Merken

# 4/7 The energy critical wave equation

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Erkannte Entitäten

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00:04

did you and so many I knew it and little bit of the summary of what we've done so far and by the way after today or at least soon after the notes will be posted on the I made his website will be the notes up to today the other ones are not ready OK so new it done so far is that we introduced the energy critical wave equation in the focus in case of a the essay good models in which to try to study the long-term behavioral solutions for lots of and the so the 1st thing we did this week we dealt with the so-called the ground state conjecture so the ground state this in an elliptic solutions which has minimal energy among ended among and the solutions and so it's a static solutions both the nonlinear way and so we proved to the result that that if the energies small smaller then the energy of the the ground states now 2 things can happen in the gradient is bigger than the gradient of the grounds that then you brought people than directions the great and the smaller than you exist globally as scattering both times and this was down through the same concentration compactness rigidity theorem the purpose of that was what we with they then than now we're going to gold trying to understand how the dynamics behaves as if to go above the ground look at state level energy and so for that was started in the range of cases and this is what we're going to today and will has shown that there is a completely soliton resolution in that sector for solutions that remained on the on the last time I introduced this season I told the the channel energy minister which is and is the way to measure person then there is the key to winning this tell me be a recall the bastard so 1st I start recalling and it'll calculation this is just as nothing more than an integration it says that the interval from aid to infinity now that the artist inside the media are always remember we're doing all this in 3 D OK and I made some comments as to what happens in the so there is a simple calculation that the DDR what expressions where the artists inside equals exactly DVR are worthy artists outside minds this constant term page squares reflecting the end point and it's important that it's a miners here so this is all the smaller and as a consequence this thing channel managing quality tells us that for salute rail solution of the linear wave equation in 3 for the possibility negative when you look at their energy outside 4 are bigger than absolute value people they you get that that's controlled from below by this expression which corresponds to the time equals 0 can and this is true for all T for all time you remain having this block of energy outside the obvious people you possibly on the knee and the proof of this is a straightforward and integration argument minutes completely mentoring so

05:09

now we're going to see the key prayer .period industries will solitaire resolution is that this kind of channel energy exists also and then only accepting provided you're not the 1 sold which is docked In the desolate characterization of OK and using this you can do In the job of proving the soliton Russell so I introduced this notation that's convenient for us where we truncate solution outside so all of these arguments are doing are done by going outside because the inside is unknowable to us but we can't tinker with the outside Union that's very well adapted to find it's been property so the Dilbert if you 0 4 hour because and capital and then we fill it in as a constant in the middle anyone until there were just brutal they can 0 in and the point is that the age 1 grows old to or of new functions is exactly the outside part of the age 1 critic In the 1st proposition is used global and dying solution of the nonlinear way the correct equation in 3 OK and suppose that both of these limits for both for negative time and for positive Timor 0 for some then the consequence of this is that you have 2 options for the data it's a the combatant supported all 4 are equals W except for comebacks show for our the data has to be either 0 that's the only way you can't have both of these limits to be easier for some OK and of course if you 0 you want is compactly supported in that saying the ball of reviews are this is true because as a 0 thereby fine it's been an airfare it equals W. outside the Wolverines are this is also true because we're W this so this is the way in which we can capture the only OK so the 1st part is to prove this so the proof of this it's sort of an elliptic proof in in so you are as if you were to pretend that you're standing in a thick equally and in this is in some sense but some philosophical similarity with moving claimed method of newness Indiana pocket but it's of course Wentling hyperbolic statements that OK so there too the 1st llama since the fall suppose that you it's 0 I wanted to make .period

08:44

little comment here it's a trivial common but they will help us understand this of course is this is indeed if it's through foreign are 0 it's true for any are bigger than ours because of the small OK to just keep that in mind it's none of them went to the same are I can only make larger docket so now

