Birkhoff normal form for nonlinear wave equations

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Birkhoff normal form for nonlinear wave equations
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Many theorems on global existence of small amplitude solutions of nonlinear wave equations in ^n depend upon a competition between the time decay of solutions and the degree of the nonlinearity. Decay estimates are more effective when inessential nonlinear terms are able to be removed through a well-chosen transformation. Most wave equations that arise in a physical context can be considered as Hamiltonian PDEs, that is, partial differential equations that can be formulated as a Hamiltonian system. In this talk, we construct Birkhoff normal forms transformations for the class of wave equations which are Hamiltonian PDEs and null forms, giving a new proof via canonical transformations of the global existence theorems for null form wave equations of S. Klainerman, J. Shatah and other, in space dimensions n \geq 3.The critical case n = 2 is also under consideration. These results are work-in-progress with A. French and C. - R. Yang
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a the about I want to tell In the long term work to end the war to end thank you very much for the organizers for having me and the attendants from for coming it's actually lovely spot and spending the great conference especially because of the variety of topics that have been discussed in various parts of come up that makes it the police keep on so but I also bullets here should tell you what I'm going to talk about and also tell you 1st maybe my collaborators this amended French from who is a student of Michael Taylor who was with me at the master for 3 years has moved to Hereford and Cherianne who came to the fields and McMaster and it's a little bit but I say it's a little bit humbling to not be talking about a new ceremonial German entitled the surges as well as the chateau and wanted to From about 30 years ago if not a little bit more so OK I'm sorry and I've adjusted and and I to my justification for doing this well or maybe it's just the shift slight shift in point of view I the advantages but at least it articulates something in in a different language so it's nothing new here and I apologize for that but but sometimes it's good to have to revisit "quotation mark all the things something to tell you about a nominee wave equation I'd like to talk about transformation theory because I think that's what we do not believe in PTE is it is changing variables OK we change X to Y and we change you sometimes do you square you squared minus the body at you but we do not sorry he so surgery because of all the time but most of us don't and and relativity you were still just doing all local the local coordinate transformations but you're not changing of bonnet space to another quarter notes and bonds and so let's part so we're starting to see that little bit was really rather undeveloped so here's a story example where you change coordinate the mnemonic space to other work in about physical eastern neighborhood of and I wouldn't know what the Hambletonian framework and I'm 1998 estimates to make everything work so that's the format of the 1 of the show now it's been implicit since day 1 of his conference and now I'm an articulated and I'm not sure anybody else articulated so I'm talking on the wrong day but OK why change variables well let's just look at the scalar roadies Ziad article disease where it has an exact solution all small data or any that for small data we think the zeros eagled epsilon thinking on small and Zia tea isn't exactly a 1 1 minus epsilon and has at times singularity 1 upon politcal selects the best of what was called the "quotation mark radical because the square and then he sells requests lies ahead covering people who cover the slides there's equivocation W's the WTO and its exact solution is root of epsilon squared over 1 minus to absence insecurity to start with the 1st started with data Epsilon and then the time existence is 1 from to actions were that I have an order amendment longer existence time and that's the point Cu was last tested court but it's pretty obvious certainly there women's room and also doesn't change if you at least not essentially if you move things around a little bit like Natalie near-term and it makes little complicated nosy interview complex and higher returns .period nest normally and without special details change the fact that singularities Kennedy so was talk about the real problem which is among a new wave equation so I'd like to talk like I like to talk about the equations which were doing this 2 time derivatives it still Paulson so really starting with a with a background linear where equation that Salinas Asian and 0 0 1 0 to be a solution then of some Mama near-term which lawmakers more precise later but let's say that it is at least order M minus 1 in its variables so the "quotation mark case would be square variables he would be Cuba and also because problem so initial data at time 0 g an initial momentum H and some basic question is sort of something people spend their careers studying is what's the summer resistance if I give you some some classifications maybe a civil space that's what I would I'll tell you what my disease going to be later for several scenes later several possible then you want to know how long the solution last and of course the best result is at last for all time and then there's so that I'm really working the class of small data all-time Lionel global existence OK so this was worked on in the 19 eighties I think seduced thesis is on this and then a lot of work of these fundamental enormous there's work Mandarin that and the time and Joe Walsh at Tottenham and others and probably not people in the room so you can show at me but do it after