equations in Eulerian coordinates, you're going to see the transport equation, right? The transport along the vector field. But when you look at the Lagrangian coordinates, this transport disappears because this is built into the coordinates. So here you have transport. But here you have no transport. So you might say, oh, we should go for Lagrangian coordinates, right? However, when you look at Lagrangian coordinates, what happens is as you move along the particle path, your frame of reference will be twisted, right? On the other hand, here you end up having to solve a Laplace equation. But if you solve a Laplace equation in a twisted frame, that's as if you were to solve a Laplace equation in variable coefficients. So you need to do a lot of work to move this Laplacian from one coordinate frame to the other. So what you have gained in one place, you have lost in the other place. So here you have the disadvantage that you have a twisted or transported frame. So from this perspective, neither of these two coordinate systems are ideal. And I will not tell you about either of these two coordinate frames. Instead, I'll tell you about a third choice, which is what I'll call conformal or polymorphic coordinates. And what is the motivation for, in the first place, for looking for another system of coordinates here? It's a very simple motivation, namely that if you look at these equations in here, the one object that seems maybe harder to grasp in here is the Dirichlet to Neumann map. It's non-local, pseudo-differential variable coefficients, all of that. Is it possible to diagonalize the Dirichlet to Neumann map? It would be great if instead of having to work with the Dirichlet to Neumann map as a variable coefficient to a differential operator, we could simply work with this operator. And so you ask, can I make a change of coordinates so that the Dirichlet to Neumann map becomes as simple as possible? And the idea there is very simple, namely, let's see, I'm going to save for now the board on the top. So the idea is that you can have your fluid domain, which is omega of t. And you can conformally map this into a model domain, which could be a strip, maybe if you have the bottom, or could be a half plane if you did not have a bottom. And in two dimensions, such a map will exist. It will be a holomorphic map between these two domains. It will be given by the Riemann mapping theorem. And so in this domain, you will know what the Dirichlet to Neumann map is. It will be a fixed object. It will not depend on anything. It will depend just on the choice of this domain. And as long as you keep this domain fixed, that's sort of a God-given object. Whereas here, it was dependent on your configuration. And so this is what I mean by conformal coordinates. Historically, I think the conformal coordinates are maybe close to 100 years old. And initially, maybe my knowledge of history is incomplete. But as I understand it, initially, conformal coordinates were used in order to study special solutions to water wave equations, namely traveling waves. So traveling wave is a wave that keeps its profile over time, just moves with a constant speed. There was also a second form of using this conformal coordinates. So implicitly in what I was telling you, the conformal coordinates are coordinates where the variable that changes is the domain. So you're modeling the domain in a different way. And this means essentially you're working with the variable eta. But you can also sort of twist the roles of the variables eta and psi and work with psi in modeling this change of coordinates. And then you get something that's called the hodograph transform. That's another way of looking at this conformal method. This is not something that I will mention about. And since I'm talking about the holomorphic coordinates, in terms of using the conformal coordinates to study the water wave equations, let me give you a few names. So the first reference that we found using this conformal coordinates for water waves is a Russian mathematician of Sianikov. That was in the 60s, I think, possibly early 70s. And then independently, CJ Wu and Dachenko and Kuznetsov and Spector and Zaharov computed the evolution of the two-dimensional water waves in conformal coordinates. Their booking keeping was sort of different. And so they ended up with slightly different looking sets of equations. I think CJ wrote the equations more like a second-order evolution in time. And the Zaharov group, they wrote the equations as a system more akin to this system that you see in here. But in any case, the idea was there to exploit this conformal invariance of the Laplacian into space dimensions. Now, that brings me to an important remark here, which is that this method only works so nicely in two-space dimensions. And that's one reason for me to stick to two-space dimensions. So CJ also has some version of this that applies in three-space dimensions, but it's far less intuitive and elegant, just because it's in two dimensions that any two domains are conformally equivalent. All right, so now that I got to describe for you a little bit the water wave equations, what I want to briefly outline are what I think of as the main problems that