09:14

let me be it let me think you the proposition and Let me call the art and you are you may recall that I mentioned earlier in the proof of this out energy lower bounds therefore solution offered in linear wave equation in 3 D which Israel if multiplied by ah I get the solution of the 1 week have in this is something that we think is in the back of pop so the statement so I call that the the 0 hour but this should be the ones the corresponding Koshy date In a my assumption is in the outer part the H 1 critical to from some Member is small enough Duncan is the not dealt anomaly is sufficiently small and of course if I have any the guy can always make this true Bible far out no matter what the economic advantages goes 0 by finding "quotation mark then I can bounce this out H 1 grizzled 2 were In terms of the not to attend the United are not to fire this corresponds to the fact that the nonlinear it's easy to the fish OK and the 2nd part is that the differences in views here can be controlled by this and this in turn can be controlled by by "quotation mark so let me 1st point out that this is bound to to follow from this 1 "quotation mark but the fact that this follows from that is nothing more than the fundamental theorem and push each OK so this part follows from this by fundamentalism accountants and coach "quotation mark now how do you pass from this to that remember the easy the 0 0 Over the 0 over our square in the summer you on right the 0 override 5th the squared as the 0 times and the 0 2 the 4th over hours Way "quotation mark the appeal of 1 of the easier it is 0 2 the 4th over our square it is the 0 squared the where are squarely in the 0 squared over our is hard times 0 are and we know there are times you 0 R is controlled by this kind of expression and so therefore it will be dealt the not and so that's how you get the company's square will cancel this is exactly the calculation I started out which showing Out of Joint again just to see

12:40

it again in it's this calculation I'm particularly this thing is not like that 2 although the preliminaries but I think there's still something to show right now and if things haven't gone away and this is this is quite an important so the idea is that long information from the solution of the nonlinear wave equation of infinity on both time directions gives you information .period OK this is this is what's happening so far how help prevent also this is for the passage from there topping called 2nd 1 so we to prove the 1st inequality way into the corresponding thing with this we did this with the nonlinear solutions will also do it With the lean years for the linear solution I have this out the energy in equal that's precisely now choral area show that the either for the possible or negative I get these lower about you can and now I truncated are 0 and let the till L with a correspondingly news OK not here I use

14:24

that in the local theory of the Koshy problem I know that if this Delta 0 is small enough that linear solution is close to the nonlinear and that's the good thing about going on the exterior which because of the exterior can make things small and therefore use the of local theory coaching problem and then passed to the stage where I haven't got a perturbation by fine it's been appropriate I think so this difference is bounded by this to the fears that comes from the local theater the Koshy problem and this to the by definition it that's my definition of unit to kill you want it to end now you know what that Renault are connected to it the notice are times you know and by the calculation I started out with this equals so is the only quality look now I know my OK so at this to the difference is controlled by this so this warning to secular is bounded by twice this plus look at this triangle inequality so this difference is small and there's so each 1 of them as this kind about where this is the bound for the difference with but I'm just using it the the money using a times B less than twice a squared plus B square but but by buying finding

16:38

speed of propagation having truncated doesn't affect either the nonlinear or the linear solutions in this region N now I use my lower back on this linear thing given by the Alberta Energy call which is 240 deposit their tyranny the combined this and I had a 1 happens so before I have to know how "quotation mark if I go to the initial and no I have this inequality for years at the possible for the negative but both of those limits on this thing as 0 so I choose the boss at the award to negative according to which 1 is used in this inequality had for either 1 the limit is 0 so I passes limit therefore what I have is that this is bounded by OK because this 2 guys don't have peace of mind past team and 2 pieces of paper no I stood this into this term and if you think about it this is the term I really want to get that was the stated term in the inequality was expressed in terms of the but when you translate saving the intangible you get back and now this is the 5th power and this is the 1 power but this thing is small because is bounded by the governor so the 5th power is bounded by a small concentrations 1 power and I can't so this is the end of the perks of spy print so that it will back to the statement that you see that

18:38

this is what we have proof we know that this is small and then we think that this is OK so you must and will have to use smaller not no I avoid that by only working with the outside where I can make things more yeah I'm now working with large they don't care about the size so you have to deal with this and you will be in a dozen effectiveness looking the 2nd statement isn't this this and that and the 0 is like before and I can show that has a limit L but I can tell you at what speed the limitations is attained so approved both in and there's a few steps involved just the patient the 1st thing is I've eventually if I'm going to have a limit is on the rebound ready forget limit will be found but I can't do that at 1 stage fly 1st cigar approved it weak growth condition at thing and growth that I prove is by the power went Of course by taking them not small can and improve the 110 storing up so that 1 but 1 television Kilkenny so why is this too well this is just a triangular cooperation some kind of iterative you it in this style the the elliptic theory works and the 3 because most of the hotel under Georgia so at that time we go to 2 influence 1 we want the compared with the previous stage refused to to the end we triangle inequality but now remember we have a good Bild for the difference my good bound for the differences this means the wine comes from from this term and that the difference the firm has is seen on that was the last found in my let no 1 but not all I have to do it's just don't most small enough so I said this is bounded by 2 to 1 110 this is a number that is bigger than 1 this is another 1 that is being I just used this small enough so that this is true OK and then I did it 3 at integrating the this that the 0 2 2 and is bound by 2 to the N and R 0 abound went to today and I was eager to the ones so I get the desired growth In this dynamic sequence but I have my different system that allows you to estimated things at the close to each other so from the dyadic estimate that passed in general it has July now they still doesn't give me the existence of the but let me know improved to get the existence of you can so Soeharto approves well we know this difference but we know also that this is smaller than the power 1 1 day after to the 5th is 1 right it it's said it one-half and so so you saying that we had to minus-5 stands which is due to the 3 and In that's a conversion series in and so I can just telescope thing this summer's finance limiting so I guess I got my limit existence along the dynamic sequence but I know that the oscillation ionic intervals hasn't changed too much so I get that moment exists "quotation mark but now that the limit exists I improved my 1st now it's bound because the limit exists so In my bounds did the 0 2 5 here gets replaced by sea and this a pair now I let it ii detail some and you see that the 0 AA-minus L is bounded by 1 of square despite Guns Cocking said so let's go back to go on