the talks because time is running in practice you're keeping track and so just that really really short here is here is in existence in global existence suppose that one-half times the dimension minus 1 times and as to what the court nominee area was order and minus 1 for Hambletonian reasons so and minus 1 one-dimensional and sometimes a menace to the that's bigger than 1 then for small Koshy data data appropriate solar space resistance is infinite and as a balance as everyone knows between solution spreading out indicating which is dispersion certainly not a strong asserting that its burden of sorts and the nominee effects trying to focus of things together and that's the small enough Sobel of space the dispersion wins and just I know everyone in this room knows that this for most people in noses but was the linear deteriorate tho it was the decay rate for the linear wave equation which for small data were reflected in a victory for the long-awaited equation and it's of course US security too In minus 1 over to which reflects that 1st factor which is part of the story so was just do them
justice calculate if M is equal to 3 so the nominee areas and minus 1 is quite erratic then who House then when we satisfy the hypothesis well I need and minus 1 and minus 2 to be bigger than to lose so and that any bigger than 3 which is bad luck for us to live in 3 states will according to we 3 space dimensions reminds me a funny joke about find little to do with the flu for reasons for reasons of time but if em for if you had a higher nonlinear then you do the calculation and integrated into which is a word also which is good news for us living at 3 space but a little bit better because 2 is also an assistant manager and then the borderline cases have these almost Global Resistance leader exponential time of existence in the water and the 2 when when the equality in those 2 dimensions and to nonlinear cases OK so I haven't said anything wrong and that's moralists how things stand well transformation theory
it tell you there's an interest from going Zita W. Wright is interest in taking quite a quite Radic nonlinear and making a change of variables may be in large based may be in neighborhood of our space so that you no longer have quite erratic term but only have Cuba terms and this is an ideas developed slightly after the the things in the 1st slide but is still alive today because this summer must Moody has some work here but a and that some of these ideas had had entered in hyperbolic situation with histories of top Muriel which which revisit the idea of society to change the variable or essentially changes of variables to get rid of them that the hub obstacle of a quadratic Germany equations and so I think I'd like to prefer to say that in the following way it 1st special nonlinear set aside a special condition which surgical no condition then Enford dimensions 3 that was the key 1 you can make tea existence time equal infinity but by transformation would get rid of the of the Quartet term in favor of more complex CU terms and in the dimension to that extended it's a borderline case expansion time existence so the idea is change variables and 1 thing that strikes me is 0 and because Will we trial of noticed the number of tries if you make a change variables certainly in the style that of this theory it's hard to make it again that is you start with the easy not nearly as maybe polynomial there's something and you make a change of variables In it's complicated non-local and just try again it's not so much a tribute to what I want to do is introduce Hambletonian formal for form of formalism which would Help helps the idea of making changes the variables that have to do a rigorously but it gives you a lot His you systematic way of of doing this so let's see what happened well I can do In fact the probably don't want to Arianna the ones that interest me the ones really comes from physics the ones that come from all around him so I have enlisted in the quake in could anomaly and nominee wave equations that come from that the principle of the section but you has some very pretty infringes the principle of 1 direction with the word more and and then action functional is a time of a Lagrangian and the Groningen I want to start at the wave equation so there's a quadratic Cottonwood Rongen plus other order terms and the reason my nominee Aridi initially was order and minus 1 is because I want this term to be order and and satisfy some small conditions worse for small art and then once you have this form of wave equation but you can effect the others are the transformer at least locally near 0 where you really just to satisfy the following for the classical formerly formalism you could take a variation of formerly with respect to you you'll find key and you call that P and that starts out with a U T a U . plus higher returns in this algebra or the implicit function theorem but locally in each exon T 2 come up with a new all you you and PT where every time in all right in your embassy you take you have to say that I've that's a function of P and Israel's function for a year so the village the transformed and