are interesting to look at. And I should say from the beginning that there are lots and lots of interesting problems in here. I think we have only touched the tip of the iceberg so far. This problems are not easy. And you see some reasons why these problems are not easy, because on one hand your equations are, well, they are dispersive. They are fully nonlinear equations. They have some, if you want, some gauge invariance. You have a number of parameters at your disposal, the surface tension, the gravity. You can throw in more parameters if you're not happy with this too. And then you have also the matter of whether you have a bottom or not for your fluid. So lots of freedom in choosing your equations. And this is why sometimes, actually, when you look at the literature, it's hard to say what is what, because people assume that you know what they are talking about from the beginning. And it takes a while to place this or that result into the broader context. But at least that's one problem that I have encountered. So on my list of interesting questions, of course, the first one for any PD you might want to look at is the question of local well-posedness. And you can ask for the local well-posedness in many contexts. You can ask for local well-posedness for small data. You can ask for local well-posedness for large data. You can ask for local well-posedness in all of this context that I have discussed. Maybe one interesting question in terms of local well-posedness is to look at low-regularity solutions and to try to understand what is a good local theory for these problems. Then the second question, once you have your local solutions, you can ask, how long are these solutions lasting? And one easier setup that one can consider to start with is what happens if you have small data. And what happens if you have small data, you expect your solutions to last longer in time. So maybe lifespan bounds for small data. Another related question to this is the following. So I was just telling you that these equations are dispersive. And so if you start with the input at some point, you expect a concentrated input. You expect this input to resolve into waves and these waves to travel in different directions. And there is some implicit decay in this picture because one wave is spreading into many waves and decaying in time. And so one common problem when you look at nonlinear equations is what is the game between dispersive decay and nonlinear growth? And so taking this into account, another interesting question in here would be maybe to look at long time solutions, small, and localized data. And notice that these two questions are sort of different from each other because here you start with your waves concentrated and they spread out. Whereas here, maybe you're not assuming that your wave is concentrated to start with. So interaction of waves can occur at later times. And it can happen not at a single time, but at multiple times depending on where your waves have originated. You might ask here for something stronger. You might ask for global solutions depending on the problem you're looking at. You might or might not get global solutions. And of course, the opposite of getting long time solutions is the question of blow up. And maybe here I'll say a few words more about blow up now because I'll never say anything about it again during this lecture. So what are your enemies here in terms of blow up? What are the things that could go wrong? And you know what could go wrong for a typical nonlinear PDE, which is that your solution grows, right? It grows and it loses regularity eventually. So that would be an obvious enemy here. So growth loss of regularity. But this is not all that could happen for water wave equations. Something else that could happen for water wave is something that you have surely experienced yourself going to the beach, which is that waves can overturn. So you can have maybe a picture like this where the two waves will eventually touch. And here is where when they touch, you will lose this nice structure and you'll have some turbulence appearing. So this is what you might call splash singularities. I will say something about this a little bit later. And finally, there's one third enemy that you have to be aware of here. I was telling you that in order for the linearized equation at least to be well-posed, you need to have this Taylor sign condition. And so that's the third thing that you need to track, whether the Taylor sign condition remains satisfied or not. And what you expect is that, well, at least my understanding of it, is that when at the point where the Taylor sign condition becomes violated, you get some instability of the water surface. And you're beginning to see the white foam that you see when you go to the beach. And so let me write here simply Taylor where the sign of the normal derivative of the pressure goes wrong. So you need to worry about these three objects separately. And this is sort of a specific of the water wave equations. And I'll say something a little bit more about the splash and about the Taylor sign condition, maybe not much about the growth and loss of regularity. And maybe the last item on my list, and I should say last