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1/7 The energy critical wave equation

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1/7 The energy critical wave equation
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The theory of nonlinear dispersive equations has seen a tremendous development in the last 35 years. The initial works studied the behavior of special solutions such as traveling waves and solitons. Then, there was a systematic study of the well-posedness theory (in the sense of Hadamard) using extensively tools from harmonic analysis. This yielded many optimal results on the short-time well-posedness and small data global well-posedness of many classical problems. The last 25 years have seen a lot of interest in the study, for nonlinear dispersive equations, of the long-time behavior of solutions, for large data. Issues like blow-up, global existence, scattering and long-time asymptotic behavior have come to the forefront, especially in critical problems. In these lectures we will concentrate on the energy critical nonlinear wave equation, in the focusing case. The dynamics in the defocusing case were studied extensively in the period 1990-2000, culminating in the result that all large data in the energy space yield global solutions which scatter. The focusing case is very different since one can have finite time blow-up, even for solutions which remain bounded in the energy norm, and solutions which exist and remain bounded in the energy norm for all time, but do not scatter, for instance traveling wave solutions, and other fascinating nonlinear phenomena. In these lectures I will explain the progress in the last 10 years, in the program of obtaining a complete understanding of the dynamics of solutions which remain bounded in the energy space. This has recently led to a proof of soliton resolution, in the non-radial case, along a well-chosen sequence of times. This will be one of the highlights of the lectures. It is hoped that the results obtained for this equation will be a model for what to strive for in the study of other critical nonlinear dispersive equations.
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now did you and you and I want so the title of the talks the energy critical wave equation and today's lecture will be organized into parts of the 1st parts would be an overview the whole sequence of lectures and kind of putting some perspective into that and then the 2nd part of today's lecture will be the beginning of the actual work where I will set up some of the technical tools that will be needed for the rest of the election OK so so this talks on the energy critical wave equation there are set in the framework of trying to I understand the global properties of the dynamics move large solutions 2 nonlinear the specific questions basically the the background to this is an attempt to describe the long-term behavior of large solutions to this specific questions then that I will live start out served by the end goal and this is coming so that in the end goal all of these studies are aware that we call this has become known as the summit and the solution conjecture so that's it I'm not really a conjecture about moral of philosophy I think it's through was 1st noted in the mid-Sixties explicitly in the workers of can and cruise call on the correct this equation and it was coming to Warsaw 1 of the 1st things attempts doing computational experiments and this was 1 of the big successes and closely connected to this was another big success of this story which is the paper of the pasta with which was in the same direction the bitter so what this this soliton resolution conjecture that is the belief that for nonlinear this person hyperbolic equations and the long term icing topics of solutions and described by what the other known as coherent structure and this is the belief that speeding come to be known as the Silicon Solution projection 2 of the more specific about this and I think that To systematically establish this as 1 of the big challenges in impeding and what does this is mean what this this conjecture saying In loosely speaking it says that in order to understand the long-time asymptotic but most hyperbolic and this person equations it would need to know is that asymptotic killing time so there be some intermediate regime of times we cannot say anything and then eventually that the asymptotic results into some all modulated solid tones so this traveling waves that are scaled and translated plus a free radiation Thurman does nothing but the solution of associated new problems but something goes and so somehow it gives a complete description of the long-term isn't so the solutions may behave very strangely in ways which cannot be described for intermediate times but eventually this a simplification and that's what this is so this is 1 of those saying this is a conjecture the possibilities and simplification and you have a very complex dynamics but eventually it simplifies to this In superposition of nonlinear objects in the near objects so until very recently the only cases in which such as in politics could be proved was for integral equations so integral equations are nonlinear this person equations that can be reduced to a collection of linear equations and examples of this are the correct the race equation the modified correct the Rees equation the key witness in the showing their equation 1 space dimension and there are not that many more examples In the allocates when is
such a result had proved there was also imperturbable regimes where start from small solutions or from solutions that are already close to facilitate the way than Greg "quotation mark now in 2012 and the thing with the attack and Merrily we were able to we established this asymptotic behavior in this specific case a real solutions of the energy critical wave equation in 3 space dimensions and that we did this in 2 stages the first one who would prove difficult position for a well-chosen sequence of times and in the 2nd stage would provide for all sequences and I'll have occasion to discuss the differences as we go on so let let's say no get more specific and described the nonlinear energy critical wave equation he this we have the usually new wave equation in the subtract his nonlinear take term with the specific known narrative and we ask that the initial data b in the space age . 1 0 functions with the great in and time derivative it is isn't helped OK and the dimension history 4 5 6 I is time interval and the origin belongs to this time so 1 thing I want to point out right away isn't this equation the rate the way we have here is a focusing the equation we have in mind the last lesson which is a positive operator here and we have minus the nonlinear so there's a competition between the many apart and the nonlinear part of the equation and that results in a focusing effect now because we're in the energy critical setting the strength of the blessing of the nonlinear editing same and so there is a real competition between the linear part and we will see this year more explicitly with former In a few so in the 1st thing it is the small data fearing on In this problem small data he held in a global solutions which scatter so when the scattering means we will say there are no linear solutions captors if it exists for all large time and this time tends to it's a plus infinity the solution behaves like a solution of the correspondingly new course and we will have a formal definition for the and I think it is so small data theory goes back to 2 the pioneering words of cattle you need issue happy times he several other now there is also a notebook Siliguri space in which we put the solution In this that Laura this place role office all embedding in this nonlinear wave equation theory and have I will be more specific about that in the 2nd part of the action and it may help you when certainly helps me to keep track of this numbers by focusing in the case of 3 the 3 dimensions this is In the 1st in election will stay treat all dimensions and then there was a specific to the three-dimensional case I would this the fractions so why do we call this equation energy critical and it's because it's critical with respect to the scaling in energy space so if we have a solution you know and the scale at the value of X over land at the Overland and then we multiplied by the correct power of land that depends on the preliminary interface landed the miners and miners to over to then that's also and this depends on the specific formalin nonlinear and there normally the initial data is independent of his online so we cannot make giving the smaller lodged by moving the scaling and that's what we call this energy this