puts us into gives us the Hambletonian which is a bona-fide Hambletonian and it's a function of UNP an Israeli Justice form and you evaluate every time you CUT in order on and as I said you have to use that implicit function course at a changing from you you got to UP but that's not achieve burial bonnets space that's point like you the X M P O X said and algebra locally so this is a cost Hamilton in system in government coordinates where are you God is equal to the gradient of age with the specter of was inspected the gradient In this case the star who was better able to an inner product and P out is minus expected you that's a first-order Oracle Systems equations equivalent to the above along the way was not the only way to make a 1st for a system but it is an elegant way making system the carries through certain certain other structure which I like to use OK let's do it for the money wave equation because I was doing something kind of general issue and very soft but if l is equal to that the quarter added term plus the high-order terms the come from the the the Of the press release action L 2 is the Lagrangian quadratic is the difference between Canada potential energy list linear version of another Federal energy and then you do that exercise with H & H H 2 which is the sum of financial kinetic and potential energy plus a changed are which related the P by the change of variables and they are all border am as I'm indicating where the superscript and then you differentiate as I told you on the previous slide and to get an which is among linearity order and minus 1 years so that's Everything I think consistent so far so I'm sorry let's do something simple I was flying the plane waves it tells me I have to find resources sold Fourier Transform is that of the linear equation this is what evidence theories about gives you a dispersion relation dispersion relation tells you but where the dual like Conyers but in with the speed 1 like is the same as a you like common stock just rabbits low-income and then here's what we want to do but I want to make a transformation From from disease which is the vector you better function UPA which lives in some summer space of usable space to a new variable Susie prime I know what I insist that this lives in the same monarch space and I wanted that transformation to be the following a 1 1 to be economical well it's because I'm going to designed transformations to eliminate things and I if I make a general change a variable of vector field that involves the Jacoby and general changes the rules and it's harder to win over a relationship which involves a transformation and student phobia then due to just if you think about the present just worry about transformation is also being canonical allows you to sit within or is it allows you to change variables to to design and Woody said designer change variables without the Jacoby and bothering you divert the fact that its canonical is dealing with some out religious and that tells me that the tells us that we can come to a new system of equations Fawzi Prime which is just like the old system of equations but with a new Hambletonian where you just composed while here has to be the inverse of z in order to get the new equations and so the new Hambletonian is going to be changing variables with its With its orders of of nominee and to be in beer government formed or capital and it means that you retain at every turn between 2 and then you buy this process you don't change true by the way the notice what I'm thinking but you change 3 and higher and then you choose what and an annual you order at the residual be endless once order and that will remain and you're insist upon a change of variables having these terms being only resident for example 0 sometimes there are no residences in the 1 0 0 was good for example if you get rid of everything that's even better sometimes you can't so each and retains the most only resonant terms so in this in this process is only resonant terms and resident turned little ones which you could say plus some commute with the squad erratic Hambletonian another way to say that is the time they are preserved the same action in roles that to quit at the linear way freedom president OK it's over to call a reduction of the normal form and dynamic system is part of averaging and we all want to do it once in a stock effect it's work-in-progress is in my 2nd side so that's only all we've done so far OK so when am is equal to 3 and I want to consider the resonances those are known as 3 wave interactions were triad interactions and those of you who you are and Buchanan find by a formalism which has to do with the army the wave number so I have to preserve and conserve momentum so I need to read waited all season are and I should have said that the right who is so confusing that the third-quarter foot Mike is a vector In all are Ken Anderson space sorry it's happened space and manager says that I consider romantic and then to be resident the frequencies have to add or subtract so a try resonance is 1 with frequencies adding and subtracting 6 3 3 and I'm OK and then back to the TV I know infinitely many almost entirely many papers have been written about this on a formal level but that the pedia I'd like to understand and the construction of such economic transformation and its mapping properties on which so that's based on which monarch space and then you usually think of these these as discrete indices of a young of the dynamical systems finite dimensional system finally indices or on a contract manifold in the discreet but hearing continue and continue plays a role in what were once talked so when exes it's important where is where equation on the Taurus is going to be a different story on not doing that when season Warren and continuous variable and things actually sometimes Harvard that's 1 thing to find so here's something about wave equation try interactions proposition is suppose I have a try interaction then the 3 Fourier Transform vectors each C 1 is elected the 3 actors involved in those 3 modes linear approval of proposition is the picture so far the
