 Last time I gave you a brief introduction to water waves and today for the first five minutes I'll remind you just a few things from last time. So remember that we're looking at the equations of motion for a fluid in some domain omega of t and we're not looking at general fluid in this in this fluid domain but we're looking at the fluid that's either irrotational or has constant vorticity and because of that the equations of motion for the fluid are reduced to equations of motions on this free interface and on this free interface you're looking at two objects first you're looking at the motion of the interface itself and second you're looking at the velocity potential on this interface. So you have this surface gamma of t and your unknowns will be first the interface gamma of t and secondly the velocity potential the velocity potential we called it phi restricted to gamma of t. So you expect this water wave equations to be a system a system describing the motion of these two variables. Now in studying this evolution there's a couple of parameters that I introduced last time which I'll remind you so one parameter that appears is g which is the gravity a second parameter that I will use is sigma this is the surface tension. The third parameter that we'll use in one case at least will be the c which I'll use for the vorticity. So one of the models I wanted to tell you about is a model of fluid in this domain which has constant vorticity rather than b irrotational and the last parameter that will come up will be h which will be the depth of the fluid. So I'll mention two interesting situations one when the depth is infinite and the other one the depth is finite. Now for my purposes here the bottom of the fluid is going to be flat and somebody asked me about this last time what happens when you trade a flat interface for a non-flat interface is all in terms of the long-time behavior of the flow. For the short-time behavior it makes no difference but for the long-time behavior it is important that the bottom is flat. All right and now in terms of this four parameters in here let me remind you what are the four problems that we were looking at. So the first problem was when gravity is greater than zero surface tension is equal to zero the depth well vorticity is zero if you want and the depth is infinite so these are the pure gravity waves in an infinite in infinite depth. The second problem was when gravity is equal to zero sigma greater than zero again c is equal to zero and h is infinite these are capillary waves of infinite depth so waves which are driven by by the surface tension. The third problem was when we have gravity and we have no surface tension and now we were going to allow for non-zero vorticity and again in infinite depth. Here the vorticity does not have a sign and the reason for this is because you can always change the sign of the vorticity by reversing the time direction. Introducing vorticity in here somehow breaks the symmetry between waves that move to the left and waves that move to the left to the right and so this means that the equations are no longer symmetric with respect to time reversal otherwise the water wave equations are fully reversible and when you change the time direction you get the very same thing but not in this case. Okay and the fourth problem that I will mention is the case when you have gravity again no surface tension so sigma is equal to zero no vorticity but now the depth of the fluid is less than infinite so still gravity waves but gravity waves on a finite bottom and maybe I should also remind you that all of this projects are joint work with Mihail Ifrim and then in the first project one part of it is also joined with John Hunter and on the last project this is joint work also with Benjamin Harrop Griffiths. Now one thing that I will not do today is write down four sets of equations for you that would take way too much time but what I will try to do is to focus more on the first model just because it's simpler and maybe this is 2d everything that I say is 2d so I'll focus more on the first model and and maybe tell you what what changes to when you go to the other models and the first idea that I wanted to introduce to you in terms of studying this models which I mentioned last time is the use of holomorphic coordinates and of course this is not not our idea this is as I said last time almost a hundred years old but it's an idea that turns out to be very useful in working with the this equations and studying their long time dynamics and Claude's question answering also Claude's question this this is also the reason we stay in two dimensions because these coordinates are well suited to the two dimensional problem so the idea here is the following you have your fluid domain either with finite or infinite bottom and part of the problem when you're looking at this domain is that you have to deal with the Dirichlet to Neumann map associated to the Laplace equation in this domain that's so so the differential operator is not a multiplier has variable coefficients and further the coefficients depend on the unknown functions in in your problem so that's difficult to deal not impossible to deal in this problem but difficult to deal with and so it would be more it would be nicer if one could diagonalize this Dirichlet to Neumann operator and so the idea is instead of working with this domain you pick up a flat domain and I'm drawing this flat domain as a strip but really you have two scenarios one when you have finite bottom and here you have your water domain and here you have a strip and the other scenario is where you have infinite bottom so you push the bottom out and then here you have your infinite bottom here you have a half plane so on the right here you're either working on a half plane or on a strip and so by the Riemann mapping theorem you can find a conformal transformational conformal map between these two domains just to set the notations I'll use here maybe the letter zeta which is alpha plus i beta and here I'll use the letter z and the coordinates are going to be denoted by x plus i y so z is a function of zeta and instead of writing our equation in this domain I'm going to reparameterize the domain and