24:41

good to so it's proven

24:47

conclusion you can see where he is a very simple considerations there's little time here look so I was true

25:03

all winter now I will approve

25:09

Proposition 1 Proposition 1 says that if those 2 limits and the going to class and minus infinity 0 0 your only choice is that your date compact supported or that it is W outside OK that's what we have to prove will use both so here we distinguish the limit is Al what happens depends on the on who if Ali then we have to have combined and if l is different from 0 will get a plus a minus doubled correct so listless prove that so the 1st case it fairly close in Rome "quotation mark so we 1st thought out With this a difference bound that we had before which use dealt not so small that this is less than the 4 "quotation mark now we come we use it triangle inequality and that we see that the 0 . 2 1 plus 1 is bigger than three-fourths at 2 could just to try the call and now I reiterate that I put the 4th of here and it arranged so the 0 hour is less than 4 Thursday the ending 0 2 to the N R now I use the analogy 0 since Ali 0 1 I think the difference with Allied longer and indifference and as get this pick a squad arrived To do this 2 an hour square which is for an hour square and now I've put debt put in here and get the 0 artistry to the minds and our squared for all and 4 are larger than ours here and then goes to infinity go to infinity and therefore we 0 is school but to support to support the I still have to catch me 1 but for everyone I have this my 1st bounded the whole thing In terms of the 0 so if the 0 0 the integral the 1 has squared that has to be 0 so everyone also has to be OK that's the case eloquence the day's L is not 0 it said basically the same so we we just have to show that there is a lot science such as this thing has components support With the that's what this means so now I I recall only formula for them and of course we know how or when this politics of this public because it's completely explicitly so for large are if I skated but will by land my Lambert I get that this difference is bounded by CEO who are over our for our this is from the phone book From this system or whatever land is chosen so now how to use it the Lambert too much air you can just look at and how I choose them so the land that is elsewhere there were 3 and how do I choose plus or minus W according to whether L is positive or negative and all my choices are made so what so one-sided children landed on the sign I get this inequality 4 are large where the relationship between land and Allen is this 1 about the way things are easier scale and maybe anyplace you by miners you so that I can assume that this inequality holds and now I have to show that 4 are large use them that is the difference the case too fast and they have no choice but to be actually so how do we do that OK I let middle-aged BU minus W and capitalized to the archives age remember I always multiplied by R and passed that equally I will show that the