equation is focusing and it has a worked that there 2 important conservation laws with this equation in the energy space the first one is the Energy yet yeah the secularism momentum when when describing you discuss that later but the energy is given by this expression and you see that there a competition between the 2 the so-called linear part of the energy and the nonlinear part this 1 comes with the plot is also comes with a mine so before going on I should talk at a little bit about that the focusing on so it ended the focusing critical wave equation what we do is we change the sign in the non-league instead of minors you To the 5th foot mocked plus you to the ship the power 5 history three-dimensional In that case then there is no longer a competition between them 2 linear part of the operator and the nonlinear piping in fact cooperated and the resulting energy has the plus side so it's but that the focusing equation was studied extensively through the 18th 19th and early 2000 and there's been a tremendous amount of work in that very important works some names that I can mention in this connection strewn on the lackeys show to capital and skinny engineering of sulfur but reinjured and the whole region and if you look at that combined results Of these people which have took a long time to come through or and this is a big body of work that in the end result can be summarized in recently it says that any large state that gives rise to a solution that exists globally in time and which Scott the asymptotic says only that it has been the asymptotic settling her quit so in particular has no solitary waves traveling waves and everything to scatter the Prince the summary Of course it took a long time to get to this summer but today we can just this is the supper so what we're gonna do now In this lectures is understanding the focusing case in which the dynamics is considerably more complicated so let's go back to the focusing case the 1st thing that you if you notice is that if you forget about the space of the captives and that considered the founder of this and consider functions just we have an amazingly constructive solution here this expression it's a solution in the three-dimensional case and the equals 1 something catastrophic happens the becomes identically plus so there's definitely find at breakdown in finding find dying breed now you could say well but this solution clearly is not in the great recently knocking the energy space but the thing is that the 4 wave equations we have financed the propagation and so hidden 28 that solutions at times 0 let's say in a box of use to at times 1 in the book celebrated 1 it's exactly equal to the To the solution and so therefore you have find Energy Solutions whose normal tends to infinity OK and this is what 1 calls type 1 blue OK and by the way to anticipate some of the things that will be thinking about styling it is still unknown well 1 loss at infinity can occur it's an open we will see that at least in the Reagan case that cannot happen now the interesting thing about the energy critical case is that there is not only this type 1 solutions which is Our rise by only but there's also what we call piped to some those solutions the breakdown in finding that and then was norm remains by and we call those type 2 books on resulted in imaginative terminology but also noted that they're not mutually exclusive .period Type 1 means that the limit is infinite type 2 means that the Supreme abounded but it could happen 0 particularly that for 1 sequence of times you go to fit him for another sequence of country remained that would be mixed asymptotic and then again in the general up to infinity will this kind of thing is not understood in the general case but in the radial case are makes simple politics in the fight and this is something we will support OK now the 1st examples of type 2 solutions were constructed by creator slide the taro in the three-dimensional case than in the four-dimensional case by playing in hell and recently there was injured in the fire damage I think also the 6 dimension look at this go note should we expect soliton resolution for this so once you think about this in just a few seconds of thought tells you that you can only happen for solutions that remain bound in the H 1 person to norm because if you have mystical position for linear solutions you clearly have the net and for the soliton party also has found because of is very scales translates of a single function and if the solitaire resolution is going to hold you will have to to have held so to understand soliton resolution for this equation where it will restrict ourselves to the solutions that abounded in the energy the company and so Our what are some examples of solutions which remain bound in the energy norm and that exists for all let's start looking at that time there's a certain geology you've what are the possible objects that arise thank you 1 of the so the 1st thing is scattering so it's so I've scattering Solutions formerly their solutions for which there is a final time of existence is infinite and such that there is any Sunday dying each 1 proposal to fasten the difference of our nonlinear solutions and the linear solution corresponding to the state that goes to 0 In a process because that's what we call scattering solutions and it follows that it would be more specific about this in the 2nd and clearly before scattering solutions that age 1 cross out to nor remains behind because it is so full Romania welcome so for any a small initial data as I said from the local polls in this theory we have boundaries H 1 consultant so now go
to other examples solutions in the H 1 presumptive norm that exists for all time and so like before we forgot about the time and I want the space than let's forgo all the time derivatives if you forget about the time the that this what you're left is this nonlinear elliptic inquiries foreign equals 3 years subplot and cupolas skewed to the 5th equal to 0 OK Of course consider non-zero solutions or otherwise the 0 solution is not now this nonlinear elliptic equation is that an equation with a very long history In it their roles in the solution of the amount of problems in the French and you 3 so this is stated in the Crawford Bentley show and many other so that they are the amount of problem is a problem or whether you have a compact Riemannian manifold in the mentioned 3 can you make a conformal a change of the rhetoric so that the resulting in 3 cases has constant scalar curvature and it's in the solution of this problem that the study of these equations could now we will call this the class of non-zero solutions to this elliptic equation this is the name so what are examples of objects in inside the 1st "quotation mark well is this guy so this specific function as a solution to the nonlinear the ThinkEquity so we certainly see some features of what the solutions might look like from this now stationary solutions obviously the cat what because if you have a linear solution and you look at their energy restricted to a fixed set lights into the Union killed and you let that tend to time tend to infinity that will always go to see this will be difficult to shore therefore for static solutions that's constant so will not be going to the wrong so they don't let's quick way of saying that static solutions those kept no so we see already this example offered a global solution that doesn't scare so this this this stationary solutions has Sunday interesting properties the 1st it is that after the sign and scaling this is the only non-zero radials solutions to this and that and this is a theory that goes back to the work of forward I have in the mid 60's and then begin this union In the late seventies contempt so for Reagan solutions there they then it and the other thing and this is the fundamental work will be listening members is that after translation and scaling this are the only known negative and this is the so-called moving that the troops there now nevertheless and this is important to us I think at that at the time when win then found this it was regarded more as a curiosity but we will see that this is important to us and they are variable sign normal relations and they're not just a few there many this whole continue and thinks construction was functional and thick and you could see almost nothing about what the solutions looked like from his construction then more recently than being almost all but guarantee story have constructed other examples that the and much more hands-on from which 1 can read properties there W the remaining from the previous life the thing we have all these characterizations but it also has original characters and if you look at this level embedding that's a L 6 is contained Greenland in 0 2 and 3 With a best constant not news realize that this OK and that this is a result of the government aplenty although In the radial case there was already proof by I police in the 1920 no this is a variation characterization results in the in the fact that this is the analytic solution knows elliptic solution with the least amount of energy and because of this it's called the grounds "quotation mark so in in
