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7/7 The energy critical wave equation

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7/7 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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the did you you have the right to live and I if so what I would do analysis recapitulates what we did last time from Kent so that we can see the logic hopefully something I would do this on the born and then we will go back this is where we stopped working but then I would remind you where we wanted to prove when some fucking so just a year after yes then only for those with the notes will be available on the Web is starting next week the eventually like movement it tonight anyway but there it is the work of the well-intentioned but and there is they 1 proof that will not be on the Web because there's no time some of them that they showed less time Watkins look at the list the situation is we have a solution this is high and this is the .period here and the solution remains bounded as we approach the and now we want to prove that the composition and into some of London later told the Times then for the Asian carriages just obtained by the week limits and then that an error that goes to 0 for properly selected this is our aim so the 1st step I was in control of a lot of time at the time of the UK so we we have the regular solutions up up up up up and by that I mean it's a solution that continues use then we know that we have such a thing he says that this support I will feel people at Monday's rematch theme is contained in the office of the Clerk of the Course in on conference excellent with take good week limit of the you would think the solution would endanger kind going forward and the difference will be supported in the by saying to the property "quotation mark cleanup using the fact that Disney is a regular solution we can control the Fox was the Flex lives on the boundary of the court and abounded :colon dissolution equals the regulars this property look and we did this by combining strict testimony with the uniform bonuses convinced that I was the 1st part of the 2nd bottom of Wallstreet as of tomorrow moreover testimony but the so basically we looked at points at times back near the ones in the tomb small Oh and we're in a situation of course and 0 I belongs to the singular says that the solution cannot be continued locally Beyond 0 for negatives time and we know that there's finally many of such singular points so it is concentrate near each water and so that the city Morrow testament says that the interval between Timpone and the 2 the International a work of art makes less than King well compared to the world yes the at the time and money in scenes of apples and Kenny Kenny liberated signing remember we prove this by going to assessing the variables and introducing some similar energy and computing some similar regularized energy and computing the derivative so the 3rd step is the 1st the compensation but but but but so why is this an interesting this is an interesting estimate because the power of the long is lessons perhaps because it if we had the power 1 this would just be the boundedness the term which is our hypothesis but the fact that we get better than that tells us that In some sounds in some avid says this is going to 0 and we have this logarithmic rate at which it is willing the square root of OK so the 1st composition follows from the 1st and reliable Oregon cancel the real wearable argument so I took little clinical visited argument I can find the sequencing you end up spending 2 0 sets in at the back of a and and but at the time the but not only that I give an enhancement of it using a hard moved to Mexico so I I can find maybe your pictures him mediation use of color you can see that in America the president so I stick my and the running thirds and intended to fire out of points they can find 2 times the 1 hand and the 2 such that this Fremont 0 less How less the on the other hand while over island 20 the number of things that I don't think the brawl top parliament steal ends less than month that will the something integral part X less than please 2 Ct the the fastest where
I equals 1 so that's some kind of improved to going to see a thing that and the argument to prepare to prove this is said using the week that 1 woman equality for the maximum operas OK and that the reason I can always puts T here I can go to the lot by using the its identity and passed the EU equals the which is a regular solution so that contribution is now so this is where the problem is that less than he did freely this year for NEC NEC was hit yet and we want to make it 10 or something of that for any so the correct lodgings for the this history but the sequence that's a lot of times does not depend on the sea OK enticing damn people read what I'm doing in my little corner that's the 1st that's at the rear part of the baby it is the I think it's the most people in the eye 2 times the Indians in in which indicated this recently that he had been here since the president's is standard along with tenancy and 5 minutes out of town and would town towers fixed and so the government has not indicated this is all just sort of go around he used the woman guy coming in the the call OK so that is it is but the OK but I just think the average I think the Supreme effect so this is what my Delaware argument give said essentially a reliable site content and now I use money conclusions is still to show by analyzing and Allison from time to time the but this implies that if and because his condition at time no other star plot but I hope the completion