Merken
6/7 The energy critical wave equation
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00:04
do you could have been and so the last time show them how to extract the scattering compliance In the present time In other today and the next time we will sketch the proof for the decomposition in the Financial Times up King OK so we have a solution that they will lose up to the time of 1 minutes had remains bounded in each 1 per cent to up to 10 1 and we want to now produced decomposition into OK so that's the thus the task so In the notes so there is a 1st step in which we through the only compositions With a weaker Aaron will like and then we will go through several stages To improve their error can and you know that to improve the error sometimes who will have to slightly tune this sequence of time and at some point we don't the sequence of anymore but then we show but the and that properties I I hope you know the 1st step it is then inspired by work on wave man this is the goes back to to work of the lackeys of we've maps and said what we call a moral it's best for more overstatement the fact that there is a way to use ideas from maps for the energy critical wave equation goes back to work that we did the court and slow so this is the approach that's presented in but today I'm going to present a different approach which I think is silly more concise conceptually and it allows I think hopefully for more progress as we move along move 10 so because of that I will use the blackboard from the beginning and ending the notice the other proof you will to read them "quotation mark so In the past at the end of the book but up the but the and that kind of thing that Bill Gates who can always make it to the 1 by scheme the problem told him from the following name I think order the crucial more and more of its assistance but the best part of although the forum in which we're going to do it he did not recognize this is coming from tomorrow at St. Andrews and in fact I think that you can recover his usual war with identities even in the wake of gains from this point the so the claim is that the pal this soon the blast sequence to but the but the bank but at the I and then the origin is a singular point so it's appointed where even locally and cannot continue this solution anymore here we introduced the notion of so what that means is that it is not a regular .period than a regular .period is a point where if you take a integral also were small balls around them .period you get that to be uniformly small for old times to those of regular points and this is a singular points was a point where this concentration that's what that means and the men may assume that there's it's faced by 1 from all the other Cingular .period since the final Romania can was pretty if this is just so as not to have to write more and more OK then part of the constancy tests that foreign 0 at nite and came to water the amendment was happy so this a half a symbolic and it depends on the fact that they are the singular points are at distance further what otherwise you have to go sufficiently close who could so we have an instrument and that the estimate is now the internal front teeth to warn of an integral X less than 1 would but the the following and expressions but but the the but that but the but the but but but but the but but but but but but so we get a logarithmic estimates but what's crucial here is that well 1st of all the interval is fine that's already very good what this thing interesting here and that is that the power there is less than 1 the so the original proved had the power threequarters we can get power onehalf using this approach no why is the power 1 have interesting instead of wise the about power less than 1 interesting it's because but in the final analysis all of this suppose that this was just bound this is another integral and then of course then develop up to to 1 minus the would have behaved like that than 1 1 legacy to Apollo 1 but somehow we're only giving the power were have so that means that in some sense that in integral vanishing with the rate has singled the "quotation mark 1 and the powers to it's a dimension defense please so the number of is different and mentioned 3 and this it has to do with the scaling quit that's exactly what the now in Waveland there is no way the sector doesn't appear in that's because there's no factory in front when you we have OK listen that's how
09:06
have this thing is work but the proof of this group of them when the show now is a very From my point of view is a very satisfying 1 because it uses they ideas that will already seen In that come from the work of Maryland myself where we showed that there cannot be a compact solution which has so similar OK and if you recall the proof of that which was given give him a month ago or something like that and it is induced in the introduction of sensible call OK and then some integration by parts no the reason 1 could do that In the compact situation was that they boundary values on the inverted called the inverted called him but at about the values of a compact solutions are forced to be and therefore we had some bounded terms that disappeared because that was not in this case we can no longer assume that the thing is 0 so we have to somehow handled these boundaries and at other times are huge we can't so so under prove that in the plane the and so the 1st then is what we call the energy flux estimate put up and we recall we have addressed some solution that those uppity equal to 1 so that is the what we call we recall that there isn't he the which of his regular solution equal to to 1 of the at the time so this is not 1 that blows out but it's 1 that moves continuously up the plant's equal to 1 With the property that this support I on the view of the 9 concealed the that is contained up up up up up up up up in the vertical so that outside the inverted cones fires solution looks like a regular and the we saw that that this was a consequence of fine it's been perfect so what you do is you consider that we claim antecedent to what and he now souls the nonlinear wave equation with that band without any initial data that kind 1 and that's the way OK so we have the objects thousands of singers is a regular solution we have dared to facts but the the of that at the cost of more than half of the year but but soon the the arrow and you are 1 of the nation's means the pair you you the the the OK it's just a short so this is L finally but as I'm more moreover the spacetime moralistic as normal but but but at the back as part of his life because this is a regular solutions sold by them final blow up criterion that enormous is fine because it doesn't blow OK Is this but so then you just want to see him in the form of a minorities everywhere because this is regulars for canceling the system what In local will this theory gives so that the next remark I'm going to do about the Is there as a consequence of this to property but the and the the the remarkable fact that this is a trivial right from the 2 other properties and the like was right because 6 thinking the the Pena Nieto the 5 times the the you do there Normie next year caution shorts you get they'll tell them zeal to and then the other 1 is another 1 and the other 1 is in addition to the OK but now this implies the meeting was 6 new loans to L 1 I think team In fact and the boundary of the course and the 2nd that's what Standard trees and if you have read the teetering between a log you don't need to even invoke a the the fundamental welcome but on the boundary of the call 6 equals to the 6 but is supports so so therefore now we've proven that Newton 6 the we don't hear a lot of time at bat the and this is huge this new tells of the law because now we combine this with the flux John right down the flanks well maybe out of rights the latest explicitly here but but but but but but at the at the back look at that this is the same and his writing his heaquarters phenomena ride the energy flux urgently floods in the following identity but socialist state and by pop back the warning to less than 1 the energy fluxes fundamental in this country and where equations but the but you know that well known identity at up at the I am at the end of but am I the but but I but but