2008 with Merrili established what would we call the ground state conjecture for the wages question and this says that a solution of the nonlinear wave equation whose energy strictly below the energy of W and satisfies the following day ,comma is the gradient is smaller than the gradient of W then it exists forever and scatters in both by and scatters in both times that is the grade bigger than embraced in the final time both times and finally in the case of equality here does not arise that there is no such functions because of radiation so the part of the energy space where the energy is smaller than the energies of W 1 can understand completely the Hunter and then there are those of a threshold case when the energy equals the energy of W and this was completely describing the work of the character that followed shortly after this so that in their proof of this grounds stated conjecture a week we obtain this as a result of that method and redeveloped to study the longtime behavior of solutions critical dispersant equations and this method worked in the focusing cases and focusing cases below the threshold on the grounds and this method has become a standard to tool to understand the global impact behavior of solutions build advanced the from the from the beginning when we and applied this 2 of 2 the nonlinear wave equation we realized that the nonlinear wave equation was very suited To his methods fit with perfectly and in the back of our minds was always the idea that we wanted to eventually see if we could find a model for which the pro soliton resolution and the fact that this In a nonlinear wave equation fitted so well with this method made us decide to studied in this case so that's how we came to know only new wave equation that I have to say that the I have no expertise and wave equation prior to this but it was kind of forced by what was happening coming to I think it is also Frank didn't particularly expertise in this direction start exploring and area had had floods will side with visions of particularly it's not that we have decided we're now going to study nonlinear way questions can came to us that this was what we needed to do OK so let me analysis explaining what other known scattering solutions there is other solution of the nonlinear elliptic equation can be made viewed as traveling waves that don't try that was speed of traveling easier now we're gonna go to true To troops traveling waves "quotation mark so what are the driving winds there also announced scattering and there are obtained by doing a Lorentz transformations all this elliptic solutions so you know you observed that the nonlinear wave equation is invariant and the Lorentz transformations and now defined them in a few minutes and then you you understand that this analytic solution as a solution of the the evolution equations and non-linear way question and then you do the Lorentz transformation to and what you end up with it a traveling waves and it has to travel at speeds strictly less than the speed of light so the and directions have to have length strictly less than 1 so in our in our formulation the speed of light is 1 look at this the somewhere normalizing things so the soldiers and traveling wave solutions and here we explicitly view them as a traveling wave solution in the direction given by now and this is the formula of watching to them from the elliptic solutions to get the traveling waves look and on described as discussed this a little bit it 2 meanwhile in more detail so it is hard these are examples of traveling waves and we will see that it can be proven that these are the only track but that is already at so
the the ultimate goal passing mention worst of established sold and sold them resolution and so let's say in explaining what what should soliton resolution being for nonlinear way going for the energy critical loneliness and waving questions so we have a solution who whose energy nor remains bounded until the final time of existence and you want to show that the number J and analytic solutions cage directions elevator which of the speed of light and that have the property that take any sequence of times converted into the final time of existence which many find there are infinite many confined scaling factors the center's exchange and this have to be arranged so that the difference in solitary waves Societe other and that's expressed in terms of the source of a of parameters condition so we'll get back to that a little bit later on and ln linear solution which is the radiation term such at our solution equals the sum of money latest solitary waves plaster Asian terror plus an error that goes to 0 in the energy Northwest Passage paeans vote to teach so this is a very concrete description of what he would like to 2 and the subject of this lecture sister cities how far we get into so as I mentioned before in the radio came and in 3 dimensions as has been proved in the work well with due care and rural In in dimensions 305 for this had been improved for the final time case you close to W in paper and also with you Merrell and as so now I'm just what I'm going to do In the remaining part of this hour is give you a broad sketch of some of these results and came for and later pick which are the large state that results in the northern region In then the rest of his lectures will be trying to prove these results and in there is part of the series the aim is to reach up to the relocations and then in the next spot in In the June July is to go to the number so and leaving the non-rated case for the Anderson incentive for you guys to return but so that is so as I was saying that there this the composition was 1st proven for a well-chosen sequence of times and then for any sequence and I'll try to explain both methods so let me through the result for all sequences times will not prove sketch an argument about how the proof go OK so this was using what he called the channel of energy argument which is that there a powerful argument in this in this problem so what 1 that he respects both what happens as you approach the final time of existence that leads to the same assault resolution is of course the energy cannot be lost his conservative but 1 and that the Saudis the sum of solid tools has a fixed amount of energy depending on who is known of this elliptic solutions so there has to be some mechanism for which the energy appears so how can antigens appear cannot disappear in size what happens is that moves out to finish so this soliton resolutions phenomena only holds when you have infestation domains and what happens is that energy just gets displaced Towards Infinity find party just get the sultan parts and then the radiation by takes care of the energy moving out in 2 but of course this is that only heuristics there's no proof in there so we have to find a way to quantify and to measure it mathematics and that's what this channel of energy method dust so that there is a In mainly in the range of cases the main factor and the main factor please don't during this to carefully let me summarize it in the following way suppose I have a globally defined for both kinds radio solution after the nonlinear wave equation in dimensions 3 In suppose I know that this isn't W or minus W or scale version then there always has to be easier for positive time or for negative energy left outside the line call for all time this what assessed there's always this lower bound is always a block of energy outside like and this is this energy is being