in particular I'm going to reparameterize the surface using this conformal map and the main point of this is again to diagonalize the Dirichlet to Neumann map and so what are going to be the new variables under this holomorphic coordinates so one variable here is going to be the conformal map itself that will describe the shape of the domain but we don't need the full conformal map to describe this domain we just need the restriction of this conformal map to the top in here so this is let's call this line beta is equal to 0 this would be beta is equal to minus h and here this is y is equal to minus h and by the way so in this conformal map we're looking at not all possible conformal maps so looking at maps that take the bottom to the bottom and the top to the top and the infinities to the appropriate infinities and when you do this you're really left with the one single degree of freedom in the finite bottom case and this is the horizontal translations of this domain so you have one ambiguity in here whereas when you look at the infinite bottom case you have two degrees of freedom you have horizontal translations and scaling if you want in the choice of this conformal map and you want to fix this two degrees of freedom so that your equation is uniquely determined okay so coming back to our variables the first variable will be the function z of alpha which is z the conformal map itself restricted to the top restricted to the region where beta is equal to 0 and then our second variable so this describes the surface and our second variable should describe the as I said before the velocity potential on the top but now it's very convenient to think of this velocity potential also as coming from a holomorphic function in this domain and then the simple observation maybe let me write it here is that since the velocity potential phi satisfies the Laplace equation this means that it has a harmonic conjugate and the harmonic conjugate turns out to be exactly the stream function that I also introduced last time associated to the fluid and so this means that if I'm looking at phi plus i theta this will be a holomorphic function in the fluid domain in here okay but now I already have my change of coordinates from here to here so composing with this change of coordinates I can think of this as a single holomorphic function of the variable zeta in in in this domain so this requires fixing the the conformal transformation from here to here and lastly I don't again also in terms of q I don't need to look at all values of q in this strip or in this half plane but I only need to look at q on the top that will determine what happens on the bottom and so my second variable will be q of alpha which is phi plus i theta restricted to the top to beta is equal to zero so z describes the surface and q describes the velocity potential together with its complex conjugate now in terms of the size of these objects naively in a reasonable setting you think of q as being something bounded but z will not be something bounded alright and so because of this because infinity is mapped into infinity and because of this it is convenient to replace this variable z by a different variable where you take out the leading part so let me find that w of alpha to be z of alpha minus alpha and I want to assume that this guy is bounded and you will immediately realize that the moment I'm assuming that this guy is bounding bounded I have removed scaling from the degrees of freedom in here right because alpha if I rescale z and not rescale alpha then this will no longer be a bounded function okay so I'm still left to the horizontal translations and that has to do with constants and constants are much easier to handle in this theory now what is the state space what are the class of functions for z and q that we're looking at here and so the important observation is you don't want to look at arbitrary functions z and q z and q instead will will be functions which are defined on the top but which have holomorphic extensions in the fluid in the model domain okay so z and q have holomorphic maybe let me use w in here and you'll see in a moment why in in model domain and furthermore they have some some decay conditions which vary depending on they have in some sense some boundary conditions on the bottom and to describe this boundary conditions I'll have to discriminate between finite depth and infinite depth so if the depth is infinite then I expect w and q to go to zero more like their holomorphic extensions at as if you want beta goes to minus infinity so I have holomorphic functions in a half plane and they go to zero at infinity and if instead the depth is finite then this functions w and q of course they'll not be equal to zero in here that will be way too much to ask because if you have a holomorphic function that's equal to zero here has to be zero everywhere right but instead the piece of information that you get for q because of the boundary condition on the bottom and for w by construction is that w and q are real on the bottom okay so from here on our functions will be functions which live in this class they have holomorphic extensions in the model domain and they have either decay at on the bottom or they are real on the bottom and because of these conditions what in particular what happens on the top uniquely determines what happens in the entire domain and so I'm going to make this a definition we call such functions this is a slight abuse of terminology holomorphic okay normally you call holomorphic functions functions in open domain but here we're going to just to keep terminology short to call homomorphic functions functions which are the traces on the top of holomorphic functions which satisfy certain boundary condition on the bottom and carefully observing here that if you look at the space of holomorphic functions that's obviously an algebra okay but in the infinite depth case this is a complex algebra whereas in the finite bottom case this is no longer a complex algebra it's only a real algebra so multiplication I is not the holomorphic function in the finite bottom case just for the sake of okay and of course you can also describe this functions in in this class of holomorphic functions in