30:33

fire not is larger 10 hours use larger than this capital are wrong this out the interval for that age is bounded by a small ponds the 0 squared the Arazi and this constant this chosen to be small enough that if this inequalities through if we do that previous arguments that we used in level 1 this guy will have to be 0 large so the whole thing is not to prove this thing and the idea is that a little age satisfies not the nonlinear wave equation because it's a difference with W but the equation that the difference in the size of this like a nonlinear wave equation corresponding to the and linear equation in your eyes the roundup the so that's what we do so verifies this equation which is a little linear equation around now the next idea is that in this linear equation you want to show the think over this still is a perturbation of the linear but you can't quite do that unless the potentially small and but Of course we have space-time here so it is the way to make it small amount is again to chop it off on a cold In a call on this but of course is that lose time independence they have to be every more watchful but it would order call and the coldness truncated large enough that is placed large enough to the side then the space-time norms of Dublin restricted to this call will be small enough that in fact this is a part of the region of the leniently and finally being still applies we're confronted the protected both the potential and the data and use financed it's been the propagation to see that it's the same as we have as if we haven't truncated the potential and positions all completed local provided we are outside of it look and so was so this is how we truncate the that the the potential and we obtain a local theory the coaching problem we show that again the nonlinear solution is a small perturbation of the linear solutions will use the out energy lower bounds for linear equations and then we get to good and so that's the end of this part of probe this is the money Of course solely for a Southern all is a nonlinear equations with small data on Friday because of its what is the nonlinear equations with with the potential terror and smoothly but both the potential and the data small this is independently of whether we started from something larger and this is the kind of beautiful 4 OK so the 1st proposition tells us that you cannot have it both look limits going to 0 and the series of combatants supported where compactly supported perturbation of I'm not going to study each 1 of those case the case is left out by the 1st proposed so I assume and that it is not a compact support perturbations Doc so what then I claim that I have this general of energy the temple this dispersant abounded for people said there were 14 OK and how it warned if those 2 limits in the 1st Proposition 5 0 then I'm compact support Compaq and supported the W this is ruled out by assumptions so I have to be compatible otherwise 1 of those 2 elements isn't it before those to limit is non-zero an easy argument as we get OK so all I need to do now is dealing with a case when using everyone is compact

36:03

so that is really 1 of the of combat support but of course it cannot be easy it's 0 I can't do it and that there is no law about possible forms OK so I think it would ruin the picture took to show you the idea this role is the diameter of the support bodies 1 OK that's the definition definition is a little bit a strange because sets of measures 0 cannot count that's all rents we're dealing a new 1 is just an 2 functions a set of measures here doesn't claim but that's about it really this is just the diameter of the support 20 now since functionaries of compact support but it is not 0 this role some positive news which is fine OK get that's all views so how am I going to proceed and willing to proceed by looking at the very edge support so I'm going to look in this life 10 now in this body no matter how small I take going in here there's always some energy in here because this role is the true diameter I can't make it any small so I will always pick out some some energy going into the inside on the other hand by having gone a little bit inside the energy is certainly no 0 but it's also small and that's how I proceed now I am in a small data problems and I have my linear solutions posted only and then I use the channel energy for you convinced some sort let's go to the proof so that's the idea without them do they write details look what services you have now he is A 1 important point you remember that that time 0 have had these miners a times 8 squared terms that comes because of the fact that they did not have the H & H 1 grizzled to normal puts the there because the solution is 0 here that there can be absorbed by the social compactly supported the that turned that appears is always bounded from below by a constant this is an important factor Florida the finally the 3rd proposition deals with the case where I a bit where I have a pullback support protrusions .period the cases there and so far I haven't dealt with so let me explain this thing so it is hoped a compact support perturbations W but is not identically 0 meaning I'm not exactly I'm only W. fire OK that's that this situation so I'm on the at call the diameter of the support of this the as before so let me 1st made the 2nd group before leaving the 1st the 2nd conclusion tells me that role is large enough so if it is equal to W for a very very long piece then I still have the out the lower bound in the Czech OK so it is for a very long bit and W I have this channel energy but In 1 of the 1st bit on this thing said OK supposedly the cause W almost every area of 1st starting from a very small then if I have all the time either the 40 positive 414 negative the size of the support will grow only knew and so if I waited long enough I'm in a situation where I can apply With less state similarly proving so the 2nd but are it is proven by using this as linear as the questionable W. truncated and that's why you need to make the odds you large To make all the time Norman Small and Think Small so 2nd part will already basically seen to proceed so is the 1st part that has known to it for the 2nd part where we won the support of linearly and we will linear eyes around W but without truncating the at all now we won't trunk Inc How do we get this small but only but by only argued for small talk this will get smaller so W its globally defined therefore for any final time if I'm close enough to the that and then I exist at the time miners Arnold are not for any of where the cost of living perturbations that's provided that the data is close so this is a perturbation equation for a nonlinear equations what I call the longtime the it cited depending on yes yes now I make their pride what was our problem can have a need to go

44:08

back to see what our the prime minister are prime is again something almost the edge all that support look at it I'd taken in our prime all moves on the edge In its 4 are bigger than our prime the Denver have the growth of support for all positive time for a link

