 Thank you very much for the organizers for having me and for the attendance for coming. It's actually a lovely spot and it's been a great conference especially because of the variety of topics that have been discussed and the variety of points of view that have come up. That makes it definitely keeps us on our toes. So I also am, well let's see, I should tell you what I'm going to talk about and also tell you first maybe my collaborators. This is Amanda French who moved from, who is a student of Michael Taylor who was with me at McMaster for three years and I was moved to Haverford and Chi Ru Yang who came to the fields and then to McMaster. And it's a little bit, what can I say, it's a little bit humbling to not be talking about a new theorem but an old theorem and in fact an old theorem of Sergio's as well as Chateau and Hormander and some others from about 30 years ago if not a little bit more. So okay, I'm sorry. And to my justification for doing this, well maybe it's just a slight shift in point of view. I don't know if there are advantages but at least it articulates something in a different language. So it's nothing new here and I apologize for that but sometimes it's good to revisit older things. So I'm going to tell you about a nonlinear wave equation. I'd like to talk to you about transformation theory because I think that's what we do not do in PDEs is change variables. Okay, we change x to y and we change u sometimes to u squared or u squared minus d by dx of u but we do not, sorry, ugr so Sergio does it all the time but most of us don't and in relativity you are still just doing local coordinate transformations but you're not changing a bonnock space to another coordinates in bonnock space. And so that's part of, so we're starting to do that a little bit but it's really rather undeveloped. So here's a toy example where you change coordinates in a bonnock space to other coordinates in a bonnock space, at least a neighborhood of a bonnock space. And I'm going to do it in a Hamiltonian framework and then I need estimates to make everything work so that's the format of the show. Now it's been implicit since day one of this conference and I'm going to articulate it, I'm not sure anybody else articulated it so I'm talking on the wrong day but okay, why change variables? Well let's just look at scalar ODE's, z dot is equal to z squared has an exact solution, oh for small data or any data but for small data we think z of zero is equal to epsilon, think of epsilon small and z of t is exactly epsilon upon one minus epsilon t and it has a time of singularity one upon epsilon so that's the, that's the what was called the quadratic blow up because z squared and then you can always read my slides, I hate covering people who cover their slides, there's the cubic case, w is equal to w cubed and its exact solution is root of epsilon squared over one minus two epsilon squared t, if I start with, if I started with data at epsilon and then the time of existence is one upon two epsilon squared that is I have an order of magnitude longer existence time and that's the point, cubic lasts faster than quadratic. Okay it's pretty obvious, certainly to everybody in this room and also it doesn't change if you, at least not essentially if you move things around a little bit like I can add a linear term and it makes it a little complicated now z and w are complex and higher order terms don't, normally without special details change the fact that singularities can occur. So let's talk about the real problem which is a non-linear wave equation. So I'd like to talk, I'd like to talk about equations which look like this, two time derivatives equals Laplacian, so really starting with a, with the background linear wave equation and that's the linearization at zero, I want zero to be a solution, then some non-linear term which I'll make more precise later but let's say that it is at least order m minus one in its variables, so like the quadratic case would be square in its variables and cubic would be cubic and I want to solve the cushy problem, so initial data at time equals zero g and initial momentum h and so basic question in PDE is sort of something people spend their careers studying is what's the time of existence? If I give g and h in some, some class of functions, maybe a sub-live space that's what I will use, I'll tell you what my z is going to be later or several z's later, several possible, then you want to know how long does the solution last and of course the best result is if it lasts for all time and so I'm really working in the class of small data all time, you know, global existence. Okay so this was worked on in the 1980s, I think Sergio's thesis is on this and then a lot of work of 80s was fundamental and on this there's work of Hormander at the time and Jalal Shatat at the time and others and I'm probably leaving off people in the room so you can shout at me but do it after the talk so we don't waste, because time is running and Frank is here keeping track and so just real and really short here is here is an existence theorem, a global existence theorem. Suppose that one half times the dimension minus one times m minus two, well the non-moniarity was order m minus one for Hamiltonian reasons so n minus one, dimension minus one times m minus two, if that's bigger