a in a Fourier fashion and so in again I will discuss the two scenarios if the depth is equal is is infinite then these functions have the property that the Fourier transform view is supported in the region where psi is smaller than 0 so it has only negative frequencies and if you have a function that has only negative frequencies then the real and the imaginary part of this two functions are related and precisely the the relation between these two is that the imaginary part of you is the Hilbert transform of the real part of you so we're looking at functions specifically which satisfy this relation and in this context that you can also talk about anti-holomorphic functions and anti- holomorphic functions are functions which are satisfied the same condition but have anti-holomorphic extensions of course if functions holomorphic then its complex conjugate will be anti-holomorphic so you have the same relation but with the minus sign and then the interesting object to work within here is the projector the projector to negative frequencies okay so P is the projector to negative frequencies and you can relate this projector to the Hilbert transform so P will be 1 minus identity i times the Hilbert transform divided by 2 and here I should caution you that various parts of the literature is different conventions for the sign in the Hilbert transform so if you're if you might be slightly confused in here this may be because I'm just using a different convention from one you have seen in the past so this is what happens in the infinite depth case now in the finite depth case so when the depth is finite the situation changes a little bit because now instead of solving the Laplace equation in the infinite domain you solve the Laplace equation in a strip and so this relation will be replaced by the following relation so now you no longer have the Fourier transform of you supported in sine negative instead you're going to have the following relation you had of I'll try hard to get the signs right minus C is equal to e to the power I think h c times u hat of C bar so this is the relation you're looking at so now these functions have both positive and negative frequencies and instead of having this relation which involves the Hilbert transform you're going to have a different relation that's the imaginary part of you is equal to something that also depends on height till h of a real part of you and this is what's called the tilbert transform apparently I learned this name from my collaborators but when I tried to look it up in the literature I could not really find it except in some software package so but it's a good name it fits Hilbert Hilbert apparently tilbert is also a name I found by during my search okay and and this operator is defined by its symbol it's a multiplier is minus I hyperbolic tangent of h psi so don't confuse the two h's right let me connect the two and so then the projector you can also talk about the anti holomorphic functions with the opposite sign obviously and so you have a similar projector to make to to to this space of holomorphic functions I'll write it down for you so p this depends on h of u is equal to one half this is never going to be used but just for your curiosity well of u plus one half I one plus I to the transform inverse this is real part of you and this is the imaginary part of you maybe when you talk when you think about this this projector I should say in this case this the reason we call this a project this is project this is an orthogonal projector in L2 so here you're thinking of the L2 setting but here it doesn't make so much sense to think of the L2 setting because in particular this guy will be unbounded in L2 because you're inverting this symbol and so here you're you have to work with the different Hilbert space I'll denote it by this with a norm norm of u in this space will be equal to norm of imaginary well h times real part of you in L2 plus norm of imaginary part of you in L2 okay so for imaginary part use L2 but for the real part use this multiplier and what this multiplier means this is still an L2 norm at high frequencies but when you look at this guy at low frequencies at low frequency this is like a differentiation operator so you're at low frequencies here you're you're only controlling the derivative of you and in particular this means that you you don't have any access to the constants in the function you okay but this this space will be very nice and consistent with what I want to tell you later all right and this this projector here and this projector here will play absolutely identical roles in writing down the equation so when I write down the equations in particular for one case 1 and case 4 the equations will be exactly the same with just different meanings for everything that I'm writing okay so one thing that let's say one thing that I will not have time to to to do for you is to show you how one actually derives the equations in holomorphic coordinates so instead of that I'll just write down the equations and then I'll spend some time explaining their their structure so we have these two variables W and Q these are holomorphic variables and so in this space of holomorphic functions we're going to look at the following evolution so wt plus f times 1 plus w alpha is equal to 0 so this is the equation for the evolution of w and the equation for the evolution of Q f Q alpha minus ijw plus the projection of Q alpha square over j I'll explain in a moment all the notations plus i sigma projection of w alpha alpha divided by j to the power one half one plus w alpha minus its complex conjugate just for brevity is equal to 0 so this is an equation that this is a system of equations that describes at least the cases one three and four in here and if you wanted to also look at the case when you have a nonzero vorticity instead of 0 here you're going to put the vorticity times some expression and here you're going to have also a linear term in the vorticity times some longer expression all right so this is for the case three my green color coding if you want and important to remember to to to to observe in here is that this corrections when you have vorticity also includes linear terms and because it includes