18:26
at the back of the bus but by the time the bank but but but but but but but but but the but but but but but but but but but but as but at the back but back but but but but but but but but but but but but but but but but but but because this is the standard flux identity and you proved this by calculating the appeal of the density of the energy and then using the fundamental phenomena that even of this and nothing more no 1 of the big difficulties in the focusing wave equation comes from the fact that this object thing here is not known to have it can't possibly change and this is what causes a lot of the difficulties that we see but this observation tells you that you can control the Macon Ga and that's me 18 we can control the negative part because it's exactly on the boundary of the call where the you're blocked solution equals the regulars From sold the corollary of this some of the other is that but by the day but but but but but but but but but at the back of the bus but but but but but but but but but but but but but but but but but this is so 2 but but but why why true well we use this identity by the fact that they age 1 grizzled to found in this film is bound history is balance by the previous things that aren't 6 3 respondents what I haven't about just worry that exists square here looking at those plants you this a square I have time that you have do this and is the cause 10 yeah this this year as it should be yeah squared because here warrant because I knew that I could of can clearly this thing is not because they have a look at just the question How can completing the square with the OK and another in the way of writing this which will something be His name at the top of the and 2 inches of other introduced some notation I hope you can read this at the back at the back of the pack but back where my definition With the right this is tangential part of the but at the time of the past this is the definition of the word and the reason why this follows from that is that if you were just expand this where you get with what is it so it's telling you some control on the boundary of these objects now I will need 1 of the term which is a hard term here the so I will add this to the corollary but at the top but I the president also a moment we can know this and needs approval I where did you can't but that it you see things were very neatly here because let meaning fine AP at the back of the To be this function this is a functional New Mex and where you hired in function OK so let's calculate the gradient of the function of X at 1 but but but but but but but but but at this whole just change of OK so you see that exactly the gradient of this effort is the flux and since the fluxes in by this gradient is innocent and I can use hard and on the boundary of the cold 1 the equal access look at Brokers who now we got the 3 things that we won "quotation mark so do they call this the 1 yeah I didn't vote when I'm going to go to Step 1 prime that 1 prime introduces sell similar course look at which would then already when we study Compaq In
27:19
the ground state conjecture who can so this will be the same but the but the top of the past decade so now as opposed to exit is so exits less than 1 man's clean it was In between these 2 sets of elections I was giving another set of elections where there was a blackboard with the hole and I lost that he raises went down the hall in both thing ground and threw to a halt to the new so inside the home the while the extra work .period wins the but why is less than 1 and this is drying the long 21 invest so are no universe now is the S Y variables and this is as I mentioned In previous lectures this is that now and along the parabolic similar variables there were introduced by India and Co and that's what we use here so we'll be fine MW of wine and add to it 1 last plea to water hands you will then asked this was just like we did that earlier in this 1 half power but 1 minus the is responsible for the coefficient 1 half in front of the what is this really accomplish 2 can will get to happen this is just information is not something that's supposed to be on the "quotation mark his soul with now introduce some Wayne's back In this way it obviously blows up and lie equals to 1 which corresponds to the boundary of the call and that's why when we were dealing with compact objects we needed to have things have vanished .period above it but now all over the news regularize this way but but but but but but and we will work with this week instead and then choose Epsilon approach so now we have S goes from 0 to infinity but this change in the world where dingoes between 0 1 and schools 0 2 1 and why that someone because we're looking inside orchids and now I write the equation 4 W which we have already seen the act equation is the same as before we just think back at the time but but but get this is the equation adopt new so it is that it is the wave equation but it has long been wave equation that you cannot the voice but they have some very interesting features firstly elliptic part the generates and why boost 1 To the Virginia the question and the 2nd the thing is that there is this this extra term here it was the site of news conference this properly as a gradient and ingenious so it's a 2nd over there in pit and that's a kind of parabolic type 2 knows when this becomes a little bit parabolic In some similar on the anyway so this is the question and they're just gonna I tell you to things lap this weekend computer intensity but at the end of the day but the other day so the yesterday the W would like well I'm sorry but that should lower than a year I think that it would be a only thing we need like this 1 of the lessons of incident hope that should be OK now what is so who look at this expression him you see that it corresponds exactly to what they have sentiment that is how to divide by the hour 1 might stick to the 3 have Bloomberg and so that's the point that this this expression inside the business of similar kind OK the president of the voices I have the legal and I have not done anything this is just the formula there's no actually in the sport of it that and this this equation has the role doesn't have the OK which is singular and equals to 1 and I'll tell you when the cups and then I but the main thing why this trend is not simply a growth this year but you can so
36:25
From this still formulations actually had they each 1 over his behalf Hawkins because that says the changing measure and in the end we know what the want is we we know what the White innocent as gives you the right factor this is comes from the fact that each 1 of these objects once he waited appropriately is enough and for you you have to use the regular hiding in call the In Max Hopper now I tell you what the flux gives together with "quotation mark the Flex estimate together with sense that it gives that then the girl from 0 to finish the integral white folks to 1 of the SW compliance and squares the statement by the end it's lesson why is that because well DVDs has this term saying In this 1 was a change the variables corresponds exactly to the flux response to the 1st and this 1 corresponds to this this is a change of heart OK so you see the Valdez values may be bad but there's something good about the S 3 with the and finally might might claim boils down to the following day the but but but but but but but but and this is just the change of the rules and the formula for the you just have to change our arrivals in the y India the BTS changes variables produces the factor 1 or 1 1 mistake because that is the law this is all In fact fit the incident so no How the true the testimony In men 10 people now what we do is to introduce an energy policy that until who will play so identifying need absolutely this is an entity which now depends on but the back