than one then for small cosi data, data in an appropriate sub-live space time existence is infinite and as a balance as everyone knows between solution spreading out and decaying which is a dispersion certainly not a strong dispersion but a dispersion of sorts and the non-linear effects trying to focus things and put things together and if the solution is small and that sub-live space the dispersion wins and just I know everyone in this room knows this or most people in the room knows this but what's the linear decay rate for the linear wave equation which for small data will be reflected in the decay rate for the non-linear wave equation and it's of course c over t to n minus one over two which reflects that first factor which is part of the story. So let's just do that just just calculate if m is equal to three so the non-moniarity is m minus one is quadratic then how do then when do we satisfy the hypothesis well I need n minus one n minus two to be bigger than two so n better be bigger than three which is bad luck for us we live in three space well according to us we live in three space dimensions that reminds me of a funny joke about Feynman but I'll tell you afterwards too for reasons for reasons of time but if m were four if you had a higher non-moniarity then you do the calculation and n is greater than two which is also which is good news for us living in three space but a little bit bad because two is also interesting to mention and then the borderline cases have these almost global existence where you get exponential time of existence in the border in the two when you get equality in those two dimensions and two non-linear cases okay so I haven't said anything wrong yet that's more or less how things stand well transformation theory it tells you there's an interest from going z to w right there's an interest in taking a quadratic non-moniarity and making a change of variables maybe in bonnock space maybe in a neighborhood of bonnock space so that you no longer have quadratic terms but only have cubic terms and this was an idea that was developed slightly after the things on the first slide but it's still alive today because there's some Masmoudi has some work here and some of his ideas have entered in hyperbolic situation with Pussetieri and Châtin a couple years ago which which revisit the idea of changes the variables or essentially changes the variables to get rid of that that the obstacle of a quadratic term in the equations and so I think I'd like to prefer to say that in the following way if for special non-linearities to satisfy a special condition which Sergio called a null condition then for dimension three that was the key one you can make t existence time be equal infinity by a transformation which got rid of the of the quadratic term in favor of more complex cubic terms and in dimension two that extended it's a borderline case and you got exponential time existence so the idea is change variables and one thing that strikes me though because we try a number of us a number of us try is if you make a change of variables certainly in the style that of this theorem it's hard to make it again that is you start with an easy non- linearity that's maybe polynomials or something and you make a change of variables and it's complicated and non-local and just try again it's not so not so trivial so what I want to do is introduce Hamiltonian formal form formalism which which helps helps the idea of making changes of variables but you have to do it rigorously but it gives you a gives you a systematic way of of doing this so let's see what I can do well I can't do every end in fact we probably don't want to do every end the ones that interest me are the ones really come from physics they're ones that come from a Lagrangian so I'm interested in a quay in a not only a non-linear wave equations that come from the the principle of least action actually that sounds very pretty in French so the principle de moindre action I like the word moindre and and then the action functional is a time integral of a Lagrangian and the Lagrangian I want to start at the wave equation so there's a quadratic part in the Lagrangian plus other order terms and the reason my non-linearity initially was order m minus 1 is because I want this term to be order m and it satisfies some smallness conditions for small argument and then once you have this form of wave equation you can affect a Lagrangian transform at least locally near zero where you really just do satisfy the follow the form the classical formula formalism you can take a variation of L formally with respect to ut you'll find P and you call that P and that starts out with a ut a u dot plus a higher other terms and this is algebra well or the implicit function theorem but locally in each x and t to come up with a new coordinates u and P where every time in the Lagrangian you're going to see a ut you have to say though that's a function of P and as well as function of gradient u so the Legendre transform and then puts us into gives us a Hamiltonian which is a bonafide Hamiltonian and it's a function of u and P and it's really just this form and you evaluate every time you see a ut in the Lagrangian as I said you have to use that implicit function theorem of course that's a changing from u u dot to up but that's not a