linear terms it will change the dispersion relation for the problem and I'll comment more on that later so what are the objects in this long expression so j is a natural object this is the Jacobian of the change of coordinates from one setting to the other so it's one plus w alpha square and then f is equal to the projection of q alpha minus q alpha bar divided by j now if you if you just look at these this equation let's say when I looked at this equations for the first time look at them you pretty much see nothing okay you see that they're nonlinear equations you see that they're complicated you can simplify them by setting some of the parameters to zero so for instance if you set to the vorticity to zero and the surface tension to zero you left only with this part of the equations which might seem more manageable and it is more manageable I'm getting to that yes so what you don't see in this equation is the Dirichlet to Neumann map and the reason you don't see the Dirichlet to Neumann map is that in these equations we're talking about the Dirichlet to Neumann map on the model domain and on the model domain the Dirichlet to Neumann map is just the absolute value of d okay and so in some sense the role of the Dirichlet to Neumann map in this equation is played by all of these projectors that you see projectors time the derivative that's something very close to Dirichlet to Neumann map all right one thing that might be misleading actually in this formulation is the following you look at the Euler equations and what is the the leading term in the Euler equation is a transport right and then there's some coupling because of the incompressibility condition you might say oh this looks like a transport equation have to be careful because this f in here is a holomorphic function it's a projection in particular this it cannot be real right if it were real it had to be constant had to be zero actually so this is not a transport equation not yet but still eventually we want to think of this as some sort of transport equation and when you have a fully nonlinear equation how do you understand its character well the obvious thing to do is to linearize it right and you look at the linearization try to understand what what type of equation is the linearized equation and this is the next thing I'll try to show you and I could actually use this board and of course if I wanted to write the all the full linearized equations my choice of this board would be very poor but first first thing you want to do with the linearized equations is you want to understand what is the leading part of the linearized equations so let me call by w and q the linearized variables okay and so now I would like to write down an equation for dt of w and q and in first approximation one thing that you're going to observe in here is that the expressions that you're going to get on the right will contain the following variables they'll contain of course w alpha q alpha maybe you'll see a little bit of w and q maybe you'll see also some w alpha bar and q alpha bar right there's complex conjugates arising in here complex conjugates are anti-holomorphic we're not very much worried about the anti-holomorphic components because those get killed by the projections so really the leading part when you compute the linearized equation is the part that has the derivative and so at the level of the linearized equation this will look like some matrix times d alpha w q plus some lower order terms and so when you see a system like this you'll think oh this is this may well be something hyperbolic would like this to be something hyperbolic given that we started with the Euler equation and so important object is to understand what this matrix is now this matrix actually is not very hard to compute what you have to do is you have to differentiate this equations with respect to a big w and big q but you only care about the differentiation with respect to the holomorphic component not with respect to anti-holomorphic component so for instance when I'm looking at j I only care about the one plus w alpha component of j when I'm looking at this expression I'm going to care only about q alpha and the one plus w alpha plus from from j so it's an easy exercise to do the computation and so I'll write down this matrix for you and I'll write the write it down in a form that's easy to to read okay hopefully I can fit it and so so what you get in here you inherit the f obviously but this f gets coupled with another guy q alpha bar over j in some fraction here this is a little bit long so I can divide it by j and here 1 plus over 1 plus w alpha bar minus q alpha square divided by j times 1 plus w alpha it seems tedious but believe me this is worth it over j and then plus q alpha over j all right so we want to understand what what kind of matrix is this right in particular want to understand what are its eigenvalues and one obvious remark in here is that in this matrix you have a multiple of the identity right the multiple of the identity is the object that contains this guys right and now what happens with the remaining matrix suppose I take this guys out well the remaining matrix first of all has trace 0 okay and then it's very easy to see that it also has determinant 0 so the remaining matrix has a double 0 eigenvalue so but of course it's not diagonalizable right it's going to be if you diagonalize it you're going to get the obvious Jordan block and so this matrix has a double eigenvalue and the double eigenvalue is given by this object f plus q alpha bar over j we're not yet out of the woods we'd like this to be a hyperbolic equation so we'd like this object to be real right so first thing we do always is we baptize it so I'm going to use the letter B for this and maybe I can put that on this board actually I'll write it down here because I might want to save it for later okay so B is going to turn out to be again a brief computation two times the real part of the projection of q alpha over j okay so this is indeed a real object so this means that our problem has at heart is a hyperbolic problem but the hyperbolic one with the double speed this is no surprise right because if you think of the Euler equations everything gets