of test of time error at handling the zeroing in on 6 at and so how they do that well I analyze the different possibilities of the profile and you stay equality I mean it is useless this facts and according to the scaling of the profile I'd can get either so similar solution which is ruled out by a fear of Maryland myself or and if the concentration is fast some self similar I can produce a traveling with and ideas and the composition except that the air only urinals 6 and now I have to work to make the Arab then caught up with me the 2nd part of the conversation but now the 2nd has 2 parts the 1st part is that there is a corollary over the whenever being I can get abound on the only 2 months but but come at of AFP up at noon tomorrow morning 1st so that the 2 today this is in and who is going so whenever I have in the composition without L 6 error going to 0 I get this improved and this is a consequence of this six-term going to 0 and on the flux estimates and all of the known estimates for the solidarity OK so as the 1st part now the 2nd part question the remember we used in the composition and inserted absolute values is the triangle inequality and to norms we controlled either by Hellas quality or by knowing this and then there were word term and the that involved the solid tones and that goes by that point was bounced that we know in the sultan's rest the year the is here and so here in the In a number not yet I know this is you see here this what the correct statement here is his whenever you have a sequence of team for which we have soul and the composition With L 6 ever to 0 the to officials not the property of 2 times that of each individual the broker the 2nd part it's a very alarming it can at this point we use a very elaborate won't so basically what we know is we multiplied the equation by you we right 0 which integrate and use the fundamental theory and control terms the terms of the 2 endpoints are the key ions In those stairs we control using this kind of thing and then we have a lateral terms that come from integration parts of the latter outside and those we control by the control of the Fox OK and what is their conclusions the conclusion that is that 1 over key to the Clinton plan but the at the development team the and the active and an integral part of Bank make-up of the past and grant is cleared up 90 of the team and come up with maintains 6 up since this from the they were banned from rural most and now we use it a 2nd military are you going to hear the 2nd available human but about but all but knowing this of so let's say I'm
lost their new and instead but but the fact that some of the best part of the this implies but I can find a time TN In between somewhere in here OK but I will find the time here somewhere in between as part of time at bat and let me put the 2 things together that know understands it's it's not that but but but but but but but but but but but but but but but but but but but but but but but but but this instant and no this quantity notice that there the other integrating could be negative OK this is not a costly .period but what they get is that Minju Clinton integral over X less than the graduate clean where and when news of the thinking you the squared minus you to the 6th place in the lyrics that is less than or equal to can so these are the things that they do but this is my 2nd real "quotation mark now I'm going to do indeed now I'm this sequence of time the ends because I have been the 1st prosperity I can do it the composition without 6 more going to see and I also have the other to norm we going to see Chris 1 in place OK then and when to use 2 properties of of the solitary work where waves which it was the difference between what Germany which was to start this diaries that the different points as the different honey the yeah here is this the end that somewhere in between the 2 this is the 1st permanent is not necessary somewhere between the 2 this the same statement is the same statement but the from because it's an old fool around the same time you know because I'm going to have a 2nd property that is because they write that both of them have hold that the same time Of that that's the point always use those 2 times to use and sometimes as a OK so this is the logic I know if purely as a traveling where I know the following we think it turned this is a consequence of the unit ThinkEquity and the 2nd 1 is just the fact that a traveling wave in the direction of focus the L eyes appearing Monday composition Ireland limits In Of the CIA and over the time at the end where the CIA and is it in center where the the traveling waves sentence OK now with this information I can use this part to deduce that it this holds for you once they have a solid that I will they have a solid and the composition he must hold also for the air because in the soliton parting the neutrality of In announcing the should the U.S. stance but they are not that that's the that's the proof but you have to ask us to the point where he worked with From the the 1st reaction is that the 1 . 1 that's going to see in but I'm going to use it in this way you interested in seeing that I will be in the final minute Qin and this 1 is the envy 11 PM pockets and the II is a different thing here maybe ecology OK but I forgot about the other 2 times now that I produced my 3rd time I can formally in there so this tells me that the error has refine them he said that while the of union's integrated into the also known this is .period was in next best what I have and that's what I gained my by my real variable arguments what would you it also denied put the the the I misspoke I hope that "quotation mark so that the the 1st thing is that now this property is inherited by the error but the error "quotation mark goes to 0 in on 6 so I'm going to be able to take out that so that's the 1st thing to do the 2nd thing now is that this will be inherited also by the because the L. is essentially X over the you can and because this is 0 all the terms coming from the solitary waves get kill 10 and so then I get to mine transparencies I can get them and and so I get to this