change of variable bonac space that's point-wise u of x and P of x it's a an algebra locally so this is of course a Hamiltonian system in Darbou coordinates where u dot is equal to the gradient of h with respect to with respect to P gradient in this case it's Darbou with respect to the L2 and inner product and P dot is minus gradient specter u and it's a first order system of equations equivalent to the above non-linear wave equation it's not the only way to make a first-order system but it is an elegant way of making first-order system that carries through certain certain other structure which I like to use okay let's do it for the non-linear wave equation because I was just doing something kind of general ish and very soft but if L is equal to that quadratic term plus the higher order terms they come from the the the the principle of this action L2 is the Lagrangian quadratic is the difference between kinetic potential energy at least the linear version of kinetic potential energy and then you do that exercise with h and h is h2 which is the sum of potential with kinetic and potential energy plus a changed r which is related to P by that change of variables and they are of order m as I'm indicating by the superscript and then you differentiate as I told you on the previous slide n to get n which is a non-linearity and then it's order m minus 1 non-linearity so that's everything's I think consistent so far so I'm sorry let's do something simple let's find the plane waves it tells me I have to find so it says solve and Fourier transform it's the linear equation this is what averaging theory is about gives you a dispersion relation dispersion relation tells you where the dual light cone is in with a speed 1 the light cone is the same as the dual light cone but still just remember it's the dual light cone and then here's what we want to do I want to make a transformation from from z which is the vector u vector function up which lives in some bonnet some box space it'll be a so-called space to a new variable z prime and I want I insist that this lives in the same bonnet space and I want to that transformation to be the following I want it one to be canonical well it's because I'm going to design transformations to eliminate things and if I make a general change of variable of vector field that involves the Jacobian of a general change of variables and it's harder to invert a relationship which involves a transformation and its Jacobian then to just think about the trans just worry about the transformation itself so being canonical allows you to sit within it allows you to change variables to design what do you say designer change of variables without the Jacobian bothering you the fact that it's canonical is dealing with some algebra of the Jacobian and that tells me that that tells us that we can come to a new system equations for z prime which is just like the old system of equations for z but with a new Hamiltonian where you just compose well here it has to be the inverse z in order to get the new equations and so the new Hamiltonian is going to be change of variables with its with its orders of of nonlinearity and to be in Birkhoff normal form to order capital M it means that you retain every term between 2 and m by this process you don't change 2 by the way in at least what I'm thinking but you change 3 and higher and then you choose what m is and then your your order the the residual will be m plus 1th order and that will remain and you're insist upon your change of variables having these terms of being only resonant terms for example zero sometimes there are no resonances and you want zero and zero is good for example if you get rid of everything that's even better sometimes you can so each is em retains at most only resonant terms so in this in this parenthesis only resonant terms and resonant terms are the ones which you could say plus on commute with this quadratic Hamiltonian another way to say that is that they preserve the same action integrals that the quadrat that the linear wave equation preserves okay it's a vertical reduction to Birkhoff normal form and in dynamical system this part of averaging theory and we only want to do it once in this talk in fact it's work in progress as I said in my second slide so that's all we've done so far okay so when m is equal to 3 and I want to consider the resonances those are known as three wave interactions or triad interactions and those you you you can find by a formalism which has to do with the wave number so I have to preserve I have to conserve momentum so I need three wave oh c is in r n I should have said I could write that because it's a little confusing the the coordinates but my c is a vector in r n and this is space sorry it's a hat of space and that just says that I conserve momentum and then to be resonant the frequencies have to add or subtract so a triad resonance is one with frequencies adding and subtracting three three fold and okay and then back to the PDE you know infinitely many almost infinitely many papers have been written about this on a formal level but back to the PDE I'd like to understand the construction of such a canonical transformation and its mapping properties on which soba space on which bonac space and then you usually think of these these as discrete indices of a of a