transported with the same speed now what is the meaning of this speed B that we see in here one thing I was telling you about last time was the fact that you have this gauge freedom having to do with the changes of coordinates and in particular we choose some non-standard coordinates in here so this B really is the velocity that comes from the change of coordinates if you want B is the difference between our coordinates and the Lagrangian coordinates in the same way as the standard velocity is the difference between the Eulerian coordinates and the Lagrangian coordinates okay so so so far so good and I'll tell you that we were really happy when we got this guy to be real when we were computing the equations because that meant we are not making any any big mistakes but now coming back to our matrix in here one thing you want to you always want to do when you see a matrix like this with the double eigenvalues you want to diagonalize of course will not be diagonalizable but you want to put it in some good Jordan form right and there's there's one very simple change of variable that does that and so that's from w and q going to the pair of variables w and r where small r is equal to q minus q alpha 1 plus w alpha times w and this idea of diagonalizing things of course is not new and you maybe eventually will see how to implement this not only at the level of the linearized equation but also at the level of the full equation so anticipating a little bit for those of you who are familiar with these things I'll say that this is very closely related to ideas that first arose in non-linear wave equations in the work of Alina who sort of pioneered this idea of choosing the good variable when you solve the equation and then work of David Land closer to water waves work of Alazard Burkin Zuley again for for water waves equation so you want to work with the right variables you want to diagonalize your system properly okay so what happens when you diagonalize your system properly so let me write down a little bit more of the linearized system so you have dt so suppose I now write the system for w and r so I'll have the system dt w plus b d alpha w plus and the next term I'll get in here is r alpha divided by 1 plus w bar alpha and equals and here I'm not going to elaborate this this will be some lower order terms and and at least in the sense of local theory but more interesting it will be to write the equation for r dt of r plus b the alpha of r so we start with the transport which is the leading part and here comes the coupling term with the first equation and this coupling term has to do with the with the gravity let me make sure I get the correct sign and this will be i times g plus a g yeah g plus a 1 plus w alpha multiplied by w and this is again some lower order terms okay so so in order here you see the the main features of of this equation the first feature the leading feature is the transport the second feature is the coupling between these two between these two equations and if you want to think of something to call a coupling coefficient that should not be this coefficient or this coefficient but really should be the product of these two coefficients and the product of these two coefficients in here if I multiply them and I can write that here I look at g plus a divided by j all right and this is the object that I have that plays a key role in this analysis and that I have mentioned yesterday and it's it's a relatively easy computation to show that this is the normal derivative of the pressure restricted to the free boundary so the sign of this quantity the fact that this g plus a quantity is positive is place a key role in the well-postedness of this system and that's what's called the Taylor stability condition so you want this guy to be positive all right else this equation is no longer locally well-post yes yes I'm getting to that so a is equal to two times the imaginary part of a projection and this projection is of r times r alpha bar sorry and and I was waiting and r is q alpha over one plus w alpha okay so you first my hand a little bit here I had to give something away before it was ready but in any case so let me finish this this discussion in the following way so if you look at this equation one obvious symmetry of this equation is the translation the invariance with respect to spatial translations right and since this equation is invariant with respect to spatial translations then you have one special solution to the linearized equation and that's the derivative of the solution so if I'm looking at w alpha and q alpha this object will solve the linearized equation all right but if I want to study this object properly using the linearized equation then I should diagonalize it right so what do I get when I diagonalize it well w alpha stays unchanged okay but again a very simple computation will show you that when you diagonalize this you're going to get the pair of variables w alpha and q alpha divided by one plus w alpha so this means that w alpha and q alpha is really not the right variable to study this equation in instead you want to study this pair of variables and that justifies giving a name to this guy in the first place all right and once you give it a name you have to understand what it is right and this turns out to have a very simple interpretation this is really nothing but the two components of r are the velocity vector field uh the two components of the velocity vector field so r represents the velocity vector field written in a complex form okay um exactly if you want to write it down r would be u minus i v where u is the horizontal velocity v is the vertical velocity so one thing we take away from this this computation is that um even though this equation is written as an equation in w and q it's natural in the first place to differentiate it because it's fully non-linear and when you differentiate it and diagonalize it then you get an equation a system of equations in these two variables w alpha and r and this would be the good variable salla alina let's say okay um and so to relate this perhaps with work that was done by people in the audience let's say using uh using the uh your Eulerian setting let's say in the Eulerian setting