moved away "quotation mark and this is what captures this fact so this is a dynamical characterization of this in Dublin which doesn't for those of you is it is obvious that this cannot hold can you use a simple calculation because W. behaves like 1 over absolute value X at infinity in space and then you see that this just around so you're not WEU have this property what is a highly used OK OK 1st III going to tell you how it through how 1 proves this thing things and here again don't pay that much attention to the slaying of there's a property offer solutions rail solutions of the linear wave equation In 3 space dimensions that's important here and this is an elementary property in the sense that the the thing that number could approve consensus the basic property and what it says Is there you have a radial solution in the energy space then is for all positive times overall negative times and any number of iron ore that you choose there is always some energy left this a lower back now there is an exception there's 1 that everyone kind of solution for which this doesn't hold and that's there what corresponds to the Newtonian attention which is 1 of which of course outside the light cone is that a proper solution to the energy 5th away can't and for that it fails but what the real mathematical statement is is that once you prefer to do the orthogonal projection to the compliment often Newtonian potential lenders system "quotation mark so that's the meaning of the state now however we use this in the nonlinear pro the way you use it in the nonlinear problem so this is the property of the new solutions that called for all iron ore I suppose we have our solution Our global solution that's fun for all time both positive and negative I'd choose nominal that's very big so that the energy at times 0 outside of all readers are not is very and I can always do that it it is so small then the small data theory applies and then the nonlinear solution is close to the corresponding and thereby finance speed the propagation of the truncation even look at it outside that light cone doesn't and then I can use the linear and I will give this proof in the and this is just but I'm just the kind of giving viewers a taste for what we do you have a source from the EU should held shall also write off of you and then we recently been and so the remains among the the celestial what not that there you here I I some time chalk after at home at the so they're the same so you actually think that you will not also and you the figure of influenza activity greater on normal because of but it has happened to him and he gives me the bans that they want outside Brent and so it's a cheap trick but that's is a very powerful tool that the player is the change in the U.S. is well aware that he is also the closing you but we only care what outside of the he said somehow what goes inside delightful so who we decided we don't and we somehow managed with information on this is the point of this trend Road look so and that was only in the region case 3 the there's a corresponding inequality that this is what we used in this study results near the ground state in dimensions strain 5 and that's we take our 0 equals 0 over there and we can only take hours you 0 and then you have the corresponding in equal and at but this notice that its claimed only for all the dimensions computer so let me just
explained the proof Of these results that same time go up for it and the other cases so the 1st thing it is said that asked that the solution is bounded so it has a weak St tends to 1 anyone can show that this week limit is independent of the sequence of you need to get an argument that that's the case and then it is you look at the linear solution with this data time 1 and that's what the radiation it's a lesson easy description in this case of the radiation then of of course you look at the nonlinear solution with the same day that time equals 1 and now being near the solution because everything is a nice 1 they can use a small lead the theory that time equals to 1 and now we look at this nonlinear solution and this nonlinear solution is what we call that the regular part of and what you can see in from finance fees propagation is that there is a clear comparison outside this truncated in inverted :colon the 2 things are equal and all the bad action is happening inside the lights ,comma so the president who shot at him drop another picture but so outsiders solutions in and that the whole block this happening from what's happening this achievement on the In the so so we now think this is the singular part of the solution and we break it up into blocks this blocks are normally new solutions and but their localized In space-time and they have certain north of another the properties with respect to each other so that you can think of them as independent and this Arvo Peart uniformly then nonlinear profiles of associated to about major are profoundly legal within just that the governor's blocks and then there is an error and error is small but only in a in a week for instance you can think of it as tending to weekly in the energy space but not necessarily Strong so now what we want to see is that each 1 of these blocks has to be a double that's that's terrible that's what the soliton resolutions so it was the blocks is not that we use our dynamical characterization "quotation mark doubling and then but clearly hit but there is no energy there because you might establish concentrated so all of his books have to be there and now all that remains is to prove that the error term has to go to 0 in the energy and warmth and for that we use the 2nd of these out of jail or what we can use in the 1 what and that's the end of April With so you have of course to take this with a grain of salt because the proof was rather long survival as we as we will see 2 was in the lead so now let me explain the other argument they argument that worked for only 1 sequence of time so in this argument proceeds in a different way In the 1st the 1st part of this argument at that but it's to show that there can't be any energy even near the boundary what in this region as he approached and wallop ,comma that's the approach developed time there can be no energy so in fact that the energy has to be In something that OK so that's the 1st part of the year and a our original proof of this fact thing and without the energy the inequality then you combine this with very light what I realized and that is in this context there is nothing more than the overdrive identity that was used by apologizing to shoulder the only reason solutions and there's wave equation and books and this and this and that to us it was a very light now we have this formula we added and you get this very nice facts that the teetering at then plus an error is a time to do and if you combine that with the fact that the energy is in the region like that you can conclude "quotation mark I'm sorry what do do 10 Thompson have seemed to have lost let's the market so
when you conclude from this it said the choose means all the time derivative inside the lights called voters Wilkinson some average and this is where you need to know To to a convenient sequence of time but of course it would make us choose our means will 2 0 doesn't mean that ordinary here means going during the year by means it's 4 by at the rear argument for such and result she from the very classical art and so from means there's some ability will to subside so for such subsequent things have period within the goes and they have the derivatives that goes to 0 it means that each block has to be done in the past but what our time independent of allusions of elliptic equally and lipstick equations solutions in the rail case are doubled and I have a different proved that his blocks and ugly but only for a sequence the and then he concluded using the weaker of the 2 out the energy he's so I'm going to now go to the discussion of any of the other dimensions and and then on radio case and will do it rather quickly so that we can then start with the actual detailed From the study concluded so what happened after that well at least I mentioned there other applications of these techniques and the but here let me 1st started with the fact that was approved by court myself and science and is but in this sort of paralyzed this for a while and it's the fact that this out the energy inequalities even with the ones with our 0 equals 0 hour falls in either look at it just false can write them contracts some of the book had it in his life and work in now this sector was the ones were as you the "quotation mark is equal to 0 which and the weaker ones the following peculiar features if it had been mentioned in isn't even but of the form for 8 12 and so on their true for this kind of data and for the other even dimensions as were the opposite kind "quotation mark and it has nothing to do but to accept things we consists of a are prepared good now for all dimensions when lorry His new inside which