the but but but but but but but but but but but but but at the back of the but but I am but but but but but at the back of but but and last Monday finish Clinton said None this is the natural at this if I instill human role this would be the natural energy associated to the 2 the sequel but people you multiplied by indeed W DS integrated by parts and assume that all the boundary times of 0 or you would get exactly this you have to do DSW times rock multiplied an integrated parts of just what you always do for the wave equation that's how you deduce the energy OK if you do that you get the energy for Epsilon and all about terms it's appeared in there and the differential in colleges OK now unfortunately the about terms don't disappear for us because the thing is you and 2nd we don't know that when actually 0 this expression is fine because its singular wife coastal what why should be fine this the 1st leg of a operation not the 2nd 1 is these 2 terms combined will want to tune in to this year it included Saudi so you have to be a little patience when calculating but then everything cancelation and but for us this news broke it's the singular medicinal herbs along the boundary in the case so we were working with compact objects they were 0 the boundary and we could use that easy the match and that's what we did anything in the paper you know "quotation mark Council put the upside so the interesting thing is the time derivative the S3 with etched right that's what you need to look at I know the amazing thing is that there is a clean formula so in the very nice things that happens here 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 but but back
45:51
you can solicit the formula and things through the surprised when you could actually comfortably and it's a very compact so let's look at it and I saw how the prove 1st the the same as the lessee would do trying to show that the energy for the wave equation is constant and time you multiply the equation by W Times direction but that the rest of the time in the history of the government of corruption and integral part years playing epsilon 2nd and know it's a power 1 strain because the president is simple this is 1 of the units that than some of the signs of change but it is the power 1 because it if you look at it it's not the same explanation what yeah well dash which is convinced that we think about it is but it would be up to less option would you get is a With a natural squared squared with the kids None of it is that tho it is not the reason that it is absolutely what happens with this when you go to 1 equal to 1 it's exactly Epson's clearance work so that the way you prove this is you multiply by the DSW times right an indignant and of course if you can do it at home take your time because you will most of the time to get it wrong and then you had no life experience can but this is the 4th "quotation mark it's so now let's understand this formula be what we want what's himself 1st let's assume that W. was 0 the back do not like the old case when W is there on the boundary DVDs that news also 0 because the cylinder so this term would disappear OK this term if you make excellent equal to 0 posters this term disappear and this time you get 1 plus an squared saying that 1 now when we integrated if the excellent as the energy is bound you get in the middle of this thing and then develop this thing is exactly what you're trying to control "quotation mark so this is the right kind of thing we want to integrate and so the best thing to integrate the severity of the fundamental thing OK In our case when In arcades wellintegrated in this term it's the 1 we already know from the left this the integral vista is converted by the flux so that somehow what saves here you know the flux and therefore I think it wrote to him in the over commerce there was a selection but said I'm not going to make absolute 0 make absolutely right that I have more of this is a citizen of this is the same as in in the 1st year here and there in the and if you get it inbounded the ball on the way but in the case this long predecessor managing to integrated and then by this magical identity it controlled his interval because this is I'm sorry because this is the a positive wanted which is what you really want to control so that's how the magic words so now we're now what do we do do when epsilon is not when we don't have the didn't boundary OK so the 1st observation and reuse all the blackboards and then move on back to spare the 1st alteration in but at the at the back of the pack yet some their energy funded by 1 or why because the quality of life before he was only 1 of his remark by up now look but at the time of the but at the back of because of this said in the definition this is Roy actually "quotation mark and the rest is covered by where there is act the 100 docket and this term when I integrated this Bowden so if I do an integral here that the role of this is going to be the Epsilon and the 20 points which is 1 of the Epsilon this it's a constant diet 1 over children and then I have this to say this is the 1 that I like and this 1 is a horrible but it has the redeeming feature that there is an absolute square In where there's an actual squared I'm stroke as mixed he has a great deal and the media this is now amenities coach With His so with integrated between us as well and Aniston at time and use the fundamental thing for kids then I 1 less Atkins where integral between wireless to integral wireless and 1 the doubling squared no 1 was consumed square government squared to 293 hands are will be less than equal to see over Epsilon that comes From this firm In this track and then but but but but but but but but but but but the pop at but but but
54:46
but but but but but but at the back of the car that so now going and due caution shorts keeping this week can but the absolute square that will keep would this term which is that the 1 and only and the other 1 I leave alone and then I put the small constantly in front of this 1 that they absorbed when I left and what they are left with it is the following see you actually plus it now I here I will get an absolute to the 4th when it's quick and in the denominator I haven't won over absolute cute right so I get an excellent to that a wanderer and that's then I have that the length of the and because I'm not using anything on that I'm just doing it and why but the but and that's it and maybe instead of once year I have won over any after a hit that went bust but but at the time but but but but but but but but but but OK suggested caution short side I kept this 2 things to him you by them he calls this season it is a it comes from the coaches for clues on way With the quality of its last fall New news innovative since the symbol for something smaller than 1 OK and I don't care what constantly habits is a universe so all I'm trying to say here is that they use in little bits in the cold here an increase in the and the constant the aforementioned what it does the otherwise less than once likened throughout the 1 and the gradient is enough OK so this is not an Infiniti Pro so the White is not that it doesn't hurt what hurts me is that they can't integrating but now I'm going so was I had his bones the 1st thing I say cameras this thing after all is bigger than but it's only bad them and it only grows if I bounded from below I can throw it away but but but but but but but but but but but but now I just choose excellence but the but a now but can you make that 2 terms equal and approved so if this OK you have to have because after all the competitions and thing but you have to think that it will work write but the key In the key point that we realized later was that the fact that the DVDs this double this direct relationship with the wants the focus of this has to do with the scaling the question all right and it is the work also for forward maps with this kind of argument gives you control the more with state estimates in ways woods you know In the work of the servants and tire wear Taliban very 2 this is