the reason that the term goes away is again because of the L to Norm going to 0 Indiana so I can get rid of 2 of the and so the offshoot of this they said they can conclude this to profit on the air this it's exactly what's inherited from the past and the other 1 is what's inherited from this you I can throw away because of the 2 properties In the other terms that all the solid those can be taken away because of this and I'm just left when I With the absence pick and since I have a Supreme Court if I take the warning to 0 I catch the can and so I have this and they have OK and now this to inequalities that will allow me to show further properties here Hawkins With this analysis on from this to mean equal so is it clear how I got to those 2 equal OK it's all this stuff that I wrote to the board got me to those 2 OK so that's been my what the gains so far OK so we have to think now here From this I can show that it follows that the difference of this too well to norms ghost Rehnquist said reversed triangle but the because of this is I think the article is bigger than that so I I I concluded but this thing which is always known I because integrating lower X less than is less than or equal to 0 by this and that so therefore I had I abstained this is just trying to To get in combining the 2 St OK now from this says that and the ground is bigger than a equal to 0 on if I go to a small every regions beginning landed on any region X land the PEN will be less than that in and of the whole gradient was that tells me that if I drink From the ball by any small amount the energy boosters the energy and if the spatial imaging is concentrating on the other I know that implies but the tangential energy is going to wise that well if I go to the integral for x less than landed the union this is clear but instead of choosing a landmark I know chooses sequence land and tending to 1 and the X land the vexed transforms into X over how long that sequence and then I get from the 2 inequalities again if you compare this comment From this and that this is just a simple thing now I also know that this goes to 0 because but this and therefore when excess mortar landed the on this thing goes to 0 also the team and into the team get control of the a clear it 1 and then we would this is what we have to do in the next but to do that we have to little to work with him To
improve to be able to show that in the whole energy you want to 0 is the whole extras OK so we have to kill the concentration what sort let me read statement the claim is that there isn't it a new sequence teasing manner such that we have been the composition with an error which besides being in a 6 has the tangential participating going 2 0 the gradient outside always goes to 0 in the composition of its everything is localized when I go and it'll be insiders also 0 because it is proved that the same is true for the world to wrong on the key and this is expression which isn't here and that goes to was the other thing I have that I know proved to that's what we just pray precisely this this will prove that also so who was the 1st such as full of people use the exist such as the assumption is that they we have uniform abundance of the age 1 person to as the most Europe which is the .period eventually then we find the secrecy 7 and the collection of solitary waves such that the words I removed the regular part the difference is the sum of the solitary waves modulated plus Americans who have goes to 0 in this OK and now what I want is that the Arab world to zeroing in energy but then have to work to more steps to compare the uniqueness of taking control of Hollinger International was the last to use this is that the used to this say to you for good work an section was thinking that's at the point of is that the meeting him to there it and analysts that is at opposite dignity it is what we have in him so the is missing or I'm thinking about he was right but that before we can do that we have to do 1 extra step the extra step is that we want to show that when we applied at the nearest so that this linear solution corresponding to this data has tickets were willing so we need to 1st before showing that ghost Xi'an energy where they need to show that goes 0 to 0 in this person said To that we have to have a new mechanism to show that I'm going to 0 in energy and there will be a new general manager so I will I can go there in the steps all so the next step and now I don't have to change my sequence of times anymore this properties are announced to show that from this automatically falls then solution it goes to 0 in this process OK so the proof of this it's very analogous To the proof of the extraction of the scattering profile that didn't on Monday the effect is the same ideas that go into that so what we gonna do is do the profile the composition and that this error and sure that all the provinces you and that's what it means to go to the end of this person I can't but I will not repeat all that because it's so similar to the 1 thing that we've seen on the council In the the OK but what I