picture OK is transparent but it's not that transparent won a redo that on a blackboard that make it a little bit more clear but what I want to say is that the the residents said is an intersection of light comes so icons dual like those that do will alike ,comma positive going forward backwards going negative you make it out of these are endless 1 vectors In space-time c 0 is the space-time .period sober index are the components of the and was 1 vector and his works C 0 is able to plus or minus the length of CSE that's the dispersion relation OK now let interpret that picture for you remember I need all made of sea of 100 kilometer 1 plus or minus made too plus or minus omega-3 to me when it vanishes that's what it means to be resident but the end of my problem concerns of momentum so also I have to have these 3 vectors the and 2 0 so since then the since the army the video the convention is that lawmakers always positive I can't trust was plant so at least 1 can be minus and then it's the same as to minuses in order to have a nontrivial residents except for a 0 0 special and that's and since Omega's monotone increasing because it's only is equal to see riding on the seat it scientific extremely like that tells me that this is the big 1 and then the 2 smaller ones in life that are catching up so was drawn that in 1 dimension although of big so I would be blackboard here so here it is New Media Access here is the c axis and then some and here 61 and then here's the dual icons so for years and over here this is of course for nobody doubts that but I'm happy to add Dixie's so I I would say that I go along here and this is going to be minus 62 and then this distance here is minus 6 3 so let's the 1 perceived to 253 people and that tells me that this height is of course a major and the thing that my applied mathematician friends talk about taught me is to understand the 3 the trial residents take the dispersion relation which is by the way over here as well and men and put up here and so if I go from C to C 3 that's going from here to here on this copy of dispersion relation right that's a 2nd call and so this site is omega-3 and so that's all major 1 is able to make it 2 Pleasant McCaffrey's completely clear the whole line so everything's three-way present OK but I want to make that but that's a picture and higher dimensions so any higher dimensions so this is the 1st component any higher dimension is now he is the only other components and I think it's best if I fix C 2 years so we don't have computers minus the 2 here and I'm going to make a resonance by moving C-1 1 around in a circle with a sphere with a fixed radius In the would C 3 has for assortment of over here this is going to be 1 this is minor succeed to Jerusalem minus the 3 that and then I do the same picture above 61 hydroxy to its history and the result is that here's the big warlike tone that's above C 1 hears the little like ,comma that's ability to end this height from here to here is C 3 and is completely clear but only touch on the axis and that means the role :colon that's said picture Reims yes we also have a version of the trying and 1 of the site of some of the National Flood said he saw the the of that this strike well this is disconcerting Emmental but up here at the intersection of the 2 graphs of the of the dispersion relation is the residents I need to satisfy the lawmakers alone exceeds yeah and so they to have to touch and that's not just a problem have the acceleration of the fire Campbell was the reconciliation of the yesterday that there is no guarantee that the homicide site the softening of the for the wave equation you can say that again this is a picture was picture yeah it's a picture that absolutely OK so proposition approved now that everything which is resonant pulling right to proves invective the only approved OK I have to make canonical transformations that the 2nd thing because there's about obstacle to doing this again just do anything the trial might say touch upon 1 that's the idea that they're actually there are mechanisms don't over History by which you can play a construct and canonical transformations in 1 of the famous ones is To the use the fact that any flow is a canonical transformations of solar Hambletonian system 6 times that move points around that is phenomenal and so you so sold the idea which really coffee is finding Zileri Hambletonian system my wave equation is H. trying to solve for K In hindsight actually foresight told you that k should be cubic if you wanna Richard removed quite terms you're equations and flow that can buy little and former Hambletonian vector field with the care and flow that K and the solution map is going to be a kind of transformation so in order to do that on ways to see what case should be it's useful to put the problem in complex eclectic audience which is things that many people in the many people use for many cases infected think candy had scored it's very similar to this in his talk and then if I rate rewrite h the regular come regular Hambletonian From a U P 2 z is canonical essentially and in the zinc Warner's has former which is modified but nice so the wave equation looks like this this is the way linear where equations like that then all those terms involving other M the equals 3 and 4 and 5 they are now multi linear convolution operators witches on other people's stocks of stop looking something like this and the CAP Q which depend on them Fourier Transform variables are the interaction coefficients and I wanted this to be multilingual largest noted down there the Hambletonian
satisfies all condition this is minor modification of the statement and I'll explain their relation in a minute but if that interaction coefficient These Are the Good Friday it 3 I have seen three-season I have C with 3 other indices if those interaction coefficients