another way of looking at booking and full of fish have nothing to do with it as for the rest of the way world millions snow this the scheming playing against secret because but otherwise you get other aid other numbers then in admitted that the thing that is a little bit and from the final mission now but it is also that then what is the expression of opinions here this is because this will always appear so what I would say the is that you are right to appoint this can be used in subcritical cases that affect the Zeiger Maryland used In some subcritical case now what it buys you is unclear In anywhere I think this is a new perspective on on on this kind of opinion if you look at the paper I will flash a slight snag so well all that was hoping 1 thing were going to do next the reason I used the plus equal to 1 here it's because the calculations word in my paper with at the place equal to 1 but going on it's more convenient to instead of using instead of using but this picture to use this picture but this being the blowup .period who is just a little bit ,comma and the effect it has is that this minus here becomes a plus because pointing in the opposite direction something other than that it's all the same and I didn't want to redo all the contributions sewers for that's OK OK so this is what I'm saying here In reminding all the singular said then the week but now that blow up .period this sequel is 0 OK so the picture is that so the
1:03:44
Esteban now had the power threefourths with God now with power onehalf but they are equally good for women do next but but 1 half is better than 3 4 4 because smaller and hopefully this further improvements and the only thing different now is himself 1 minus the have and that's the point of often using that picture of what is nice and warm ministry and the sign is the opposite OK so I'll go through the roof here
1:04:25
"quotation mark it was incomplete but it's a very lengthy thing and in the see clearly where things coming from I think this the calculation in quite the same immunity "quotation mark and federalism 1 so now we have to improve things so that's that the process is that OK you get something in the reason for this is a very nice estimated it is
1:05:02
is and saying this again here it's because of this thing growth slower now an average then this long and so has to vanish at some rates I what and so 1 can think of this it is as a Toby Aryan you so this is telling us something about the way in which is our songs and we want something out of the object itself so that's where we need to pass subsidies but the classical through various arguments says that if you have just hours ability for such subsequent you have OK and that this is the reason why we we now need to don't so it's actually said .period it's a very classical things and series at 2 2 1 OK so that let's move on so now we're going to do the actual an argument in this context the that offer of course we're going to use it in a bit of modern technology to that Winigan fall the the original thanks where used to harden into a maximal function for example that actually is comes in very handy "quotation mark so now there is some reliable type of worry the next level tells me but I can find the sequence remember the time is now going to 0 instead of 2 1 can soon I go find a sequence of times and the 2 I'm going get 2 different sequences of time and if you recall in the proof of the extraction of the scattering of fire we have to talk to sequences of times 1 was the and and the other 1 was the and miners of foam because we needed to the same integration and we needed to control both and points in and this is the reason for this is simple and what we will do it 1st produce this objects and that the averages are actually going to 0 it's not only that but something enhanced where you can also integrate the bigger in the I still think the average and a little bit bigger in Annex I still think him and those close to to you and we want to choose these guys so that they are actually separated you think of and the role and accusing each splintered into 3 parts and 1 of them is 1 extreme part and the other 1 is the other this is a very standard thing that you it in this kind of problem accepted the numerology don't pay too much attention for about 2 fractions it's that it is the point of the Florenz the Arkansas there's a separation between and there are other signs OK so we don't produced a sequence to sequences with his 2 pro broken it was important for to him there because you need that their differences in the size of each 1 the difference is the length of time and the union and the length of the interval focus on how we do this so now we're going to use this tomorrow insisting at 1st so it became a big capital J and we look at the 1 equal to 14 lines J and the 2 equals to the minister it will apply it in that case so that the 2 over T1 gives me J to the threefourths we could get jaded onehalf we'll get to this point again and then I'm sticking it in the into the interim so this is just the splitting of and the row between 2 2 4 2 the minority into tonight into equal and since the start of it smaller than this but pigeonholed there is 1 that's smaller than the miners onequarter which is 1 minus 3 causes J I'm concerned this is just a pigeonhole argument this all of them are bigger than theirs I have jails them I have bigger than jail into the minds of Porter and that's the 3 quarters but it's a smaller so 1 of the mistakes that were made if are being again contradiction OK today is 1 little J
1:10:42
like that and now I look at the sequence made of 2 to the adjacent fought the miners Jr where this has chosen new JayZ and choose a decreasing subsidies welcome to the secret sequence that's going to 0 answer the phone call the ACT with all of rights that's exactly what it was slow going focus and this is precisely the way that you use such an assistant where instead of having it he had J. here the good news there is no way to OK so this is what we knew we had to the data online and so forth and then that we can make the same decreasing and we get at this tend to now it is the 1st part no I'd Mora called G of the dysfunctional team which is the thing that I controlled by tomorrow it's estimates and Genser gave better control and now I know that this averages of due date tend to 0 that's what I chose the meeting so this is tend to 0 after passing to sub sequins and related relabeling the indices I can assume that there are 4 to the minister Clinton just a simple argument and now I look at the heart of the 2 a maximal functions of G times kind of the 2 2 2 opted to mutate look function and now I'm going to use that we that 1 warning equal so that the measure of the set of the Mangini to where the maximal function is beginning to to the mines 2 it is less than or equal to Due to the G 8 times for to the minds Jade and ingenue G which is a long trauma so that's why I have 4 here and to hear and live with look at this said this suggested we got 1 morning at all so what this tells me it said that in the set where it's big is getting very very small so there are a lot of teams in which it is small and that's how I choose my people can I have a lot of room because they have a very small fraction of the smoking and so that's why me you can't use them like that and no the air the GE remember was integrated only up to the why can't I go to see comes the because between Kerry and seek times the you equals and is a regular solution and everything's going to 0 4 and the maximal function is exactly what you need to control thinks this is exactly the maximum resisted that the national maximum corporate tenants now we have a lot to sequences of time we will 1st use just over each 1 of them and then will put them together OK so the 1st step is to use each 1 and we will get any compensation for wire and a composition for the other than we will use those to the compositions to get some improved estimates and then we will find new sequences OK so that you have to be doing things gradually he began get everything you want "quotation mark start and can give a preliminary so I suppose we have a sequence since then this condition star then we can have a preliminary decompose so we have the gene not scales centers strictly less than clear 10 Lawrence parameters which are less than 1 traveling waves such you is