want is there is a property a solutions to the linear wave equation which I want to discuss spring complete so this is kind of an site OK so this is the property of solutions new wave equation with data on each 1 herself then it tells you basically that the energy concentrates around the boundary of the line you can't say everybody will agree now to prove this statement when you are in odds dimensions 3 of my news on it is used strong warnings principles the approximate your data like Compaq supported the data for the compact and supported the thing you see where the solution lives and then the forms of land now for even dimensions this doesn't work "quotation mark now you could use that in there the radiation fields that was introduced on landing To proved is an immediate consequence of the radiation the problem is that in our proof for the existence of this radiation field used prompt the head so you don't want to do that now there is another proof of this fact that follows from standard estimates from the standard and improved this person estimates on the new wave equation that are due to them corn and his collaborators that this person the estimates for solutions of the wave equation says that if he hasn't had nice initial data in 3 the let's say the great in the case that 1 of How can that improvement of than corn and collaborators is that the way From there In light cone there's an improved trade picture In a few years that this fall in the but I don't want to use any of this proves somewhere in use of different while I want to use a different proved because it's an introduction the 2 of them the channel of energy argument is that we're going to use it in a few minutes I never the mentioning this is only interested in interesting In even dimension but the matter and we want to provide proof of his things in all dimensions and I should also mention that there is another approach in the radial case using DNA for analysis and that's In in my work with "quotation mark but the school To proof of that so this is approved for a limited time and Meryl so the 1st point is that by density attendance noted that the instances what is nothing to say next handling the use of aerial like and here is a very alive into the and of course so this a very real identity for the wave equation you Pruitt by integration parts this nothing more he and you can trust me that this at the rate coefficients for foreigners 2 million EUR energy has the 1 during OK so now I integrated this between 0 and 2 all right so this is a constant so I integrating at the linear and then rang in the 2 boundaries on this I just use the fundamental theory and that the primitive 1 East but now comes another important point in this and what fall as for the linear wave equation of course we have conservation manager but there is also a consideration of the L to cross to explain 1 just in integrating Inspector Simon the area OK and that means that since affinities conveying the image minus 1 and here it's important to have so that means that the L 2 norm Of the solution to the wave equation remains bound as the ghosts of when when the initially the has come back support so you would think that grows like the but no it doesn't it remains can you consider also directly from the formula for a solution using the coastline on the side of the use of the media and in this instance it I have a combat support OK so there's nothing for so using this From this previous inequality this term controlled because they have to normal use controlled the entities controlled the end of this term is called and this article look so what we get it is absolute value this term in strolled into terms I controlled by that remark and and the others are controlled by so that's a lower bound and now and then proceeding given up about "quotation mark In the welcome so suppose that M is the diameter of the support and then by Finance feed support is contained in emplaced and here this is valid and both even Arnold consigned them to process the usual flight at speed and let me pick and are so I'm going to stick integrally next bigger than equal to the miners are Annex less than or equal to team when X is bigger than equal to the miners are I Band X by the perception and when X is smaller than the winters are bound by the miners look no yes we think that Emmys 610 are isn't given no the 2 terrorists buying and since I had the problem and this H 1 wrong has the squares but with their hands I can combine them into the OK OK no the miners again I do that I do so let me go to the In the end terror my add and subtract that part correspondent X less than demons are things the part that I added once I use this caution sure thing and give me this the part I subtract combines with the other part To give me a line Horowitz OK so I finally end up with this I'd put together both inequalities and get like this inequality you know I devised by R and where they take the law into infinity and I are going to infinity this thing goes to 0 and this thing is 0 by financed by the finance bill that I can so that proves mine Microsoft so this is a way to show that all the energy concentrates on the of the call but I only use integration with parts the music area that OK and the system will be important for us during were so this is the kind
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you have this properties and in addition this personal posters so this mistake to know and I don't want to make 1 comment once we have and we