showed that this out outer energy inequalities even with the IRA have an analyst with history which is true in all all dimensions but they exceptional functions get to be more and more lasting dimension gets larger and larger so the projection there's always a projection that you have to do it's so a space of increasing demands asked at the convention goes to infinity as the any ambient dimension ghost infinity but the constant is always 1 half so that there is a uniformity in the Commons so using this fact my stood in case he really is where they want approved for all on the mentions the rating case out along the well-chosen sequences now for the even the mention of case there that there was a problem because out the energy inequalities do not hold then there what we did this we we 1st looked at the case of related problem which is the waves not from the wave maps are they hyperbolic and Alonzo harmonic maps and in this problem they have been known for a long time BAT under a certain symmetry assumption called a quick variance there was no cell similar In in since described here and this was ruined by an integration by parts by Christodoulou and denies that L but that an important fundamental ingredient in this proof was that the energy densities for whereabouts even they could be focusing the energy density doesn't change and now for the nonlinear wave equation in the focusing case this is the thing that things from the beginning and how views that they a energy density it was bigger than are equal to 0 it was to control the so-called flocks all this and this control the flocks that allowed for integration but far as proof of the lack of in Herbie and what we did eventually here it is L In for the With "quotation mark Lawrence and we reverse the analogy between Wade Mt and nonlinear wave equation usually that you think of the nonlinear wave equation of a simplified model for weight and the way maps more complicated but what we did is instead used the Wakeman to prove something for the nonlinear way OK so that is a very simple changes variables under which yeah solution to the nonlinear wave equation solve something that looks like a wave map equation yeah if you look at the corresponding energy it will be no negative energy density provided that the see is small enough and if there were
then we use the radiation terms that shows that on the boundary of the light :colon you have this morning's conditions for you because he had for the and then an iteration argument to go inside please come from inside and these words and the final step was use out the energy in the inequality with our equal to 0 to show them that this person term ghost and here in the four-dimensional cases the data for which it is true is for the 1 for which this time there were 3 0 but there this In the identity I even gives you the In the error and the time derivative most of them so that they comply with this but this doesn't work in dimensions 6 have because flips the the for which it is to and then it Howard Yahoo's postal and Chicago we were able to do this also in or even dimensions by combining to relay their case and not using this channel energy and I'll have occasion to prove to go back to the then so they could
there in the next the conclusion of all this is of course now for sequences times accounted for all dimensions in the case have other on radio case and they the summary of that which we will see in full in the last elections Is that for it well-chosen sequence of times 1 can also prove listening on radio someone has asked for a sequence of times the full soliton resolution for energy critical wave equation In all dimensions "quotation mark so I'll stop with this situation overview now and the words shift gears and instead of speaking in broad terms to specific so now we can but instead of continuing with style OK so there's a restart the start from the very beginning so the 1st part is the local theory of the "quotation mark the problems for the nonlinear way and I will discuss this in at some length so I started with a linear wave equation we need to draw pictures whom but so we start with a classical linear wave equation so we have a linear equations evolution equation with the right hand side initial data that an initial there today so there's a general method to solve such problems and as the 40th 1 so we've taken before it transformed space and we get for their differential equations and we can by superposition so the 40 method gives you a formula for this so there is a "quotation mark turn on the sign term want and what we call the Duhamel terms which is this integral and we summarize saying the WC which is the new solutions and the Duhamel and they're the curtain that annotation amusing is the happy is the spatial for his transfer and so "quotation mark the square root of minus the last in the it's meaning it is this the wonderfully aside adjustment applied this aim for the signing and this this thing means this compliance with those standards not so 1
of the main properties of the ninja wave equation is this fine speed propagation I'm going to draw the fictional but at the start of the 3rd World .period that 4 but so the Finance be the propagation tells me there In the right hand side age is supported away from his coma and the date W is also supportive away from his Conn then being here than W has to be easier "quotation mark the Safina's speed appropriation for the wave equation In in on dimensions there's something better which is the so-called strong principle when sets the following me into the picture so but at the back so now the support is contained in here then but In all the dimensions is this but in all dimensions the support is contained here but in all dimensions the but this is what disappointed OK and what's the difference between and even on map OK so pretty who were in three-dimensional space and we see a jet go by we hear it for a minute and then it disappears and we don't hear it anymore In that's the three-dimensional if I were in 2 dimensions and we throw of Pebble In a pond we see the ripples all the way up for consumers says later "quotation mark so this things will play an important role as we should thank so now we start with the
local theory of the "quotation mark problem for the nonlinear way an old saying from now on I will stick to no more fractions to them well there will be fractions but the specific facts there will be a force words you coming in but OK so this is our only New Wave Energy critical focusing wave equation now for foreign the study of these equations there some important artist which In the trader called sir Kansas so the Strykers estimates playing the role Of the flaws in the study of wait equations generally dispersed equations the stricken contestants they play the role that the solar estimates playing the study of the policy "quotation mark and this many circuits estimates or give you some examples of the ones that we the so it is solved sold year I'm talking about the linear wave equation that we had here so that
a traverses age and the Dublin refers to this
so the 1st estimate tells me that the soup but the LAT nor plus the l final 10 more yeah the that L Florida orange of one-half derivative and then the other version of one-half derivatives this control in terms of the initial data and the right hand side and there's the 4th and this fractional one-half there the it's important to us by the equation this is the maximal amount of derivatives and you can could solve I'm sorry fractional derivatives appear even tho our problem has no fraction of the disease you can but that's life and if they control of this space-time norms that we think of as like a soloist and there is another Strykers estimate political as to where on the last day have the same quantity but on the right instead of having fractured under the units have a L 1 0 2 the and this 1 is obviously easier for some things because there are no fractional the route but sometimes you need to also used and because we have an infraction derivatives an important technical tools is this summer mystery here this force should have been for "quotation mark it is this estimate when you apply half and half of the route that for 2 new to the affairs in behaves as if it were true derivatives in this change setting and then there is
more of a similar accident and those are used in the proofs new fix .period arguments so a small bank but there it is carried out through 6 . and to the six-point argument this is a convenient to use it and updated annotations if I is a time and the role of the L. a normal guy a moving X will called S of wine room and the L IL for next move will be the W "quotation mark look you don't mind mutations in a sometimes say it we will call the Essen on the L 5 of them Of that depends on what we're doing at the time so let me