1:16:20
the regular part the modulated solitaire displays an error but in what sense is there is an error His Oleanna as error in the sense that it says 6 almost there around 1 VH1 Crandall to but 1st emerges get the unsuccessful .period so this is a 1st step who can survive explain how you didn't hear about on the floor of the rat much of the money is not in and 6 my vegetables Crockett the man who was sufficient for me you know we have no we have to work very hard for strict will get that at the very end of this year said so what we would do well you will see that the discount at the end 1 0 and step before the last 1 and then from the stickers Florida have to move to the actual images OK but you will see this would see next time that's a lot of extra thinks I needed for the sicknesses and go with his former the center city and this is the starting point OK so how do you prove we're going to use that as inequality star remember this moreover mistyping equality but we're now we chose the intervals well so the 1st point is we take a couple functions the which is why May 3 and supported them before and now I'm going to look at you when you were in which is you don't feel of X over the years OK for truncated menu at the time .period the key areas the sequence of crimes for which I have this enhanced what now I claim that it's enough to prove that the composition for this sequence why is that because the difference this thing equals the new miners lead by the Maryland goes white wines that focus on this there are basically 2 cases suppose that we look at Exit bigger than him OK if X is bigger than the end you or the the enemy of the United equal so the Spartans 0 what happens with this guy you X is bigger than me you so this is a regular solution and so for regular solutions then truncating up to small size the surrender NGOs so that's the part where it is bigger than the How about the party breaks exists will where exit is a small Olympians this guy is equal to you because of all the Church of the so on last is but he is regular and it's in the region were Texas small Olympians on the most certainly proved this thing OK so that tells me all I have to do is handle this yeah the Brazilian I series is that a mountain where you don't need any other singular look at this and think of it as 1 issue was broken so that we be they don't interfere with each other I I want to know what was going on show I'm sorry but we have use a profile of the compressors so what we do is we do apply highly composition we decompose sequence until this blocks the about pleasant memory which stands tho the the dispersants so far so good and I'm going to be divided the team the day it has assumed that very the identically 0 or the limited most of plus or minus I always control and I have a of course should lost over the the 1st observation said the
1:21:43
0 and you 1 goes to 0 outside X bigger than the why is that because the you couple what moreorless he and outside P you is so this is really the and since I have a couple of on me me the couple for up to 3 PM this is smaller To me this is struck so this is the year 0 as an since they use of ends 0 1 are localized In X less than quantity and that gives me some control of the parameters in the province OK so this goes back to to the whole region Gerard you just have to take my word for this so what do you get for free from this Bowery Gerard the accident got put in the wrong place at the line there was Frenchspeaking that this is head so what you get is immediately from the from the localization properties again the oldest scaling parameters are controlled by a constant presence these men in this day and the translation parameters are controlled by the 7 because away from the cement nothing is happening the thing is just as so everything is controlled like that and the other thing that you get in this is that you can go hand in hand to terribly important heavily here is that this limits it is not a 0 0 4 and old 1 in the exchange that can be only 1 word the team in the land J equals the sum these are areas that were local and we will call it pregnant 1 the there's only 1 for which this is not so there is only 1 for which there is scaling is something of a that's what this means and in for that 1 if it existed that we could change the profile again To make them equal by rescheduling the prize and now by extraction this sequence is bound right because of this property so the limited always exists I'm not saying now that year less than 1 year this so I can assume extraction that all these the music now we will divide the profiles into 3 case the 1st was suppose that there is J 0 but there is a J 0 for which this through things are equal OK what we will do it in this case the associated nonlinear profile actually exactly fell similar and with compatible and because of the theory that Merlin II probe those things cannot exist so there can be no selfinterest profile this the 1st case I I will I will explain how you approved of I'm just trying to show you the big structure the 2nd case that is still that the date ending 0 but the is much smaller than remember the jet the middle of the J is always bothered by a constant identities good so there in the comparable on land adjacent small and the last cases when the game is not 0 but when it is not 0 it means that the day in Indian ghosts but because we said that we could change the team jails to be either 0 or going to OK we will see that that the profiles from Kaster can be put into the air they automatically have small and 6 OK I will explain the insect so we will not need to take care of this profile but in this case In case we will show that the allergies are less than 1 and that there is a profile is actually the term way and once we have this decomposition fall look so that once this
1:27:10
is done the result follows 4 case 3 the point is that for any linear solutions the 6 nor will mostly stood in the United States has 6 know almost as the goes to infinity and if minus the Indiana overland journey and goes to infinity that's where that profile than in your preference evaporated so each 1 goes to but we can combine them because we have a Pythagorean expansion foreign and 6 known to us currently and he said this is the proof that there's 6 no almost as you it is and speed of property despite the proof this procedures used final speech propagation and and the this so because of this thing is that it the 3rd case coming moved into the air so we just have to understand the 1st 2 case now the other point is there In the 2nd case the case only have accused there can only be financially many and that the number you can have depends only on the H 1 person too because if you recall for all nonlinear elliptic solutions the gradient has a lower bound you know about this so if you have too many of them by the Pythagorean expansion you would violate the balance of the H 1 critical too show that always gives you an upper bound on how many of the kids you can so now I'm going to try to explain why In the 1st case we have that so similar case which it's not possible and the 2nd case we get the solitary ways OK but this 1 so engaged 1 so that means that landed J 0 N equals the current and DJ 0 and is 0 not in this case is not hard to show that the CJ and 0 since they're going to the euro you can take them all equal to 0 by changing the profile and that the the linear profile and hence the nonlinear profile have support computer plussized so after using properties of this standard the properties of the profile composition and the approximation theorem you can see that the more words estimate gets inherited by this nonlinear profile OK and what we had in get is that the the the over people's wondering within the EU over to the