couldn't do this before we automatically have that there has to be soul it could be that there is no soul because if there's no solid if this person Norm goes to 0 by then and local Koshy theory and the perturbation theory that the solution will up will exist for a long time but he couldn't say that until he came from that this person so he could have been that it might the composition and I've been having some of them would be used to pay for them it's OK to stand in the head of the this is the only in comment that had the wit past approved who can
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is the last step of the approved and this is the fundamental extreme green we need to kill these guys we need to show that the energy normal and so far we know places where this energy goes to 0 but don't normally where they could be concentrating on the boundary and now we're going to use this channel energy argument 2 true but that's the case and remember that the channel energy that we used in the rail carriers allowed us to prove this dispersant property of some of the solutions which are not the sole supplier which is what we used to prove the soliton solutions the radio and we know that those things that back in and out out the energy bound fails 4 the non-retail case at this fails his counters and not only that but in even dimensions is also fails in the rain so somehow we had we will produce a channel that energy that doesn't see this country OK so what we will prove in very simple estimate for the new wave but it's an estimate that will it be interesting only when they think that when this things have some good properties limitless there is distinct from them at 1st so but I give myself up there in age 1 to which is supported in the ball with his her and I saw linear way with this thing then Alcalde the 0 the energy in the minors 1 the L to process age minus 1 I 10 and I'm allowed to use this guy because I have a complex of 4 and this are both Constantine so the statement is that for any 8 a 0 between 0 and and all the positive there is I can have a lower bound for the amount of outside eggs they than P plus 8 and all for any 8 all meetings in can't and this is about as the 1st thorough here another term that depends on economic and the problem with this 2 2 this way With the money no a priori may be completely useless because what's on the right my meaning this no interest but we will see that we are in a good situation where there's not but they say this is of the type of lower bound and those out the energy in equal they were giving the advantage of this 1 is that the strip whether those workers could in the region so the proof of this is very similar to this the concentration on the boundary of the common and that's why I did that 1 1st so that in the beginning is the same I have to here is the same I think an integrated thank you Ohio relating to make his I just play put the terms on the other side now against the infidels into the vegan exiting people think knelt and takes less than people think OK so perhaps it's not so interesting to go through the beacons of disparate but all investors but basically at the same argument as before the rest the accounting when you get 2 so
that the important thing now is this state and what would be really important for us will be the poorly would you like that moments you it said depending how you we we use the support to get the kid gloves are and then we forcibly internal corporate at time so the important thing is
is costly With Correla tells me that if I have abounded sequence which is well prepared In the sense that the energy is concentrating on the year there's 6 Norman is going to 0 the tangential energy is going to 0 In and solution is out going in this sense that at long last the DIR actions 0 and assistance 0 if I'd look at the salute of the solution with his data and this in favor of the H 1 grizzled to enormous positive than for any data between 0 and 1 so them basically the support is morally 1 because outside of 1 everything is small then we do have a true channel of energy "quotation mark so far is the date that is well prepared then there that moment of energy outside people think is controlled by the total energy and I can choose the not To me anything smaller than support "quotation mark that would be the corollary from there inequality we had before more workers explain how the coronary fall so the 1st of observation is that the will OK I'm going to I don't have this exactly compact support I only have enough to an Maryland goes to 0 I will force them complex and I do
that by a local fix some are slightly bigger than 1 that would choose later on picket captain function and look at the difference but got the function is identical to 1 in the Wolverines 1 an excellent guilders the different coasters "quotation mark because in the park outside 1 there was no energy the energy 1 2 0 an inside when I think the difference against the soldiers who cares more the next thing is that now I note that this sequence will go to 0 weekly in each 1 process while the single to their weekly in each 1 proposal to boast of their weekly just because its energy is concentrated in thing on him New energy then has to go weakened to 0 because test against functional the contribution and the thing that onions is negligible and outside I don't now says this the weekly it will go to 0 in on 2 procedure minus 1 by the village compact so they see mind 1 contribution will go to OK so deeply that the city was usually the vessel where people to not be 1 ones the arrange 1 the body next covering is 1 makes is thanks OK I I haven't evolved and is looking at the data look so this is true In this date and almost as if in a L 2 proceeds minus and now I look at the the corresponding solution to the wave and now I just use the dilemma I'm in In condition to use the dilemma because