this diagonalization happens at the level of the undifferentiated variables and it involves uh to do it properly have to use paradeferential calculus uh which uh if we like paradeferential calculus is nice and good but here it's it's it's nice that you get this diagonalization very simple and algebraically all right so um the last thing going along this line that I want to show you is the system of equations for this differentiated variables for for w alpha and r all right let's see how this works out I know that no no but it's trying to figure out how hard to push on this yeah okay do you see everything there all right one thing what which you may have noticed while computing the linearized equations is that I have neglected the effect of the surface tension and that's because that one does not play any role in this discussion about diagonalization diagonalization has nothing to do with the surface tension has nothing to do with also with vorticity so that those variables will remain the same in all cases so now I promise you the system for w alpha and r and here I'm I'm I run into a bit of problem with my notations because uh I don't want to really well let me not make things more complicated than I should be um so so here's the system dt of w alpha plus b d alpha w alpha plus one plus w one plus w bar um r alpha is equal to one plus w m just be a little bit patient alpha thank you um and then uh dt of r plus b d alpha of r um minus i times uh w minus a g w minus a and i is in front divided by one plus w alpha is equal to zero so I see this this is a pretty manageable system I wrote this system in this first case and it no longer applies in the case of finite bottom you have to make some changes there and one obvious reason why you see it it doesn't apply it has this i in here right i is bad i is not holomorphic multiplication by i makes sense in the infinite bottom case but has no meaning in the finite bottom case all right but in any case this is our system and now uh I'll I'll try to to give you some idea about what happens when you try to look at the local well postedness theory for this equation so now I'm really getting into the problem um but before I get to local well postedness I need to tell you a little bit about the function spaces and the first thing that I have to show you then is what is the energy for the system so this system has an energy uh and the energy is uh integral of w square actually this is the energy for the system above okay because it's at the level of w and q times one plus real part of w alpha plus the imaginary part of q or q alpha bar I might have some wrong signs in here so don't panic if some things are not positive okay they should be positive so two comments about this energy so there's a corresponding energy obviously in Eulerian coordinates this energy as was discovered by Zakharov plays a role of the Hamiltonian for the system one thing I'm not going to do for you is I'm not going to write down the symplectic form in holomorphic coordinates that's if you want one downside of using holomorphic coordinates the symplectic form is ugly uh symplectic form is simpler in uh Eulerian coordinates but but here it doesn't look so good so this energy really corresponds to Nether's theorem because of the time translation invariance of the system uh because uh you also have a spatial invariance with respect to spatial translations you're also going to have some horizontal momentum which I'm not going to write down so you you can play a little bit with the Hamiltonian structure if you wanted all of these problems that we're looking at are Hamiltonian and they all have the same symplectic form uh do they no uh this one has a different symplectic form uh the one with the vorticity but the other ones have the same symplectic form one thing that we read and I should say one last thing that this energy uh consists well consists of two parts that's obvious uh this part uh is like the norm of q in h one half homogeneous square okay it's exactly this well up to a constant perhaps a factor of two uh this one you you want to relate this to uh norm of w in l2 square right um you have this guy in here uh so is this positive or not um and the answer is to this is very easy or this energy will posit it will be positive at least as long as uh the simplest thing to say is that the fluid surface does not have self-intersections um in effect this energy will still be positive sometimes when you have self-intersections but you can allow some kinds of self-intersections and not others I'm not going to elaborate that but this is essentially a positive object all right the energy is positive definite all right and so one thing that we read of this energy is the kind of spaces that we need to work in in so from here we see that uh the the spaces to study our revolution in are spaces of the form l2 cross h one h one half homogeneous and this applies to the first problem this will change when you move to each of the other problems they'll have different slightly different setups and in particular this business that I was discussing before with the projector in the case of finite bottom means that in the case of finite bottom you will no longer control the low frequencies of w properly so the energy will will instead of having the l2 norm of w and here the h one half norm of q will contain norms which are with respect to this space and with respect to the associated inner product so you lose some low frequency control in the finite bottom case all right so so we you want to use this space so you might say when you look at local well positeness uh you want to work with say w q in uh l let me call this space script h which is l2 cross h one half and then when you look at the differentiated variables w alpha and r from here on for further derivatives this is the variables you want to work with so this should belong to maybe some hs and let me discuss a little bit this exponent s all right and again this discussion will be specific to problem one and in the high frequency limit also specific to three and four but not to two because two has a very different scaling at high frequency all problems one three and four have the same scaling at high frequency whereas