case In this now has to be identical easier because you make the and and the scaling gives you the best and this is true or not everywhere but sufficiently B. B you know we get that this thing is identical easier now this is the 1st or the equation so you can integrate by characteristic and as and I can't say whether it's me we have this 1st or the equation and there's only 1 kind of solution this this is the formula for the solution for some function and since the J 0 was support sees compact support no we also know that Eugene 0 it's a solution of nonlinear wave equation so this is a since equals that as a solution the nonlinear way request so this is a sound similar solution to the nonlinear wave equation with compacts but in the era of Maryland is so this profiles couldn't have been president the continent and because look at the other case too analysts say that this the 1 we're looking at is the first one now the warden is identical 0 11 1 and is much much smaller than the and 1 is Ltd now because of the fact that the land the 1 and is so much smaller than in this term does not control when you play then that will go to by the scaling and then he improved that X over TV replaces seaward L 1 goal was to see water at the end the profile is called is concentrating around seawater tea and so X over the becomes L 1 so in the limit In this nonlinear a property position you get that this guy very misses you so that means that now the 1st or the equation is this 1st on the inquiry let me a traveling wave and is a traveling waves literally have proved that the kind Emeril the traveling wave solutions and exactly solutions of the elliptic equations Lawrence transformed according to this direction so that already it you have to use that there to show that no 1 has to be less than 1 but what we showed is that if you have a traveling wave the speed necessary consists of 1 of engineers solution lipstick equation Lawrence transfer so that's how
1:34:06
we get but this guy is traveling as of this out and gives them legal position in the preliminary but the gives it for the 2 sequences of times that we constructed because a given for any sequence of plants for which this average coasters this Morrow it's tight average and the reason we want this averages instead of for each team it is that in this computation of these limits you want to take week limits and so you need to use compactness and devoted to variables to little his compact if you are contained in each misplaced time and that's the real reason why you need to pass to the averages please To be able to use that locals and the Koreans gets Strong covariance local but look at that works In fact this is something that plan we have used to indicate their enrolling our 1st in the radio came press into 2 heroes instead of 1 of now however improved abounds on the only errors using various it's OK but will have to switch the sequence of 2 so how they do this so the claim is now there is another sequence of set for this sequence of times we have composition and the error in addition to having jails 6 we are going to 0 Hassan Simon up further good progress the tangential part of the gradient who's 0 remember that In immediate outside the salon Everything goes to 0 the gradient outside the and then goes to 0 that's no news that for any slightly smaller ball it also goes the energy is concentrating near the boundary of the ball the same is true for the tea derivative and then there is the 3rd the 4th property which is fundamental which is your solution is becoming out appalling but that there is a relationship between the spatial derivatives in the find and this is precisely the expression in the and that is going to 0 she the 1st thing the chief so I will I will show you know how to do this but I to address it you question about the circuits the strictest norms doesn't come at this time what happens is that once you have this problem then you can prove that the strictest groups but not before you have this problem this is a things said because is more us but there's there's the breakthrough provides that with him for me and I remember that we threw in the profiles that were scattering profile that's the point is we have to kill them and we kill those by by using the OK the proof of that the Mohammed to that I would describe is In the same spirit as the extraction of the scattering report OK In what allows you to make that succeeded is defected to him this Prop so you go back to that proven you can extract what you're using and is percentage had few constructed OK so analysts say so that having killed over provides all right but let's not skip steps let's listen OK so that's what we're doing today that that's the last thing we want to do Is so to do this is where we need the twotime so we will what we will do it's through a further estimates at this 2 times for which we have the compressed and the only way we can process by using that you already have the the composition but fortunately it's enough to only have 6 1 1 2 so we recall that we have this that's that the kind of what we get from the Moro it's estimate using the maximal function and then we have decided to times for which we we can put the maximal function OK this is what we have and that each 1 of these 2 times therefore by the previous there we can do it a soliton decomposition where the error elbows to the focus so we have all of this all there is another the conditions and this goes a 0 in 6 OK now I will use it improved belt to estimates at those 2 2 times in which I have this decompose OK so I think it's much too long and I'm underestimated the wrong on my solution In the ball or
1:40:59
reduce the high end up only to have decomposed so in the decomposition have I have the Epsilon and I have the ability the solid and the solid those eyes that they'll to norm and took part inside this union In the quiet outside this we'll print and those ball centered at the center often Fullerton with their radios epsilon decide the supplier where the actually is number that's given now I remember that I have an apt fury bound and have been jailed the Jays I can just sit and there were fixed number that depends only on the pitch L 2 enormous disagreement the look sir this part of the the composition of the triangle in equal no longer Hungary was held the inequality to go from there to norm to this extent so I go the 6 long and into the calculation because its threedimensional themselves 6 the ball volume on the ball is 52 Q you get the wife and she the wife OK that's just the way it has the passage from 2 no here I do have there in each 1 of these ball this the Union use each 1 of the balls and the walls are is absolutely and so I get absolutely remember these guys a complete your soul and so I can treat them as if this some they have 6 normal standards of some of its success OK so this is flying and then the last hour outside I just used to the fact that this is the they get now I recognize what happens to each 1 of these guys this norm because these regular goes to 0 with concentrating this of course this 1 I know the ghost euro with them because that's what I proved them and the composition of the profit the this 1 well this is just that a uniform Const Latino hold many of you have in this was remember we have a good point was found the raise .period response and this 1 because I'm away from the center by an amount when I look at the L 6 long gain power and so I have either little of the for this the investor or excellent PIN for this term of them so the conclusion since this is for each epsilon is that this ratio losses because I had Little told the I M and actually the and but epsilon is arbitrary OK so what they now show said this helped enormously 1 2 0 with a fast here and I have in this is something you want you cannot prove it seek after the composition of but fortunately only Indians 6 India "quotation mark look no other improved now we're gonna use those have 2 times and I'm going to do with nearrealtime pipeline so too do the argument but I ideally they multiply my equation by you an integrated or excellence and the and the innocent How so because I multiply the equation by something again 0 so I got this 3 things so I get all of these things and now I'm going to integrate by bars in the and sex when integrated by parts in the bags were in here is that it is not as a you no were gap is the flux find you that's from the origins in ink and then from the key terms have given you BTU you can you and then I get what I don't do it with a different in the grid but parts which is the solid OK chernomyrdin show that this 3 perishable 2 0 faster than T 2 over T1 this is where we will undertake to at minus the 1 is of the size of each 1 right by then this is the almost from the start the not because this this is an identity when we take a secondary involved here falls here and then disappear and here if the plus involves here more than I have the great square and and I 4 and then lying at the you to 6 because Newton 520 who could fly I I really let their images histories here and then integrate and now in the OK where so so the object of this is to try to show that this in averages I Tokyo and so I do that I