dynamical system finite dimensional Hamiltonian system finite indices or on a compact manifold they would be discrete but here we're in continuum and continuum plays a role in what we're going to talk about so when x is in r n it's important where x is a wave equation on a torus is going to be a different story so i'm not doing that when x is in r n c is in r n and its continuous variable and things actually sometimes are a bit better that's one of the things we'll find so here's something about wave equation triad interactions proposition is suppose i have a triad interaction then the three Fourier transform vectors each c1 is a vector the three vectors involved in those three modes are collinear proof of the proposition is the picture so the picture okay is transparent but it's not that transparent so i want to redo that on a on a blackboard that'll make it a little bit more clear but what i want to say is the the resonance set is an intersection of light cones so dual light cones dual light cones so the dual light cone positive going forward backwards going negative you make it out of these are n plus one vectors in spacetime c zero is the spacetime component super index are the components of a n plus one vector and it's where c zero is equal to plus or minus the length of c that's the dispersion relation okay now let me interpret that picture for you remember i need omega of c of one i'm going to call omega one plus or minus omega two plus or minus omega three to be when it vanishes that's what it means to be resonant but i my problem conserves momentum so also i have to have these three vectors add to zero so uh since the the the the convention is that omega is always positive i can't have plus plus plus so at least one can be minus and then it's the same as if two are minuses in order to have a non-trivial resonance except for zero zero's special case and that till and since omega is monotone increasing because it's omega is equal to c right omega c is equal to c length of c that tells me that this is the big one and then there are two smaller ones in length that are catching up so let's draw that in one dimension i'll draw a big so i have a big blackboard here so here is um the omega axis here is the c axis and then um and here is say c one and then here's the dual light cone so if i go up to here and over to here this is of course omega one nobody doubts that but i'm have to add the c's so i would say that i go along here and this is going to be minus c two and then this distance here is minus c three so they c one plus c two plus three three equals zero and that tells me that this height is of course omega c two and the thing that my applied mathematician friends taught me is to understand a three wave a triad resonance take the dispersion relation which is by the way over here as well and and put it up here and so if i go from c two to c three that's going from here to here on this copy of the dispersion relation right that's a second copy and so this height is omega three and so that's uh omega one is equal to omega two plus omega three it's completely clear the whole line so everything's three wave resonant okay but i want to make the but that's a picture in higher dimensions so any higher dimension so this is the first component any higher dimension is now here's the all the other components and i think it's best if i fix c two here so we don't get confused minus c two here and i'm going to make a resonance by moving c one around in a circle with a sphere with a fixed radius and then i'm going to see what c three has for us so i'm going to go over here this is going to be c one this is minus c two so minus c three is that and then i do the same picture above c one i draw c two and c three and the result is this here's the big dual light cone that's above c one here's the little dual light cone that's above c two and this height from here to here is c three and it's completely clear they only touch on the axis and that means they're all collinear that's that picture yes do you really need light cones they have the impression that you have just a triangle and one of the sides is some of the the length of one side is the sum of the length of the two other sides that's this triangle i mean you that has nothing to do with light cones no well this is just conserve momentum but up here the intersection of the two graphs of the of the dispersion relation is the resonance i need to satisfy the omegas of those exceeds and so these two have to touch and that's not just a triangle yeah the the the the sides that built a triangle and what the resonance condition for the omega say that the length the longer the length of the longer side is the sum of the length of the for the wave equation you can say that again this is a picture of that this is a picture of that yeah that's a picture of that absolutely okay so proposition proof now everything which is resonant and is collinear by two proofs in fact of this you don't have to prove tonight okay um how do you make canonical transformations that's the second thing because there's a obstacle to doing this you can't just do anything ad hoc you could try you might touch upon one that's a good idea but um there are actually uh there are mechanisms built over history by which you can construct um canonical transformations