vanish for every resident trial and now that's equivalent to climbing stairs well client only has 1 way another right because you plug it you have a the polynomial nominee area of Cuba water the new make it a symbol so to speak out of its derivatives but you put in 1 way of number and has vanish but we've seen that we're in a situation where we have working the Fourier transform of of a function so we have double convolution 3 functions the 3 wave numbers the same because of the coal Indiana but any resonance all 3 resident what a resident Fourier Transform variables are just 3 multiples of some base variable and economists definition is testing the base that's basically it so without doing a general situation was to predictive situation here is a cubic Hambletonian the gives rise to and form and Bob simple stomach and think of an I put that equation and talk about removing that was economic transformation so expressed this like it did in the previous slide and Fourier
Transform variables bogus somehow little messier than you think but this is easy actually so H 3 even read off theirs a P and scoring Peabody apparatus and disease and so there are a lot you terms will address the 8 different possibilities so I put down to characteristic ones and then this distance differences this difference and the normalization ended up with the federal volition selects a term which expresses age in 48 coefficient inventory the coordinates then the formalism to eliminate I wish this is really doing the standard thing but in the continuum on function space is to find a case 3 such that it's plus on record with H 2 yes to H 3 His and the time when flow "quotation mark case rating will eliminate age 3 and will modify each for Anaheim and you do that despite try and residents and now the whole the whole wrestle .period is defined we the just show that the Hambletonian vector field actually has a well-defined solution solution map on appropriate space With maybe some continuity and smoothness of talk about what you can expect and I need it and it is a ballot cut continuous transformation of a neighborhood to enable embodies OK so what what is the solution ecological equation look like well it kind of looks like the Hambletonian with cubic terms of of disease and it's pretty factor which is the interaction coefficients which is what the new equation is telling us to do but to get rid of age you have that water is 1 of the denominators those are the things which can be small and 0 and that's what the dangerous part you have to deal with so I think it is a coefficients was interaction coefficients with these powers in the interaction coefficients with different combinations of these and the bars etc. finally I only put 2 terms down there I think 8 over so terms in this OK can let's see the 1st denominator non-zero accepted 0 serial numbers terms last fall that's fine the 2nd
denominator is this vanishes on a residence at which a true for you right here and then the null condition but tells me that the numerator vanishes exactly where the nominator actions so that is where it where it is on on cold in here 3 for a transfer vectors as the belief here limited my example with which shot which is which is the neighbors proof of my resonant conditions says that says this a sharper Kercher Koshy Schwartz in wheelchair so then I have to sold that vector field and so it happened so now I'm going to solve zeal .period is equal to the Hambletonian vector field involving K 3 which all this Hambletonian vector field but that's all it is not time time is what you do when the physics times with the wave of pigeons salt is an Exira parameter that I'm going to take the 1 start start up the ante at time when it is my transformation from colored S is probably a bad choice to the water as is analyzed by OK and then the flown out the other is that when you evaluated time 1 that's going to be my transformation and it's going to be in tells straight because I wanted normalize the CU terms of the Hambletonian and the next thing I knew was held for except this is work in progress on makin a touch it but OK tells and then the question is both the flow that exist and it's a little bit bizarre not Pt so Hamilton in vector field is differentiated the case 3 input multiplied by and a few things here it has lots of turns and walks adults now has about 32 terms but what we see what the at what the difficulty is what would have to worry about is here is a nonresident nominated 2 of them because that's the structure of the 1st but here's a resident on it and we're going to have to make sense of this evolution equations with these coefficients so let's think about it I was made out of you and P it's a L 2 enormous nothing and is not managing so if you fruit forgive me for making a non-canonical change so that I have a new function related to the old functions w who else Wong is the wave energy than as a little bit easier than kernels to be a little more years of war so the homogeneous the drug homogeneous insulin analysis it said so then we have to do a local analysis on residents rules it's not too complicated but you have to do it in patches and it turns out that the the to Colonel looks like this please other variations variants of that and here is a local estimates there's abounded part which gives you no problem but there a singular region and the singular regions I mean that even with the conditions it's not about it but view is not about you now how can that be if the numerator vanishes when the nominator engines all Hey this isn't higher dimensions if world 1 dimension enumerated nominator canceled the high-dimensional so the singular parts of when the phone fits all down the bottom little or open the top so there is there is a singular so it's not a among the thing knowledge of sector feel any reasonable wanted space that I know meeting online I don't know but anything that think where you want to be you have to do something else so what would you do well vector fields any interest in Deckerville unbounded what we need is not that the vector field was founded but it's in product with the with the position vector and Hilbert space is bound to me don't want norm to grow too fast that Colin Energy estimates so consider how if I take just standard Sobel space may take that and I and I suppose I have a solution is the is the HS Norm