1:47:55
look 1st at this but the terrorist under the surface in the room which is this guy I use caution and then the flux To power would have the EU squared the ball and have that she could abuse the hardly term but some I use the sixterm by the using of him I get the length and a little water so I get a little heed to the 2 women officials said because of the control influx gives me an undisclosed the fastest the
1:48:41
average cost for this term where do I get the gaming at the gain from the new square but we haven't had this there appears only at the In the end point of the time and and it's at the end point of the in the role where I already know the composition and so I can prove tourist I guess so the conclusion is that the average of the squandered against and that 190 also had and after passing to subsequence and remembering something that goes to 0 come before to the minors and another thing that goes to the European V8 an latest twice for "quotation mark all right so what do we do next this 1 is area can do things initially think because it's chorus upon the bigger than recalled this guy is not a cost so now we have to use another here and the argument that we used to handle this no cost point that this is an argument that Howard G. I and I have been a previous paper on wave maps In the query case because of pulled back another argument here look so I you OK to go for 5 more minutes yet were the school for 5 more minutes he would have if it polar worried to repeat next Simon and repeat the effects of the wants so I look at this guy and again I use the maximal function and I get the maximal functional words begin to the millions and is small informants and the thing the same argument as before trauma and this OK so now I've ever seen what to do with this With the added this kind of a little bit the bad stuff it we know that this thing is bound just as at least we know that I know what I'm saying is that this bound together with his average bound implies but the thing hostility this that the set of points where the object is small this substantial OK and this is a real viable the standard kind probabilistic argument but the thing that you have to do which is a little bit different than usually is that you don't know that this thing is has assigned but you replace that by the fact that the universe by OK so you have to speak the thing into a possible landing a part and how you live I I gave the argument here but let let's assume this but we believe this is a set of Indian probabilistic reliable "quotation mark the answer is this the major were big a small and this at the point where this is small is large we can combine have find as a sequence of points for where both things happened at the same time where both in favorable events happened the same but the maximal function goes to 0 and and this thing
1:53:24
here many will not go to the weather in the limit will be less than or equal because I have a signing OK so from this do things OK
1:53:39
so here I am telling you the show that the maximal functions you find sequence and which the maximal functions most 0 and the what this thing is going on because it was made smaller than 2 to the minors and for every answer is the best entry quotas name Super no so now what happens now what happens is that this sequence the end because I know there's I can do that because but just success 10 now I will the factor for a solitary wave this thing is essentially so I kill all the solitary waves he tells me that the error has to have this problem I remember the error Rosalee 106 so that they negative that of 6 sixparty this around so that he the
1:55:02
fact for any elliptic so and any Lawrence parameters this quantities for the a crucial identity that needs to be and how prove that when the country we know but this is era from ballistic equation and then you see how the Lawrence transformation it and if you patient and into the calculation it that this is all and using the
1:55:47
salinity the parameters and the fact that the and 6 Norm goes to 0 and look for the regular part there's no contribution because its regular and this it is a smaller and smaller sets of this schools then we get this this the next thing is of course we use them the more I would think but that it has already this goes because we know that this amount to boast of 0 also the volatility and whenever you know they'll 6 lost 2 0 in the you this coastal 0 I get rid of this thing then you know that this thing is going to be and now we see what the effect that has on the solid you know that for the solid demand because the traveling waves as as you you know that there's a connection between the L and this season the therefore an area of this thing is concentrated so acts over at the end is in the elderly plus a small error In the place where the Fullerton is concentrate so that tells me that when I Phelps said right when
1:57:28
I do more miscalculation on the solitaire I essentially get bit "quotation mark annually but this is here because of all to here were using where in Lawrence parameter in the translation parameters again so I think we have the other and gold 1 more the start and were so we conclude that the at at the end of this is that for this error we have this 2 facts because of the solid just you gets here and all them regular advised against and he has only you so I I'll show you next time How from this too In fact we can get all the conditions that the stated at the beginning yeah so will start with the next time and then we will see that from all of these conditions you can eliminate the profiles that To infinity in the air and so then you get in there also has a streak of small moments as Iraq and that you have to come up with an argument to show that the energy from was to assess what we're looking for and here comes a new channel in India remember that when we discuss the radial case I said said that the general manager is not true in the imminently In the delivered for the linear equation what happens is that for the linear equations once you have the additional properties in deserves her you can prove a channel of energy and I will show you how to do that in the next that's how you then show that the thing goes Caesarean but you need all these preparations to be able to get to the top the Randall matches and instruments it was meant to be helpful OK so we was stopped for thank you for your patience me the season and this way it works the same way for 3 4 5 and 6 for higher than 6 there is a problem of having to do with how much movement known immunity and then it should still be true whether the more technical and I don't think we want to be more technical and we are already at OK but it's only that they so you know commodities that you don't really need them go on to the procedures to be used to do there isn't that if there's quite a bit of but that a I should say that the better estimate that you get there the more control you will have on the sequence of times that you can pick the more control have consistency pick the more chance you have of passing to a general so that that is the real reason why we're it is desperately trying to get rid of that long it you we would like to show eventually and this infinite and this country and that may not exactly be true but that this summer a version of it will and then if that is true then you can choose a lacking unity secret and can choose a lack a sequence of times and you have much more of a chance to pass into general thing but the you know many years may pass in the field of natural and but all of this 1 where but it's been then years what questions and in a few and