but not least, because books have been written about it, is when you look at special solutions. Time to get a new chalk also. And here in the category of special solutions, I would include maybe two objects. One object would be traveling waves, solitary waves. Of course, solitary waves play a big role in the understanding of the long-term dynamics of any dispersive equation which has them. Some of the equations I will tell you about have solitary waves and some don't, and I will tell you which is what. And related sort of, you can also look at waves, you can also look at waves which have angular crests. And in particular, one of the most famous solitary waves for the water wave equation has an angular crest. There's the so-called Stokes wave, which has an angle of 120 degrees on the top. I will also not say anything about this anymore, but I saw that CJ will tell us maybe a lot more about this topic. So this is the list of questions, and I'm sure if you think a little bit more about it, you can add to it. But the three things that I will touch in my talk are the first three. So we're going to stick with the good solutions and not worry about what happens when they go bad. All right. So I was telling you about my collaborators for this work. And as I said, so far I think we have only touched the tip of the iceberg. And so far we have looked at four different problems. And at various times, I'll say things about these four different problems. So the first problem that we looked at is the problem of when you take the surface tension equal to 0, positive gravity, and infinite bottom. And as it turns out, this was a good idea, because this is the easiest of them all. We did not realize that at the time, so in some sense we got lucky. And this is joint work with the Mihaly Freeman, partially with the John Hunter. Second problem we looked at was when you reverse the rows, sigma greater than 0, g is equal to 0, and again, infinite bottom. And this is joint work with the Mihaly. The third problem we looked at in a brilliant move I have just erased the Euler equations one minute before I should have done it. We can just put the equation here. Yeah. So, ah, but I still have that board over there, yes. And so I was telling you about the role played by the vorticity and that the vorticity is transported along the flow. And I was also telling you that water waves are the waves for which the vorticity is 0, because that's where you can reduce the dynamics to how the top is moving. Well, I liked a little bit about that. There is another simple model where you can still reduce the equations to the evolution of the top. And that's the case when the vorticity is a constant. It's constant throughout the fluid domain. And the reason for that is that if the vorticity is constant, then it will stay constant as time evolves. And then you can still do the same kind of analysis, introduce some sort of modified velocity potential, and still write down some equations very similar to those equations and reduce the problem to the boundary. So this would be the case when sigma is equal to 0. You have non-zero gravity. But now the new twist is that you have constant vorticity. And again, you have infinite bottom. And so this too, as I said, are joint work with the Mihaila. And the last problem that we started to look at, and actually we finished the first paper in that direction just a couple of days ago, is the case when, again, you look at the gravity wave problem. But now you take a finite bottom. So this sort of completes the circle back to the first project because you can think of the infinite bottom problem as a limit of the finite bottom problem as the depth goes to infinity. And in effect, one of the things we were very careful about when looking at the finite bottom problem was that the results that we prove are uniform as the depth of the fluid goes to infinity, so that you recover results here as a limit of the results here. So this is something that we don't think was really done before. And in David Land's book, this is listed as one of the interesting problems. Of course, we were not proving everything about that, but still we made sure that whatever we proved survives in the infinite depth limit. All right. And I should say one thing, what is that differentiate these problems? What are some of this, why are some of these problems easier and some of these problems harder? And I will, I'll say one word about this. Well, I'll say a couple of things about this. First of all, in terms of the enemies that you face, and maybe one thing to consider is the Taylor sign condition. So the question is whether you a priori have a Taylor sign condition satisfied in these problems. And the answer is yes and no. In the case of infinite bottom with the gravity waves, we know that the Taylor sign condition always holds. This is a result of CJ Wu. What is the size? No, this is the infinite bottom. No, no, no, no, no, no, no. Independently of the size of the data, yes. And in effect, when you look at surface tension, but even with or without gravity, the proof of the Taylor sign condition remains the same. So all that really matters here is that you have an infinite bottom. So as I said, CJ gave