and one of the famous ones is to use the fact that any flow is a canonical transformation so solve the Hamiltonian system fixed time that move points around that is canonical and so you so the idea which really dates the Birkhoff is uh find an auxiliary Hamiltonian system my wave equation is h so i'm going to solve for k uh hindsight actually foresight tells you that k should be cubic if you want to remove quadratic terms in your equations and flow that k by inform a Hamiltonian vector field with the k and flow that k and the solution map is going to be your canonical transformation so in order to do that in ways to see what k should be it's useful to put the problem in complex symplectic coordinates which is things that many people uh uh many people use for many cases in fact i think kenji had coordinates very similar to this in his talk and then if i rewrite h the regular regular Hamiltonian from at up to z is canonical essentially and in the z coordinates h has form which is modified but nice so the wave equation looks like this this is the linear wave equation looks like that then all those terms involving uh the m equals three and four and five they are now multi-linear uh convolution operators which we saw in other people's talks uh looking something like this and the cpq which depend on the Fourier transform variables are the interaction coefficients and i want this to be multi-linear so i just noted it down there the Hamiltonian satisfies a null condition this is my modification of the statement and i'll explain the relation in in a minute uh if that interaction coefficient these are the if i take m equal three i'll have three z's and i'll have c with three uh indices if those interaction coefficients vanish for every resonant triad and now that's equivalent to Kleinerman's definition well Kleinerman only has one wave number right because you plug you have a polynomial non-linearity of cubic order and you and you make a symbol so to speak out of its derivatives but you put in one wave number and it has to vanish but we've seen that we're in a situation where we have we're taking the Fourier transform of a function so we have double convolution three functions there are three wave numbers they're the same because of the co-linearity that any resonance all three resonant wave resonant Fourier transform variables are just three multiples of some base variable and Kleinerman's definition is testing the base variable that's basically it so without doing a general situation let's do a particular situation here is a cubic Hamiltonian that gives rise to a null a null form and it's probably the simplest one i can think of and i'm going to put that in equation and then i'm going to talk about removing that with a canonical transformation so express this like i did in the previous slide in Fourier transform variables okay it somehow looks a little messier than i usually think but this is easy actually so h3 you can read it off there's a p and then the u squared and a p but you have to write it in z's and so there are a lot of cubic terms well i guess there are eight different possibilities so i put down two characteristic ones and then this distance difference is this difference and the normalization ended up with the factor of normalization so that's a term which expresses h in Fourier coefficient in Fourier coordinates then the formalism to eliminate h uh which is really doing the standard thing but in the continuum on a function space is to find a k3 such that its plus on bracket with h2 gives you h3 because then the time one flow of k3 will eliminate h3 and will modify h4 and higher and you do that despite triad resonances and now the whole the whole rest of the point is to find to show that the Hamiltonian vector field actually has a well-defined solution solution map on an appropriate bonnet space with maybe some continuity and smoothness i'll talk about what you can expect and i need it i need it as a bounded continuous transformation of a neighborhood to a neighborhood in bonnet spaces okay so what so what is the solution of a homological equation look like well it kind of looks like the Hamiltonian with cubic terms of of of z's and this pre-factor which is the interaction coefficients which is what the equation is telling us to do but to get rid of h you have the what are what are the denominators those are the things which can be small and zero and that's what the dangerous part that you have to deal with so think of it is coefficients with interaction coefficients with these powers the other interaction coefficients with different combinations of z's and z bars etc i only i only put two terms down there i think eight or so terms in this okay let's see the first denominator is non-zero except a zero so here are the non-resonant terms so that's cool just that's fine the second denominator is this vanishes on a resonance set which i drew for you right here and then the null condition tells me that the numerator vanishes exactly where the denominator vanishes so it is where it is on on on collinear three Fourier transform vectors and it's completely clear because i made my example with which is sharp which is uh which is uh Geneva's proof of my resonance condition it says that says it's sharp uh sharp kosche Schwartz in rm okay so then i have to solve