00:00
Folge <Mathematik>
Punkt
Sterbeziffer
HausdorffDimension
Zahlenbereich
Kombinatorische Gruppentheorie
Stichprobenfehler
Arithmetischer Ausdruck
Logarithmus
Arithmetische Folge
Nichtunterscheidbarkeit
Wellengleichung
Vorlesung/Konferenz
Abstand
Leistung <Physik>
Analysis
Schätzwert
Zentrische Streckung
Kategorie <Mathematik>
Frequenz
HelmholtzZerlegung
Energiedichte
Singularität <Mathematik>
Konzentrizität
Beweistheorie
Mereologie
Faktor <Algebra>
Ordnung <Mathematik>
09:06
Ebene
Abstimmung <Frequenz>
Punkt
Momentenproblem
Gruppenkeim
Gleichungssystem
Fluss <Mathematik>
Bilinearform
Term
Gesetz <Physik>
Physikalische Theorie
Gradient
Topologie
Zahlensystem
Negative Zahl
Regulärer Graph
Gruppe <Mathematik>
Endogene Variable
Nichtunterscheidbarkeit
Wellengleichung
Zeitrichtung
Vorlesung/Konferenz
Umkehrung <Mathematik>
Drucksondierung
Schätzwert
Lineares Funktional
Addition
Physikalischer Effekt
Kategorie <Mathematik>
Ähnlichkeitsgeometrie
Physikalisches System
Dichte <Physik>
Summengleichung
Objekt <Kategorie>
Energiedichte
Randwert
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Differenzkern
Verbandstheorie
Rechter Winkel
Partielle Integration
Beweistheorie
Mereologie
Aggregatzustand
Standardabweichung
27:17
Punkt
Wasserdampftafel
Fortsetzung <Mathematik>
Fluss <Mathematik>
Gleichungssystem
Gesetz <Physik>
Term
Inzidenzalgebra
Ausdruck <Logik>
Differential
Variable
Arithmetischer Ausdruck
Regulärer Graph
Endogene Variable
Wellengleichung
Vorlesung/Konferenz
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Leistung <Physik>
Schätzwert
Nichtlinearer Operator
Multifunktion
Siedepunkt
GrothendieckTopologie
Matching <Graphentheorie>
Mathematik
Güte der Anpassung
Schlussregel
Ähnlichkeitsgeometrie
Teilbarkeit
Objekt <Kategorie>
Energiedichte
Randwert
Quadratzahl
Menge
Rechter Winkel
Koeffizient
Mereologie
Parabel <Mathematik>
Grenzwertberechnung
45:28
Punkt
Gruppenoperation
Zahlenbereich
Fortsetzung <Mathematik>
Fluss <Mathematik>
Gleichungssystem
Term
Gradient
Ausdruck <Logik>
Richtung
Evolutionsstrategie
Arithmetischer Ausdruck
Weg <Topologie>
Einheit <Mathematik>
Vorzeichen <Mathematik>
Trennschärfe <Statistik>
Nichtunterscheidbarkeit
Wellengleichung
Vorlesung/Konferenz
Delisches Problem
Grundraum
Leistung <Physik>
Bruchrechnung
Parametersystem
Fundamentalsatz der Algebra
Dicke
Mathematik
Amenable Gruppe
Singularität <Mathematik>
Rechnen
Frequenz
Fokalpunkt
Inverser Limes
Unendlichkeit
Integral
Energiedichte
Randwert
Existenzsatz
Differenzkern
Betrag <Mathematik>
Rechter Winkel
Mereologie
Ablöseblase
HelmholtzZerlegung
Beweistheorie
Kreiszylinder
Grenzwertberechnung
1:03:39
Mittelwert
Folge <Mathematik>
Subtraktion
Mereologie
Prozess <Physik>
Punkt
Sterbeziffer
Randwert
Extrempunkt
Desintegration <Mathematik>
Kraft
Obere Schranke
Computeranimation
Übergang
Fluss <Mathematik>
Vorzeichen <Mathematik>
Mittelwert
Schätzung
Zehn
Stochastische Abhängigkeit
Gerade
Leistung <Physik>
Lineares Funktional
Parametersystem
Bruchrechnung
Dicke
Physikalischer Effekt
Logarithmus
Streuung
Klassische Physik
Reihe
Rechnen
Fokalpunkt
Integral
Objekt <Kategorie>
Differenzkern
Beweistheorie
Mereologie
Energiedichte
Term
Innerer Punkt
Beweistheorie
1:10:38
Folge <Mathematik>
Extrempunkt
Regulärer Graph
Computeranimation
Maßstab
Regulärer Graph
Mittelwert
Schätzung
Wellengleichung
Indexberechnung
Einflussgröße
Folge <Mathematik>
Schätzwert
Wellengleichung
Bruchrechnung
Zentrische Streckung
Parametersystem
Lineares Funktional
Konvexe Hülle
Indexberechnung
Fokalpunkt
HelmholtzZerlegung
Menge
Verbandstheorie
Funktion <Mathematik>
Rechter Winkel
Konditionszahl
Mereologie
HelmholtzZerlegung
1:16:17
Theorem
Folge <Mathematik>
Subtraktion
Punkt
Gewichtete Summe
Kondition <Mathematik>
Regulärer Graph
Solitärspiel
Term
Stichprobenfehler
Physikalische Theorie
Computeranimation
Algebraische Struktur
Ungleichung
Spieltheorie
Gruppe <Mathematik>
Nichtunterscheidbarkeit
Translation <Mathematik>
Gerade
Ortsoperator
Lineares Funktional
Parametersystem
Zentrische Streckung
Dispersion <Welle>
Kategorie <Mathematik>
Stellenring
Reihe
Profil <Aerodynamik>
pBlock
Frequenz
HelmholtzZerlegung
Restglied
Flächeninhalt
Differenzkern
Parametersystem
Mereologie
Term
Beweistheorie
1:27:08
Resultante
Lineare Abbildung
Theorem
Punkt
Ortsoperator
Dichte <Physik>
Wasserdampftafel
Ausbreitungsfunktion
Zahlenbereich
Gleichungssystem
Wärmeübergang
Term
Analysis
Computeranimation
Gradient
Ausdruck <Logik>
Richtung
Unendlichkeit
Wärmeausdehnung
Theorem
Wellengleichung
Inverser Limes
Pythagoreisches Zahlentripel
Elliptische Kurve
Normalvektor
Zentrische Streckung
Approximation
Kategorie <Mathematik>
Profil <Aerodynamik>
Menge
Approximation
Summengleichung
Restglied
Kompakter Raum
Beweistheorie
Energiedichte
HelmholtzZerlegung
Wärmeausdehnung
Term
Beweistheorie
Standardabweichung
1:34:03
Stereometrie
Kovarianzfunktion
Einfügungsdämpfung
Punkt
Auflösung <Mathematik>
Hochdruck
Gruppenkeim
Gleichungssystem
Soliton
Analysis
Computeranimation
Gradient
Arithmetischer Ausdruck
Nichtunterscheidbarkeit
Uniforme Struktur
Konditionszahl
Addition
Gebundener Zustand
Folge <Mathematik>
Addition
Lineares Funktional
Parametersystem
Soliton
Kategorie <Mathematik>
Stichprobenfehler
Rotationsfläche
Stellenring
Profil <Aerodynamik>
Rechnen
HelmholtzZerlegung
Randwert
Menge
Kompakter Raum
Rechter Winkel
Konditionszahl
Beweistheorie
Standardabweichung
Folge <Mathematik>
Ortsoperator
Orthogonale Funktionen
Zahlenbereich
Derivation <Algebra>
Fluss <Mathematik>
Term
Obere Schranke
Stichprobenfehler
Variable
Ungleichung
Arithmetische Folge
Mittelwert
Endogene Variable
Inverser Limes
Spezifisches Volumen
Maßerweiterung
Leistung <Physik>
Schätzwert
Division
Fokalpunkt
Menge
Objekt <Kategorie>
Energiedichte
Mereologie
HelmholtzZerlegung
Normalvektor
Residuenkalkül
Term
Grenzwertberechnung
1:47:54
Folge <Mathematik>
Subtraktion
Punkt
Wasserdampftafel
FisherInformation
Gruppenoperation
Fluss <Mathematik>
Term
Computeranimation
Fluss <Mathematik>
Flächentheorie
Spieltheorie
Mittelwert
Schätzung
Wellengleichung
Zehn
Leistung <Physik>
Parametersystem
Lineares Funktional
Dicke
Indexberechnung
Ereignishorizont
Objekt <Kategorie>
Quadratzahl
Flächeninhalt
Menge
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Titel  6/7 The energy critical wave equation 
Serientitel  Leçons Hadamard 2016  The energy critical wave equation 
Teil  06 
Anzahl der Teile  07 
Autor 
Kenig, Carlos

Lizenz 
CCNamensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/20479 
Herausgeber  Institut des Hautes Études Scientifiques (IHÉS) 
Erscheinungsjahr  2016 
Sprache  Englisch 
Inhaltliche Metadaten
Fachgebiet  Mathematik 
Abstract  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 wellposedness theory (in the sense of Hadamard) using extensively tools from harmonic analysis. This yielded many optimal results on the shorttime wellposedness and small data global wellposedness of many classical problems. The last 25 years have seen a lot of interest in the study, for nonlinear dispersive equations, of the longtime behavior of solutions, for large data. Issues like blowup, global existence, scattering and longtime 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 19902000, 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 blowup, 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 nonradial case, along a wellchosen 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. 