the final by winning by a solution a solution on an interim all I was doing that will be a function which is continues with values In each 1 and it's kinda evidence continues advancing those 2 in the interior of the of the interval and for all compact SUV in the world we would have control of this based on norms as the solution will satisfy the interval that's what we mean by look at Florida suspended standard terminology it by the way at the end on this 1st a series of lectures Al make available to people that's not OK and then but I won't give you the 2nd missed 12 political village after the and for all users who can the main result on the local kosher problem is that it is in at arbitrary initial conditions you have a unique solution which is defined maximal In the role of the Finnish which will call Imax in key t-minus antique glass will be the end points of maximum and and to prove this 1 uses the contraction principle which will be and combine it with the changeable to show that if there is a Delta north but there is a Delta notes will not such that if there s norm Of the linear park on the interval is small enough there is a unique solution and that in the With initial condition using your you want moreover it we call a this norm on their initial data we see that there a difference between the nonlinear solution and linear solution is has this kind of cool so that's what they meant that earlier said that if you're small then linear solution is close to the nominee and this is the specific guest now this is used IN gives you a solution sometime in the role and anyone to expand it to get to the maximal interval existence and to do that you have to have a little uniqueness argued that suggests that if you have overlapping in the time intervals and they in the middle of the the the union welcome to the and of course this is for in the roles which contain the origin but we can translate into anything 19 for instance I was talking about Time Warner "quotation mark now it was very important for us it's something that we call the finer title of scattering everything we have to to find their way to describe the fact that the blast is the final time for the suppose that there 1 thing that can happen is that the age 1 grows on tour almost infinite Type 1 blog but how do we described type to the wall up were the norm remains by 1st so let me just say in a few words what happens at the time is the norm remains bound but you can't continues what happens is that the gradient squared away your solution is concentrating like adults and that's why you can't continue continue 2 you know how you measure that there is this arsenal that mission In the tire of existence is fine and that means that something doesn't go up and what blows up is this dispersal it has so the LAT becomes so if there is this enormous fine and then the pluses infant but more than that this tells you more than that it considered the solution needs to scatter so scattering is also described by the estimates so there is a solution waves that set this difference courses scattering that's an assault combined in terms of this assumption so if this holds that equivalent to saying that the pluses plus infinity and solutions cat it was a good way of quantifying that the focus on saying all these things for forward time go backwards get the same then there's 1 of remarked but I want to make the fact that we know that the solutions are constructed by the Bakara iterations gives us the fine been appropriation for the nonunion Prop for free because remember that In here we included the right hand side In this fine it's been preparing as so it in the final and plus 1 in terms of UN and and you know that you and has the right support property you can conclude that you and this 1 has it and then by the Bakara contraction mapping Prince Will the 1 converses system so the solution will have the right support can dance solos if supportive you 0 1 intersection with
a symptom exactly then you are outside there and now against say the same thing that makes 2 solutions agreed In the ball exhilarating then the library outside the life but of course there for the stronger Koreans principally concerned because things get spirit of women only look so I just briefly I want to mention that this notion of solutions is 1 possible motion as other notions of solutions that she could have and they are important instead of having the LA delayed norm you could ask that the L 5 Elton relief I could Sarah Cook the immediate question is is this the equivalent added 2 notions equivalent and the answer is yes so if you if you have this property and you're quasi-military to believe In a true or in the case in the mouse cells but just in the distribution of sounds and then you can conclude that you have a solution In the Duhamel so what notice that you always have to have his auxiliary space now it's a very natural question to know if you know that you just continues with values in each 1 person to and solves the equation in the distribution of funds this a solution in this now this is true interventionist foreign higher which was approved response from it is open and the three-dimensional it's actually interesting opened pointing the let me who yes yes yes yes it should be tried but the don't no listening to the heads struggling with at work why would we want to use this space and time because when you use the space you don't need to use fraction the disadvantage enhances the exponents of different it's been team next and I will show you in a few minutes why that's a problem or where that may be a problem and which is why you may want to have both notions which OK so the next thing I want to give you is something that is the kind of furthest that you can go With this local theory of the caution analysts we call this the longtime perturbations OK and we call it the longtime perturbation level because the size of the time interval is not important in group OK and this is can this is 1 version of it technical thing but it's an important technical and that's was mentioning him suppose you have an approximate solution it has finite L final 10 enormous continues with values in each 1 of them tho and it's also equation With an F. and the efforts sufficiently small and we have a new theory why sufficiently at close To data therefore capital all but Bayou Arrow II will in general just vector you in the consists so and that this conditions in the absolute it's small enough this allows you to have a solution to the actual nonunion problem which we will call you which is close to this capital you and how close answers by and epsilon that's controlled by this and this is a very useful tool industry please let me show you what are some results that will come from using this and it's a hipper to raise the argument the use and there is no quick way of describing it but it's the 1 where you where you use it the fact that for not homogeneous for the Duhamel Hamilton you have a wide range of circuits export do you know this I'll go to the to using the proposition this proposition I think goes back to the work go and the keel staffing landing Culkin can call on the energy critical challenges equation this work such as a result was 1st specifically formula and it was important in their work and it's important that all of the other works although there is a lot of the it and therefore he Soviet science so forcefully defended the year there's a knack and that's why school long-term operative we the will city face through the phenomenon is that maybe 10 page "quotation mark and I don't want to give it because this is it I would say harmonic analysis who can consider a signatory of the the course of going gives to it's at the very technical so there are other