that vector field and so i have to so now i'm going to solve z dot is equal to Hamiltonian vector field involving k3 which i'll call this Hamiltonian vector field but but solving is not time time is what you do on the physics time is what the wave equation solves it's an auxiliary parameter that i'm going to take to one i start it starts out the identity at time one it is my transformation so i'm going to call it s that's probably a bad choice because there are a lot of s's in our lives but okay and then the flow map is when you evaluate at time one that's going to be my transformation and it's going to be and it's tau three because i want to normalize the cubic terms of the Hamiltonian and the next thing i'm going to do is tau four except this is work in progress so i'm not going to touch it but okay tau three and then the question is does the flow map exist and it's a little bit bizarre because it's not a pde so Hamiltonian vector field is this i differentiated the k3 and put multiply by i and a few things here and it has lots of terms i have lots of dots now it has about 32 terms okay but but we see what the what the difficulty is what we're going to have to worry about is here's a non-resonant denominator two of them because that's the structure of the first term but here's a resonant denominator and we're going to have to make sense of this evolution equation with these coefficients so let's think about it um z was made out of u and p it's l2 norm is not the is not the energy norm so if you forgive me for making a non canonical change so that i have a new function related to the old function w whose l2 norm is the wave energy norm then it's a little bit easier to kernels to be a little bit more easier so more so they're homogeneous they turn out to be homogeneous and so then the analysis is and so then you have to do a local analysis on the resonance set and the resonance set's not too complicated but you do have to do it in in patches and it turns out that the the the kernel looks like this uh plus uh other variation variants of that and here's a local estimate there's a bounded part which gives you no problem but there are singular regions and the singular regions mean that even with a null condition it's not bounded vector field it's not a bounded vector field now how can that be if the numerator vanishes when the denominator vanishes well hey this is in higher dimensions if we're a one dimension numerator denominator canceled but this is high dimension so the singular parts are when the cone fits all the way down to the bottom of the cone or up in the top so there is there is this singularity so it's not a a monger thing not a lipschitz vector field on any reasonable monoc space that i know maybe not none i don't know but anything you could think of where you want to be you have to do something else so what what do you do well we do pde vector fields any interesting vector field is unbounded what we need is not that the vector field is bounded but its inner product with the with the position vector in the Hilbert space is bounded because we don't want norm to grow too fast that's called an energy estimate so consider how if i take just standard so below space and i take the and i and i suppose i have a solution is the is the hs norm controlled of that solution so take d by ds of the l2 of the hs norm of that solution just to see if you have a chance of approximating and get a and getting a well-defined flow map and it's bounded by z cubed and so it will blow up for large time but for sufficiently small data you will have a time one flow map so that's what i want so but then what's the drawback hey the technology for the wave equation can fit here but it's a lot better in invariant norm so below spaces so this is not this is nice to know there's some cancellation it's unbounded vector field but it has energy estimates but it's not quite what we want to do so take angular momentum operators take dilation operators actually only want to use those two what happens under Fourier transform well you know angular momentum the jl angular momentum operator and a function of x that should be a little x sorry is equal to angular momentum operator and c of course it's it's translation it's a it's a Fourier transform invariant a dilation operator if you dilate an x you collapse in c so you just get plus an extra term so under under Fourier transform you get the same operators the operators obey the Leibniz rule with regard to these kernels that's because the kernels are designed for things which are which are which have good behavior under dilation and which are translation invariant not under each c but under simultaneous rotation rotation by c of course it's a it's a frame invariance so you then don't just take linear derivatives of your function but you all you also take angular momentum derivatives and dilation derivatives and actually let me be a little bit more precise so the function space z is where we work but I just want to make that change of variables to get to the w's so I'm going to look at a w so below space and I want w but oh w is a function of c on the Fourier transform side so I want multiplication by c angular momentum by c dilation by c that I want to be in l2 and I want those to be in l2 when you sum their terms up to order s bar and s bar is