the first proof of this. And what we ended up doing is giving an alternate proof of this. And you can think of the two proofs as one happening in the physical space and our proof happening on the Fourier side of things. The problem with constant vorticity, at least if the vorticity becomes large, we don't know that the Taylor sign conditions stay satisfied. So who knows what happens? Maybe somebody will prove it later. But at least we were not able to prove it. At least not unless maybe it sees his very small. By the way, so still coming back to this problem, you might ask, where do such waves occur? Where are they interesting? And at least according to something that I read by Adjan Konstantin, he claims that such waves occur when you have some tidal action on your fluid. And finally, the fourth case, when you have finite bottom, and then it doesn't matter what you put in here, you can ask the same question. And the question becomes different from the question in infinite bottom. And so we were very happy to be able to give a proof of the Taylor sign condition. And then we discovered that actually a proof was given before by David Lan. Fortunately, our proof is different and also seems to be more general. It applies in a broader context. So one question that one can ask in terms of this Taylor sign condition is whether the Taylor sign condition holds, even if your interface is self-intersecting. And our proof being Fourier analysis base has nothing to do with the interface self-intersecting, whereas the other proofs in the physical space do use this properties of the interface. So in other words, we can consider waves that go like this and still know that the Taylor sign condition is satisfied now. What is the meaning of that? I will leave it to your imagination. And the last thing that I wanted to mention in terms of all of these problems is that you can consider two more things, actually. Two more things. So the next thing that I was telling you that is important in the long-time analysis of these problems is whether you have solitons or not, and in particular whether you can have small solitons. And the first two problems turns out they do not have small solitons. The first one doesn't have any solitons at all. We don't know whether the second one has large solitons or not, but it definitely does not have small solitons. These two are the problems, and I will comment on this later, have small solitons. So in particular, you cannot hope to prove that you have maybe global solutions which are dispersing as time goes to infinity, because some solitons might emerge in there. And the last thing that I wanted to mention today in terms of these problems is that even as you can still consider them in two varieties, and people have considered them in two varieties, one could look at periodic problems and non-periodic problems versus non-periodic. So periodic means you look at the problems, say, on the torus, and non-periodic if you look at the problems on the real line. And you're going to see some differences. For instance, even this problem will have solitons if you look at it in the periodic case, but not solitons if you look at it on the entire real line. And of course, when you look at the periodic problem, you're not going to worry about this picture anymore because your domain is periodic, so what goes out comes back in. You cannot expect the dispersion mechanism to be so strong. All right, so I will continue next time. Thank you. Any comments? Yes, sir. Are there any particular assumptions you make on the bottom of real line? Do you have to be smooth or do you have to be flat? In all the work that I will tell you about, we'll assume that the bottom is actually flat. So as you see, there's one single case in here where we assume that there's a bottom, and we're assuming that the bottom is flat. And there's a reason for that because we want to look at the long-time dynamics of the solutions, and a bottom that's not flat will mess that up. If you want to look just at the local oposeness problem, the shape of the bottom is meaningless. If you want to look at long-time dynamics, having perturbations on the bottom will affect the low frequencies in your flow. Our question is for now. Yes, I have one, in fact. In the case where you have small soliton, is there a situation where at the normal level, people, or even very gross level, even better, long-time dynamics near one soliton, you start close to a soliton? I think there's a vast literature in that direction which I'm not all familiar with. I'm sure there's a lot more stuff done at the linearized level, so looking at the linearization around solitons. But I don't have a good... OK, OK. Yes? So why do you need this information to be holomorphic? Is this differentiable of some level? The reason you need it to be holomorphic is so that the Laplace operator remains the same in both settings. You should think of it as conformal, but conformal and holomorphic is the same here in two dimensions. Yes, question? Just to revoke for your question. For your question. Pego is here here, so Soumeishan Pego has a paper on the stability of solitons. Oh, OK. Linearized. Linearized, OK. OK. OK, so let's find the speaker again.