controlled that solutions with the DVDs of Elton of a just north Of that solution just to see if you have a chance of approximating and getting a well-defined flown out and it's mounted by killed so it will blow up a large time but positions for a sufficiently small data you will have less time on floor 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 yeah In very long some stations so this is not this is nice to know there's some consolation its undoubted vector field that has Energy estimates but is not what what quite what we want to do so take the angular mental operators take dilation operators actually want to use those to what happens in the Fourier transform Will you know angular meant the jailing momentum operator In the function of X that should be a little like is equal to any other Madam operator and the corsets this translation it's a it's a Fourier Transform variant the dilation operator if you die later XU collapsing season just get plus an extra term so underfunded Fourier transform you get the same operators operators albeit aligned role with regard to these firms that's because the kernels are designed with 4 things which are in which are which have good behavior under dilation and which translation during not under each CSE but under simultaneous Roeschley rotation see of course it's because of supreme so you then don't just take yeah linear derivatives you are function but you all you also take any lamented the derivatives and violation derivatives and actually only me a little bit more precise so I got the function space Z's where will work but I just want to make that change variables to get to the W so I'm looking to double usable space I want W but old W is a function of exceed the Fourier Transform side so what multiplication by CSE angular momenta makes the violation by sea but I want to be in to and I want those to be BNL to when you saw the terms of the order a small and as far as not a script I need
another and but the Fourier transform on the 1 Fourier Transform allows derivatives it's actually a disaster to put derivatives on a free transfer MCI because you want to know what you want a neighborhood which stays stays fixed and and if and if you are put derivatives on the Fourier Transform side people some space and wave equation propagates things and waits make them grow so if you want to stay with a space afford this here's a little bit of amusement this is an old estimated of Sergio it says the W those space coordinate version of W is bounded by the L infinity norm is bounded by this enduring norms of space but it's completely self-evident the Fourier transform his own Finnerty as well just because in fact you don't even need all those derivatives you can use less fuel angular mentioned and only 1 the elections the hardware was energy estimates In the disease space so you have to take angular mentioned them and Landers In control the disease but the basic the basic theory is there the energy estimates were and so you can force force for a ball radius Epsilon in these space solve 2 became equation the vector field the auxiliary vector field Hambletonian victory was given by Case 3 up times ones and and that will Is is your solutions Is it a flower what is afloat should flow we see infinity should have slowly differential well you know what PDA's a flow is can be continues but is rarely differential in the same functions space which was sitting but you do have a and and I can so I got squeezed in the bottom side the decoding of the flow that is there the derivative of of of the of the solution map at times s -minus the identity is bounded by disease Sobel of Normandy lose derivative and if you look at Tudor is a map you lose children so it's us for what I want to say it's reasonable considerate to think of PTA solution maps as flow smooth flows but on scales of spaces because you don't expect to have the French ability in the same space but the Jacoby is bound the space down and said sometimes it doesn't always happen sometimes will you have but that's what I think is there was a little bit of propaganda that's what I think transformation the transformations the prime is tells readers III which is flown out a time 1 achieved economic transformation of our Hambletonian so its new Hambletonian could you plug a new variables it has nosy and no H 3 anymore it only has a remainder at age 4 and the remainder at 4 even as very concrete made out some Pozen records of the case really that we have already had mind and the old age for and now for Amersfoort and we have that improve the existence of the of the of the former 0 0 of the 1980's is just happens to be but by canonical transformation so what's the point well I think I wanted to elucidate the the the Hambletonian stepped sense of what it means to be written at a trial resonance which involves 3 where numbers and 3 linear motors and in a in a nonlinear interactions and conditions which involves a lot and that was partly dilemma that there you almost trivial proposition which says that the I think that that such where members that present when numbers because linear and then the estimate which says that you can make the analytic sense of the mapping in such thing but of course the goal would be to aim for the dimension and equal to where 1 transformation is not it would be nice to make a 2nd transformation well this also is an old so it's also coming back to an old theorem I guess it was you off audiences probably better than me at the at the at the history of this but I would not visit has a has a theorem also who she has 2 papers and John law we've been at this conference has at least 1 paper on global assistance for small data in any equals to and there are transformations in another considerations I'm not sure that each 1 of these papers has exactly the same the condition for the for the 3rd order terms as as each