that I might have said they had to be supported in some already use some radios and the reasons why and think it is far bigger than 1 so this is the exactly the statement of but then I just discussed now this terrible to 0 because the miners 1 goes so I get but remember that this data is well prepared the data is well-prepared means that the team derivatives plus the the the I R is all it also means that away from the boundary of the voluntary is 1 there's no energy so I can replace this quantity 1st by the energy and but with the negative because Texas pointing outside and then I get 1 1 1802 remember this is practically the BIR times the key but the the art is practically the only key so this is practically squared and the squared is 1 half of the square plus one-half of the others and that is the only energy I have because a tangential gradient was next so the combination of those 2 things gives me that and I think are very close to 1 remained divided and then again which is what I want is the Supreme Court has not 0 I can take large enough and that 10 so all this preparation and initial data Coleman hits with the fact that signal has attended a G 4 and the basic idea of the energy channel method is that you can pass From dispersing league owing to 0 2 going to 0 in energy by using this so that's the conclusion of the probe which shown OK so of course this has been completely linear right users and many of them why can I use this for the nonlinear approach and that's where the fact that this person's norms with 2 0 if this personal nor most of Europe the linear solution and the nonlinear solution are close and that's just me the local theory the kosher problem and that its perturbations "quotation mark and that's why it was important to prove that his personal pursuant to 0 so that that's the step that allows you to pass from the new OK so that is the
conclusion of the OK so we're near the end so suppose by
contradiction that for some some sequence this doesn't go to 0 so it's bound from 10 now I have to read scale by the selection my previous inequalities can notes Standard immediate and I was very what they have this is my general manager announced so they cannot have economic times to cement and they have bigger than anything and because this is uniformly down some subsequent I have immunology and as a way put the couldn't put for book and this is a very true for any and all In 0 1 so now I'm going to tell you how I would choose the OK remember that we have in inherent in the composition we have a number beta them which indicated how it how much inside the ball the sultan's are OK but the data was the upper bound on this the 8 unaltered will be chosen slightly bigger but still less than hour what's the effect of the effect of that is that 4 x bigger than anything not we'll see this because of the concentration OK the have of the world please yes because that's a given by the energy in the air when the energy of the soliton tends to infinity as Elton's to 1 the company so that that is crucial it is absolutely essential for the money but we this is OK directions focus so I know the fine W 0 and pleasant ,comma doubly 1 and To be irradiation top-class this that is my solution where I forgot the soul look no since is a penalty is bigger than data the difference between you and this guy guy outside is small With the solid the negligible in so that's why we need to you have this out the energy information for any time now now is we don't know where the solid isn't and now I look at the solution non-linear ways required and this if things just have to do we're staying away from the other Cingular .period so let's signal now we know but there and this person or that it is willing to see so there approximation theory it gives us an expansion for the solution with state it tells us that this the solution for that plus the Epsilon the beginning solutions for this spoke of says that this person wrong goes 0 the linear and then only and solutions are close to each other and an error which becomes small OK and it becomes small world not just in the misperceptions sense but Indiana and that's the approximation now because of this inequality initially and find speed of propagation this inequality propagates in time and what we get is that you is close to W X bigger than Kamensky and because they cannot stand that's what Feinstein of propagation when Pete equals the end this is 8 of 0 which is this and this doubt is smaller than those on altered systems small parameter here that doesn't play much of a role adjusts his separating the difference sink we have based the composition for the new 1 we have a general property for health and then W N because of this the composition this guy's small this is a regular solutions on small sets its contribution is small and this is the thing that has the channel so that means that Dolly inherits a channel In this again now I can put P here the but again by the concentrations property of actually wrote to the 1 there in the end the ball radius and finance predictions of the grossly the rest of America so I have but in this region this it is all so I