problem two the surface tension term uh takes over at high frequencies all right so so what what can we say about this exponent s and uh one uh thing to look at uh is whether the problem has scaling and this problem one in particular does have scaling uh so the scaling is uh if you want written at the level of w alpha w alpha of alpha and t goes into of t and alpha goes into w alpha of lambda t and lambda square alpha and uh q and r of t and alpha goes into lambda r of lambda t lambda square alpha and once you have some scaling you can compute what is the critical exponent in here that's very easy to see so the critical exponent uh corresponds to s so um maybe let's write this here so s critical is equal to one half all right um and uh you might say oh perhaps we should be looking at solving this problem at this regularity level but then you remember that this problem is a quasi-linear problem and to my knowledge hardly any quasi-linear problem was solved at the scaling level so instead we're we're going to give up a little bit and the exponent that I'll be uh dealing with primarily today is s is equal to one so it will be one uh rather than one half half derivative about the critical level and for those of you who are familiar with what happens with the non-linear wave equation is exactly the same balance of forces the state of the art so to speak is more or less about half uh derivative above uh the scaling you can one can expand this discussion a lot I don't want to do that um so let's see all right so um maybe before I tell you about uh local rule posedness I'll tell you about the energy estimates and so one sort of twist we introduce in in this business of uh energy estimates is actually not a new twist is a twist that one has seen all over the place in connection with non-linear wave equations namely that when you control the evolution of energy you don't want to control the evolution of energy in terms of sobolev norms of the solution rather you want to control the evolution of energy in terms of point-wise norms of the solutions on grounds that this point-wise norms of the solutions are weaker objects and they really tell you that the energy will budge only when you have point-wise concentration and not just a large l2 norm or large sobolev norm and so in order to do that one thing that we do in all of these models we introduce some sort of control parameters and I will warn you that I'll oversimplify things just a tiny bit in here by less than a log to put it this way just to keep things short so we're going to use two control parameters our first control parameter will be uh what I'll call a um and this will be a scale invariant norm so this is norm of w alpha in l infinity plus norm of d one half r in l infinity okay um so so this corresponds exactly to the critical exponent one half and so this means that this constant a can appear in any combination you want in in your estimates more or less and the second control parameter that we're going to use is half a derivative higher but half a derivative higher so that would correspond to s is equal to one I'll remind you that we work in one space dimension and in one space dimension the embedding h one half into l infinity does not work but one thing we were very happy to be able to do in here is to relax this l infinity requirement to a bm o requirement because h one half homogeneous does embed into bm o so in here output be a more norms d one half w alpha in bm o plus r alpha in bm o and maybe so this is as I said half derivative higher and to get another way of understanding the the level of regularity for this I I'll remind you that when I defined that coefficient the the transport coefficient b which was somewhere did I erase it okay all right so so you see that I can rewrite this as two times the real part of r projection projection of r divided by one plus w alpha bar so the leading part of this is the real part of r okay so the fact that you control r alpha in bm o means that you're controlling the derivative of your velocity in bm o so you don't control the velocity in l infinity you only ask to control the derivative of the velocity in bm o and so if you do this then by no means you're going to have energy estimates okay for for even a transport equation and so in order to make this work we have to make use somehow of the fine cancellation structure that's buried into into the water wave equation um and so what what is the good energy estimate then for for our equations I'll write two types of energy estimates first of all energy estimates for the linearized equation and then energy estimates for for for the differentiated equation now when you look at the linearized equation what is a good energy so so you look in here and you know that the l two cross the h one half norm of w and r should be the correct object so let's define this energy a two of w and r to be equal to integral of w square plus the imaginary part of r r alpha bar um d alpha so this would be the the good energy for the linearized problem around zero um maybe I should put a g in here all right but here we're looking at the non-linear problem and the coupling between these two equations has the g plus a coefficient in it and so this g plus a should really appear in your energy and this is where it appears and so this is the natural energy for the non-linear equation now you see also another way where where this coefficient becomes important you make sure that your energy is positive definite okay and so what is the our energy estimate for this functional d dt of e two of w and r just to give you an idea of the flavor of the estimates that we're proving is smaller than and here we're going to have an implicit constant and your implicit constant depends on a and smaller than b e two of w and r okay and now in terms of n adjustments we primarily tend to think of the n adjustments for the linearized equation rather than for the original equation because there's more information if you can prove n adjustments for the linearized equation and so if you look at the original equation you can have a similar n adjustment which says that d dt of the energy same energy if you want d k w alpha and d k r smaller than a b e two of d k w