other but I think those should be the self-evident at least from the Hambletonian point of view by considering them as resonances between the plane waves and I am and it's clear that they're not going to be is not on just 1 KSE they can't be calling your listeners this for there might be planes but they're not :colon OK everyone knows OK how sorry I was states interference Susan was standard argument it is all necessary so I don't have to go through this argument because it's a sophisticated audience but just put it all on the board just so you know that I know which is if you have linear behavior became a linear equations at times and minus what what with the affair a time and minus 1 over to you also do Energy estimates in the inner space and and disease space and you're going to end up with the C 1 norm of your solution appeared this is solving wave see 1 appear but because I've increased M I have an advantage here this is why they minus 2 shows up because 1 analyst you done 1 of em amendment that that the last dams on them on the ground this is considered the coefficient and replace this with that Sobel of estimate From the 3 sides ago and put this on the right-hand side gives you and usually a down the trapping zone In In the initial data solutions space and that shows you cannot form a singularity and if you do it in in the borderline cases you don't get trappings only you do for a while and for a while you were .period on and on and
on and on and you plan ahead and few it it is believed that this was only actually it on the same side of the it's a nice idea of course you've had that idea but we haven't pursued it but why not why not 1 of the I think In this essentially a steel mill the money was spent much of what he will do that right so far there was some kind of role model will be introduced governmental I wanted to be good revisit that I did write a paper on the comparison because for small data in for analysts there is a small data scattering result and there's also the normal form so scattering you could think of the normal form it's just not very explicit total solutions linear If you go forward in time and backward by linear so Jean Condit said Wael conjugating a salute your equation linear so it's this 1 conjugation know the condition of course flow backward in time and compare with the linear equation backward is another contribution the related but more and a 3rd conjugation or even have to do it time after time particularly with ordinary water is normal form but if you have scattering why should you be a proposal for for but scattering of OK that's a hard question to answer but the 1 thing you have with the normal form you don't have a scattering is explicit madam you know what your solution does where you know how you changing so paper which compares scattering with the government or for small data but it's not a big existence picture or maybe it's something in this Iowa are our effort there was not the to compare the existing series was sort of discussed but I think it is necessary also would be worth a lot of questions comments but yeah this paper plates the Sudanese connection in right so that I mentioned that was in my slides for I think it was 1st of the 2 thousand they use the I'm in method must must Misha and yet the argument is rated a sort of left space residence time residents selects it's it's it's not a change of variables it's in its own inspection of durable formula that just talking about just that yes collection yet because it businesses the residents that would be the single the Napa seem to have you have you have listen to their for 1 interesting about that is you need regularity on the forest lots of a piece of this is it comes out that you can control it reinforces on the transformation of the city built in what was and they don't go twice they don't go twice so I .period the you have to change variables again that's a different normal form you if you change variables and you just some coefficients and then you adjust them again yet so that they is could going on you never knew the means the condition ludicrous to the content of the press you know it needed to be it needed be but my example turns out to be a the steady normal the is true form so view of the liberalization of capital you said you could find a way to break up the condition of the country the view from which he asserts that Clinton used did tend have to remove it from he will be ready to work with you that little collisions in the freezer all you but I guess it's on the physical side to I think probably it's the same probably the the condition you plug in 1 way of winning number 1 140 transfer space the body and 1 space-time exit a new test whether something is 0 but it but my sense is on when you do go to the court acknowledged the you will have more residences which were not all calling and so you have to compare say pairs before it transformed variables that will be like that suppose there's a pair of Fourier Transform variables each 1 on the light on how can there can interact will be eligible a little bit like but what 1 thing I want to say that is not exploited in this witches on contact domains like a Taurus making normal form the involves dividing a 40 series coefficient by the resonance relations and if it's 0 you cannot divide that has to stay but in the continuum it's very possible to divide by something which is 0 for example the Hilbert transform is perfect .period 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 demand the vanishing of the interaction coefficient just shouldn't it demanded that the role of the role that the that the relevant vector field after a note expressing for variables has a sufficiently regular singular formulation thank you can say that you vanishes see the effective yield has a singularity but it's relatively mild stimulant 1 yeah ,comma 1 of the few beam