versions of this proposition where instead of their final 10 wrong we use the than that "quotation mark what does some consequences of this 1st suppose we we start out with a compact set of data broken so we have not just 1 day that would take a Compaq said then we can give uniform bounds of the maximum time of existence they just defending itself on each individual they got on the complex that's the 1st thing the 2nd thing is the thing that is most of engine for continues dependence it tells you that if you have some data a sequence converting that there this a sequence all have kind of existence as at least as big as the 1 of the and that is the role they converge to and I think in the early versions of local will post and this this was absent they said this was open problem until this kind of research the local anyway all of this the results are needed in this thing so now I'm going to discuss Lorentz transformation remember we discussed the Lorentz transformations of this solutions to the traveling waves that was the previous but if so and remember that this depended on this vector now let's take the vector Al 2 B 1 0 0 OK yeah Maryland is a factor of Italy so I rector with different times OK thank you then if you look at what the Lorentz transformation that it wasn't so the X 1 variable gets changed selects 1 minus of terrace where the one-liners and squared their anyway X to Annex 3 variables are left untouched and the time variable changed so somehow it makes space but in this very specific way and services for the linear wave equation and it's true that they have a solution along with linear wave equation and performance this Lawrence transformation then you still have a solution the new wave equation the question is this is it's still a solution in the energy sphere the answer to that is yes and is given by the and you see that what you have to do is estimated at 1 grows to norms instead of paralleling I proclaimed guilty ones provided tilting is strictly less than 1 the CD there yeah I'm blows up as elbows to if the hostage OK but for rejects for which held that the sea and OK and this is a very important compactness think of this family of Lawrence transformation now the approved the if is starting to sit down and write to prove it was down to the following if I final me hat by multiplying by the way group included St. this is true and approve this is the only have to do a puncher center and then do a change of variables in frequency In this comes so it's elementary if you see the right way and also on this is continues in In in this so for each fixed L I'll be fine why that in this way which is the inverse Of the Lawrence transform from that that we had before so delivered this Lawrence France reaching jail in the and that that's all that you do OK I write this down again
they're not supposes that was it globally defined solution of the nonlinear way we can certainly be fine is Lawrence transformed by this for by the way for this is the same formula I had before that looks so horrible but when you really have a right to think ordinance is not so it is very simple no since the EU's willing time this is well defined as an element of the LA Law and this is why we use daily norm before because since the accident at the have the same power you can do change of variables in spacetime norms and see that gets preserved by the Lawrence and that was the reason why we didn't just work with final 10 minutes but we need this allocate norms because we needed to have the instruments it was an unfortunate technical aspect of the theory but anyway you can overcome overcome it so at least he had as an element of Laidlaw but now you have to prove that it's actually in the energy space and that's a proposition that would prove with the rural that if using global finance energy solution and you know that Lawrence transfer this is also the low-flying jets and they the strategy for this proof it somehow truncate the outside call again so at the beginning of the theory you wouldn't know how to prove that so we didn't and then eventually when we realize that the important thing is what happened and is happening on outside of court think we realize how to prove With so the next spot this introduction goes tool that profiled the composition which is about major our profile people so this is it you can think of that that's a completely functional thank you the machine on the other hand and it is not really that because in concrete cases you're taking some solution and breaking it up into which blocks of energy and space-time baseball it's both Inc so what is this the profile the composition what does it measure it so this profoundly compositions were 1st and started in elliptic settings going back to to interpretations of the alleles work on concentration compactness it is they were works by Stuart 3 and resist score on it was British and Kerala who 1st noticed that there is a certain also were now Latino parameters and certain pick the glory and identities the turned out to be extremely important and you can think of this interval concentration compactness In his evolution problems to see this profoundly compositions as characterizing the defect of compactness Indian batting given by the so you can think of the LA it's normal that wave equation being controlled by age 1 crossing too that's an embedding of some kind in 1 side has the route is the other 1 doesn't so users asking Is this compact incidence of course not because as a group often variances which is known compact hardly inscribed the this and the effects of compactness and it's in the thousands of friendly competent courses that's not the only way to think about it that's what you really think want to use this forest of all solutions of fire "quotation mark look so we start out with any abound sequence in which 1 person too and we start with a bunch of solutions of the linear wave equation and sequence of parameters for each day the sequencing and In the land us are from infinity the exteriors of our 3 in the DJ's and are so the exchange and PJ's under the space-time centers of this solutions of land adjacent scale for and we say that this sequence orthogonal In the parameters are such that different solutions don't see each other because of the location where the sky and that's what the meaning of this condition OK
and now we have this solution gj and we say that this is a profile the composition for this sequence is the product as a result Worrell and in note you J. L. a that scaled and translated the JDL and W. D. J L the difference between a solution for the user to answer and this summer the modulated profiles then this is limited it this is small and in what sense it's small well 1st of all the H 1 grizzled to norm is behind them where times you expressions and 2nd the space circus norm all this difference this small as goes to infinity and OK that's what we mean by saying we have a profile of the composed this precisely what I want see as saying that we have a problem and the company and it takes a while to do get used to working with this thing I have say for me it took years but anyway eventually become 2nd nature into it so battery and Gerard before any of 3 in the mood for ending 3 proved that for any round sequence you can always extracts a step sequence which has a profile the composed In the error not only the LAT Norm goes to 0 but also they'll find that wrong moves to 0 and the infinity of 16 console this kind of the circuits norms school 2 0 but not the age when President too this need not too and it's not that you can't prove it you cannot do it too and this is 1 of the reasons why this channels of energy useful because this is what we used to show errors I actually 0 in the stronger I'm going back to the 1st part of the it's alleged that it's the 1st thing usually yeah it is that you can think of it as a signal that each 1 grows up to if you that can you don't have to think of it as the end of the 1st new should figure would initialed environment that's so that's why I went this to say that this is a purely functional and take statement look at what your already this year that's a little only the completion of match but of course if you have a different equation you could do similar things look at long however these profiles jail constructed their weakness of the region's sequence so what do I mean by that for each J if I do the reverse scaling and translations on my original sequence and I think the linear solution was that as data and I take the weekly men that gives me the profile Atlanta so when these problems are are all the week limits all and translated persons who claimed they tend to lead it apply solution come the fact that this holds is a consequence of the was the amount of the parameters and the fact that for each new day less than captain this expression but remember this is the error most Weekly In each 1 across the financial should sees the feeling I feel you call goes weekly to functions and is a parameter tends to infinity and the 1st thing you could have happened if nothing new conception of the risky and that we use him this summer and he had translated the softening of that she was the last year and then need yes so a and if you remember I said profit the composition is sequence with his properties the parameters of its properties and the limit goes to 0 in the 7th and then for any such thing you can prove that this Domingos to Lira and Costa 0 week Christopher served in the functional analysis exercise "quotation mark the answer is there's the the deceased for a solution of the media did and I haven't seen anything yet and in working to have the the heartbroken now