not s because I need another index but the Fourier transform the un-phoria transform allows derivatives it's actually a disaster to put derivatives on the Fourier transform side because you want a you want a neighborhood which stays stays fixed and and if and if you put derivatives on the Fourier transform side you put weights on the space side and wave equation propagates things and weights make them grow so you want to stay with a space that looks like this here's a little bit of amusement this is a old estimate of Sergio it says that w the the space coordinate version of w is bounded by the l infinity norm is bounded by this invariant norm so below space but it's completely self-evident the Fourier transform is l infinity as well just because in fact you don't even need all those derivatives you can use less a few angular mentums and only one dilation the hard work was energy estimates in the z space so you have to take angular momentum and and lambdas and control the z's but the basic the basic theorem is there the energy estimates work and so you can force force for a ball radius epsilon in the z space solve the the k equation the the the vector field the auxiliary vector field Hamiltonian vector field given by k3 up to time one and then that flow map is your solution is it a flow well what is a flow should a flow be c infinity should a flow be differentiable well you know with PDEs a flow is can be continuous but is rarely differentiable in the same function space in which you're sitting but you do have a Jacobian and i'm sorry got squeezed into the bottom side the Jacobian of the flow that is the z derivative of of the of the solution map at time s minus the identity is bounded by z sublev norm but you lose a derivative and if you look at two derivatives of the map you lose two derivatives so it's a flow what i want to say it's reasonable consider to think of PDE solution maps as flows smooth flows but on scales of spaces because you don't expect to have differentiability in the same space but the Jacobian is bounded a space down et cetera sometimes it doesn't have to go all the way down because sometimes it can go lose half derivative but that's what i think is that was a little bit of propaganda that's what i think transformation the transformation z prime is tau 3 of z which is the flow map at time one achieved a canonical transformation of our Hamiltonian so it's new Hamiltonian remember you you you plug in your new variables it has no z no h3 anymore it only has a remainder of h4 and the remainder at 4 even is very concrete made out of some Poisson brackets of the k3 that we had already had in mind and the old h4 and now m is 4 and we have that improved existence theory of the of the former theorem of the 1980s this just happens to be by a canonical transformation so what's the point well i think i wanted to elucidate the the the Hamiltonian sense of what it means to be a triad resonance which involves three wave numbers and three linear modes and in their non-linear interaction and the null condition which involves one and that was partly the lemma that that almost trivial proposition which says that that thing that such wave numbers such resonant wave numbers have to be collinear and then the estimate which says that you can make a analytic sense of the mapping in such a thing but of course a goal would be to aim for the dimension n equal to where one transformation is not enough and it'd be nice to make a second transformation well this also is an old theorem so it's also coming back to an old theorem i guess so the audience is probably better than me at the at the history of this but alinak has a has a has a theorem also uh hoshige has two papers and Jean-Marc Delors who's been at this conference has at least one paper on global existence for small data in n equals two and there are transformations in it in other considerations i'm not sure that each one of these papers has exactly the same condition for the for the third order terms as as each other but i think those should be the self-evident at least from a Hamiltonian point of view by considering them as resonances between the plane waves and and it's clear that they're not going to be it's not going to involve just one c they can't be collinear as soon as it's for they might be in planes but they're not collinear okay everyone knows okay oh sorry i was going to stay at the state of theorem excuse me uh the standard argument is uh oh that's right so i don't have to go through this argument because it's a sophisticated audience but uh i'll just put it all on the board just so you know that i know it which is if you have linear if you have a decay of linear equations at time n minus one with the k rate time n minus one over two you also do energy estimates in the uh in the space and in the z space and uh you're going to end up with the c one norm of your solution up here this is solving wave c one norm up here but because i've increased m i have an advantage here this is why the m minus two shows up because one m is you've done one of the m minus the the the last m is on the on the ground this is considered the coefficient and you replace this with that uh sobelov estimate from that i showed you us three slides ago and put this on the right hand side it gives you and it gives you a bound a