alpha and d k r with respect to the diagonal variable so you see the same dependence on the parameter b and the implicit dependence on the parameter a and at least in the case one but we can also implement this in in the second case for the second problem these estimates are invariant with respect to scaling okay so there isn't much to improve here in terms of of the energy estimates this this is as good as they get and so if you just use this n adjustments both for the differentiated equation and for the linearized equation what you're going to end up with is the following theorem maybe i can use that board so the theorem would be that the equations one two and four so these are equations which share this scaling in the high frequency limit are locally well posed for w alpha and r belonging to this space h one so this corresponds to choosing s is equal to one in the air and some some comments are in order here so of course this is not the first time a local well-posedness theorem was proved the first local well-posedness results for water waves are due to na limov for small data in the case one and two of sianikov in some sort of gevrey setting for the problem four of sianikov was also the guy to introduce this holomorphic coordinates in the dynamic problem the same theorem and then this guy also na limov work with fairly high regularity the next important result was the result of cj u cj proved well-posedness again in high sobole of regularity but for large data and her contribution one of her contributions was to figure out the positivity of that normal derivative of the pressure then I should mention the work of al-azard burq and zuyili their initial work if I remember correctly was just a little bit above this epsilon above okay and so that's one one we were able to get the first result so we managed to beat them by an epsilon but then they came back and they said oh but this problem is also a dispersive problem it has three cards estimates and if you work hard then use those three cards estimates you can improve this result a little bit more and and and then after them it came and I'll apologize if I misspelled the name guian and the boy fair and they improved a little bit more the threshold and I don't know what the current threshold is maybe it's I'll guess it's one eight better than am I right something like that one eight better than that by using three cards estimates however our n adjustment still remains the the better one so we have the right and I just not just no no no no we did not try to use any any three cards estimates and and there's a reason why why we stuck to the n adjustments and that's because the n adjustments are the object that's crucial when you look at the problem I really wanted to look and that's the long time behavior of this equation now one thing I should say maybe a few more comments I have in here when you look at when you state a local wall post and as a result the correct setting for the this is very vague setting vague terminology if you want you have to qualify what means local wall post illness this is a quasi-linear problem so local wall postness includes existence let me write this here existence uniqueness means continuous dependence on the initial data but what you don't get because the problem is quasi-linear you don't get better you don't get any sort of uniformly continuous dependence on the initial data so this is in some sense optimal another thing we were keen to do and and we were able to do is in here we have these parameters we have let's say when you move from the first case to the first case you have the depth and we wanted to have this results uniform with respect to at least some of the parameters in here and so what I can add in here is that this result is uniform as the depth goes to infinity and it's also uniform as the vorticity goes to zero so you can take those limits if you want and have uniform estimates for them let's see another okay maybe this concludes the remark about the local wall post illness theorem and that brings me to the topic of next lecture where I'll try to tell you about the long time estimates for the equation my plan was to get to that today I'll confess but it did not happen and I did not want to rush the first part because I know that many of you have not seen much of this stuff before so if you lose it early on then the last lecture will be useless but nevertheless next time I'll try to tell you as much as I can about the long time estimates for for this problem and that's it for today this right so once you write down these sets of equations you no longer care whether your initial data came from a graph or not and maybe one thing that I was planning to show to you but again I did not have time to is the proof of the positivity of the normal derivative of the pressure our proof not cj's proof because that proof applies regardless of whether your curve has self-intersections or anything like that so one can formally continue these solutions past self-intersections there's no absolutely no issue with that except you lose the physical interpretation right what your model is no longer a wave it's just your favorite pd let's say but or most hated pd is depending but has nothing to do with the self-intersections so you say you have local world personas for one two and four but all of those cases have c equals zero so I'm a little confused about that sorry sorry sorry one three and four thank you very much one three and four right and so since you asked the question or pointed this I'll say that there is the corresponding result for problem two is also true except that you have to fiddle a little bit with the numbers because that problem has a different scaling okay can if I can make another remark I meant to make this remark early on and I forgot about it so as you see we have these four parameters that we're looking at but we're looking at just the four four of these problems and actually there is a much larger variety of problems than one can study in here that's this is one nice thing about water waves is a great playground you can find lots and lots of equations very different dispersion relations very different dynamical properties lots of interesting problems to to study