that there is also this important and Laurent expansion In this tired extremely useful if you want to look at the energy of Europe sequence there's a about missing them you can use the energies of the profiles blast the energies of the remainder anything the limits as an ghost with for each captain and the same thing is true for the L 6 0 the number H 1 and thence into us that is also true and from that 2 things if you go back to the nonlinear and the same is true for the nonunion because it's 2 4 for each 1 so there you are already seeing the nonlinear as we will see you then bye extracting such sequences and changing the profile possibly we can only and we can always assume that the jails are identical 0 or than the plus Infinity tend minds this is always the case and I'll I'll stay in London more intimate look want to make a comment here it is tempting to think that this category expansions also hold not just for H 1 puzzle to for each 1 individually it turned out that the mind is made is useful it's so the something where the energy conservation plans and you can find the countries but there is altered versions
of those it the same is very quickly we we separated the profiles into those but many have do
that here I said that we can always assumed that the identically 0 will the plus infinitely minus the so the 1st case of call the index Jake or and the 2nd case it is just terminology hooker if any terror allowed
that you do have the gory and expansions for each individual piece provided they keep the court terms a separate so you have to keep all the this scattering 1 all the scattering wants to it cannot break them up but you can break up this category I mean that in the course yeah it will the its in this the scattering ones are good enough that you can put them all individual amid non-strategic or 1 I'm good enough that you can spilled the diskette scattering will signed him but they all get added together they can be split-up then this is that I can't separate if you can use the impression that the city was alone you're not alone on the on the left hand side they have on the right-hand side I have the core and the scattering 1 but I'm piling them I'm splitting them up differently did the core was I just I can put that some inside look at normal I don't get rid of anything and just how I "quotation mark yeah so what I I will say is that at this point it becomes interesting you know whether some uniqueness in the Senate composition because it could be that they feel that the fact was that you didn't choose it correctly and that's why you didn't get the right disagree and expansion can so we said the profiling from composition was a sequence of on but why is this the only profile in the car right you could have 2 different profile legal positions in principle for the same sequence of support for the same period sequence of initial note about so the 1st part is that if you have a profile composition and you change the parameters In this way To landed filters which are this the limits exist and are not no 1 0 and so on and news change the translations and change times and you get new profiles in this way then this is also from the court and that's how we can reduce always the case when the peonies and them to commit Irrawaddy and rely on the Jedi to but to minors so there is this ambiguity In this is inevitable but this is their time ambiguity that we can understand and then there's a theory that says that this is the only clause in lemon
if you have to drop by the compositions of that same mounted sequences and this is a day and they're either both fine and the will of the infinite there the only way in which this could have happened as well described by the priest so there really is uniqueness in the profile the composition except for the ones that you can described by those transformation and this is something that they prove indicated so this is some positions on the profile of the and then the next
statement says that if you have a week limit of the kind that create the profiles this has to appear as a profile in the profit of the company To all week limits appears bro model this transfer picture takes a little bit of time to digest this this is kind of a background and now in the last few minutes we will pass to the nonlinear versions of the programs which is what using non-union pro OK and this has not only profile so I nonlinear profile associated to this new refinery it's a solution the war nonlinear wave equation which asymptotic killing looks like this year I don't think and of course this limits after extraction and we can always assume exist because we're admitting negative infinity and plus interest on the part so for any sequence after extraction limit exists is in a plus infinity minus infinity of life so
it's not that hard to see that you always have a unique nonlinear profile associated to the linear profile the company so this guy's always exists and now we re scale them and 1 can understand what time of existence of this new profile because of a simple formula and this nonlinear profiles are the building blocks for for a solution In an approximation of the before let me just say there this limit when the London Jr the day entrance to minus infinity where then keep losses plus infinity and really to minus infinity the officers so In the case is when you have scattering London J. N. over TJ on that gives you scattering properties with associated with the new profile and that's why those articles scattering Jason contends and that but let me just finish today where approximation of the that gives you away they have mounted sequence which emits a profile the composition let me take their solutions with his initials suppose that all associated not provides cash the those 1 situation all the associated nonunion provides cash then we look at this remained term which is a nonlinear solution minus the sound of the modulated millenium profiles minus the error than the linear solution corresponding to this person then this difference most of 0 and the solution UN exists for all time and so we have a way while riding our solution to the nonlinear equations as superposition of solutions To the nonlinear equations where is not only new profile plus parish so this is the easy case when all of this UGA scatter forward if we don't get the foreign you can do the same provided the stay away from the final time existence he and provided that uniformly the space-time along this person so this is the way in which this nonlinear profiles are building blocks for the meeting for the nonlinear approach and I think was stop here continue next at the time we year you go on the Internet from however 2 of the the 2 sides also threatens to undermine the unit saw a Frank it's OK you can answer that I clarified so you right let in the of much of the rest of his life it has been a close relation of the mall here they're a little more complicated and then sell it that the surprising thing but we found that very recently is that some some of them In stickers norms don't work for the interest to move to you in and if you look at the kilt style and quality and you use that space-time that will not go to the necessary so when you have to you can't use these things like the black box you have to understand which should to be good the deaths of this story is moving from various criminal profiling system from existing case hundreds of cities this official said the main consists of women wear them and that the main difference is the hepatitis right so that if you want to do this and just be an infinity of of 6 normally then it is basically decay focus but if you want including the Dallas-based I'm norms than you have thinking this contestants in the probe the register the the want of them and what you want about computers and you need that potential yes so the perturbation landmark is the fundamental 2 to prove his legal this year that's the fundamental tool to prove this approximation if the last how problems you have to go back to the person that this is the analysis of the situation and I'm going to be the favorite doesn't make mistakes the noise needed to be no it's not no need for that think time the b