trapping uh zone in in the initial data solution space and uh that shows you cannot form a singularity and if you do it in uh in the borderline case you don't get a trapping zone but you do for a while and for the while is for exponential time and then i'm done so thank you and i'm ahead of time just please it happens for a nonlinear shreddinger can you do some the same type of uh it's a nice idea uh of course we've had that idea but we haven't pursued it but why not why not why not there were papers uh in the last century by high ascii or nonlinear polynomial and at some point there was some kind of almost global existence in some i don't remember the details okay it'd be good to revisit that i did write a paper on the comparison because for small data uh for nls there's a small data scattering result and there's also the normal form so scattering you could think of as a normal form it's just not very explicit tells you the solution is linear if you go forward in time and backward by linear it says you it's a way of conjugating your equation to linear so that's that's one conjugation another conjugation of course flow backward in time and then compare with a linear equation backward there's another conjugation they're related but it's another conjugation and the third conjugation well you've been have to do it time after time you know order by order is bierkoff normal form but if you have scattering why should you bierkoff normal form but scattering is so okay that's a hard question to answer but one thing you have with the normal form that you don't have with scattering is explicit mapping you know what your solution does where you know how you're changing it so i have a paper which compares scattering with bierkoff normal form for small data but it's not a big existence picture well maybe it's it's something maybe it's something our effort there was not the uh compare the existence theory it was sort of discussed scattering but i think the existence theory also would be worth it show more questions comments well yeah there was a paper by shatani pusate where they sort of made this connection between resonance and non-convision right so that i mentioned that was in my slide four i think it was so it's a 2012 they use the method due to masmoudi shatani and yeah bierkoff normal right and it's sort of a space resonance time resonance so that's a it's a it's not a change of variables it's a it's an inspection of a duramel formula i'm just talking about just a yeah connection between resonance yeah because the resonance that would be the same resonance that would be the same and the single and you have to sing it have to you have you handle the singularity in a certain way interesting about that is you need regularity on the Fourier side so that's a piece of this is it comes out that you can control regularity on the Fourier side of the transformation at least yeah and they don't go twice they don't go twice so i'm aiming at twice you have to change variables again that's what bierkoff normal form is you you you change variables and you adjust some coefficients and then you adjust them again yes so basically for the quadratic linearity I mean the condition is correspond to the coefficient of the horizontal term zero you know it needn't be it needn't be my example it turns out to be yes yeah but is there any the analog of the two linear form so in view of the in the formulation of the bierkoff normal form it is not difficult to find out what is like a the coefficient of the the the ternary the linear terms right then the if such such a coefficient is zero then can we have a sort of like a some sort of easy formulation or condition for the the in physical side like the north the north condition the null conditions on the Fourier side oh you but I guess it's on the physical side too yeah I think probably it's the same probably it's the same the the null condition you plug in one wave wave number one one Fourier transform variable space you plug in one spacetime c and you test whether something is zero but but my sense is on when you do go to the quartic non-linearity you will have more resonances which are not all collinear and so you have to compare say pairs of Fourier transform variables it will be like that suppose there's a pair of Fourier transform variables each one on the light cone how can they how can they interact it will be algebra a little bit like that but one thing I want to say but it's not exploited in this which is on a compact domain like a torus making a normal form involves dividing a Fourier series coefficient by the resonant relation and if it's zero you cannot divide that has to stay there but in the continuum it's very possible to divide by something which is zero for example the Hilbert transform is a perfectly bounded operator which which has a singularity so I really haven't you know found a place where it's very nicely articulated but the potential is there that it's too strong to demand the vanishing of the interaction coefficient just should demand that the relevant vector field after you know expressed in Fourier variables has a sufficiently regular singular integral formulation and so you could say that even if they do vanish you see that the vector field has a singularity but it's relatively mild singularity the one I've shown you