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prize is that I went from 1 hour 16 to 1 over and this is valid for all Altes in this now I appliances and Delta over to so I did that notice that when the and goes to 0 this region disappears and this is a regular at me because this time Delta over 2 hours away from the singular time is a regular function in the energy so how have with its integral over nothing deposit this is what the child energy argument so this is a contradiction to the fact that the epsilon wins had a lower so selection Iran's couldn't have had a lower bound so they won't and if you remember in the other channel of energy arguments we've seen there's something that happens really the boss of the time and again here we don't need that because we have of going for the full time is almost 4 In this completed so this is the last step the fire in the last part of the reason we have this year which is used when that I had and the composition with an error having sought probe I have to use all the programs I have to use all the properties of the era that had up to now eating was yes because at end their otherwise I don't know if there are other ways I don't know there's his little right because the is Indian general result their right hands side convening except that for this particularly welcome message and sequence I have I have good control in this way of his so what 1 could conclude from this that this is a correct approach to this sport but everything matches the Justice Ministry will mean that you had you have the hearing which sends you it's excellent which received sequence time satisfies itself where strength you have that result for forest when you need a certain amount of reasons because of the Union of all of these things come in different places showing that from Ceciley kg PGA weight you have this right and so on those lists of properties and necessary to show that this thing and the lower house the 1st thing that very only the dispersal property the junta 0 but you need this person's property to pass a lenient enough notice so I know that this is a somewhat convoluted maybe not the easiest easiest thing to absorb that I hope but listing in some feeling for how this proof "quotation mark but there before asking for more questions I just want to make a few comments so we true in this the composition but you could ask 1st is that do you really need all this solid consistent to the the committee only need 1 this is a question that many people have asked and it turns out that you need more than 1 and as the goes to infinity this was proven in favor of Lieutenant Maryland in in the mansion fines you have Maltese sultans appearing in the asymptotic bookstore the reason for the mentioned 5 said this was very important is that fragments of details of 2 of the summit of the higher the dimensions the less fat they are and that's why they're human words in the mansion fire now in the final time blow up it's still that hasn't been proven has resigned day yes I kiss him business hasn't been proven that the community that the solutions with 2 bubbles that have finance all but the idea that this is within the reach of the techniques developed by Jesse attendance in his thesis that will will returned them among and the political then there is the question passing too other sequences of time still the 1 and have already said something about that that of repeated this is a difficult problems it has to do we have no multi solid collide and it will take some time to understand look so this is all they want to say about that again then how specific is listed as probable OK so this is something else to understand something for this problem all of his work decision was made for well I think 4 other nonlinear wave equations maybe geometric wave equations this approach also apply so for weight lapse for example for critical way cups and leave the district work hopefully eventually also for the way version of young males at some point note then in what happens without finance piece of property good so enchanting cases of correct these cases of things of that type I think this is a really good direction for the future they want to say more than that but I think that they you know this shows that things like that can be done so no 1 should have the courage to to think about the OK and the meeting have 1 more thank from that's thus the final society has really appreciate the interests of the sustained attention movement action so thank you he also commented on the fact which is the and seen before so I'm just going to tell you lose more than what you see in the it was of the year I think in some sense and that is the case because for small frequencies which in this case it's with and I have no relies for handling of it you have no voice I have no reason some of it is located at language the questions I couldn't 2 which has lots of other directions 1 of the very very long-term well I think I discusses the covered but any other the questions welcomed work was still alive this is so critical situations where there is so far nothing has happened since the beginning the game yeah I mean really is the the the horses and important questions another important future they're the tangled in on the the 1st of all you have to use the approaching this game for high-frequency right about the Slovenians but said that the some of the day in enormously should come to understand what replaces the injured outer energy equal to their lawful this year if you the loss of control the news conference I don't know but of the of all that he has 200 hundred dubious this year's you of the need to have sold more so you just makes you so if you could very loosely tied the knot that's why you need to have to be in and have to come to few questions the use of OK so you then you have to do something union against you World nation into an example of the view of many of the people who many many pro but of course this 1 because it's more work to do you don't want to do something as finishes things of work it is better to start things off L but it could have the comments on that this this is the case in fulfilling and be