44:37

so is our prime is close enough I can still make the norms small and the new as a truncated so now I saw the way equation with this data and I have got this identity In this they find time beating minus Arnold by Finance speed the virus larger than and then there's then the H don't don't live there anymore and the 0 this does for the support it says are everything is doubled so my solution will be W for are bigger than this plus the for all the stake and if you think about what this means there is the diameter of the support is smaller than the diameter of the support plus so there is something new and grow and now I I want to use the channel of energy argument to show but the growth is actually linear is not subject considers that that's another way of understanding this the outer energy equal to the force to To will in the 4 positive time "quotation mark softening and of course that's why they are restricted to a situation where that is Strong holding principle he use was a great will this case 3 times himself I'm excited him so now I have to show that's what I'm trying to show so I only have to show bigger than recalled because the other qualities then and I have new

46:53

good local theory and they have a small for deletion of the usual linear flow and so on so I get the outer Energy inequality for this year because it holds for all the news agency and this is their extra terms was saying that when you are in combat support situations you can observe this because this is a swap you can make is this more than a quarter and so this is bigger than a times that and this is what you get out of compact support look so

47:40

now I know that for all the possible for all the negative this is true now and this is is the 1st made me get foreign gene which is close to GL and then we can replace debates still thereby financed OK so I and I and

48:07

I inserted these are all that's in between to give myself a bit of room and make that small now was

48:18

I I have that I get the abound that I want is that would roll with overall but since this is true for any again for and that's so at least they get it for a very short time and now I have to 3 now what happens that the 1 thing that you have to do careful about is that this law boundaries 40 possible for the negative it could be that you went a little bit than it was true for the negative but then the next time is going to be fruity ports and then you can't continue but that is ruled out by the support prophetess if you chase out what the support properties of this implied this can't happen so that it has to be consistent even for negative for and that's why he's going with it so that requires a little bit of thinking and nephew drawings and announced skipped you can't so the meaning of this proposition if combined positions 2 and 3 is that it as long as we're will were patients were willing to wait a little bit of time we will always have the law bound unless we're plus or minus 2 because there are 3 possibilities you're compact support but no 1 0 then you always have the if you are neither compactly supported nor W. political backers supported I think you have the Louvre and the W blast a compact support object then you however that if a the positive or negative findings as soon as you wait long enough but the the size of the support this for all possibilities count current what want is for 1 I'm not look at this site it's a fixed number on that comes from "quotation mark but so the next thing I am going to do so we're gonna use in this thing together solid told solution in the race so that's the way that's going to go through profiles so I'm going to translate into this results in terms of profile just to make it easy to see how it's used OK there's no room left here this does not OK so I consider a radial non-zero profile U.S. and object of this kind and of course I don't have to go to the 0 Profiles in History will always and finally consider announcing and if you recall 1 of these 3 options are always holding after some reduction it's the time translation using a 0 it TJ an overland Jeanne in absolute value finish involving 2 busses fitted to mine "quotation mark no I'm going to assume we think that the so actually that 1 of the following 4 it is that is that is not a compact support for integration of the law or minus W or if it is a compact support perturbation the diameter of the supporters logic on

52:49

then I have no doubt the energy I have out there and you're bound for all the positive awful this lesser conclusion but it's not that the day ends is era are 0 this is nothing else than what we've already not it in the scattering situation the approved by simpler and so on but I that simple so now we have articles stated in terms of profile remember that Jay is destined in rotation he doesn't believe in to called that they will be used as an index hit the problem before "quotation mark so now let's turn to that there's soliton resolution in the ring Reagan so this is the

53:56

program similar to the extent want to digest so that you be be aerial solution of the nonunion wave equation in 3 and there's no sign just arenas then there's 3 options the 1st option is that they have financed and blow up and it's Type 1 that means that the North has a limit and the limit is injured In that case I have nothing to say OK but some different theories stuff OK parade so that's not very exciting I think there had said nothing for the 2nd stages again the blast final but I have tried to do that is to say I assume than than the norm remains In bounded as I approached the then I have the soliton resolution may have irradiation Terim and here a number of capital stayed I have science plus or minus I have scaling parameters like the 1 to land the capital J 1 much smaller than the next 1 much smaller than the next 1 and so on and they're all much smaller than the linear rate and my solution rights as In this sum of of modulated solitaire plus the recreation terror but something that goes to steel to the US so I can but it's 1 thing to be observed is that the union of a and B shows you that you can go all have makes the symptom there cannot be a sequence for which you go to class infinity and and another sequence for which you remain bound cancel has already a the and the quality regardless of his accomplices above 40 years and the composition to provide then there is the 3rd case the 3rd case is that your solution his cloak OK In this case we have this radiation terms we have James this capital J now could be 0 here there's always has to be at least 1 solidarity if you have fine law but in this case you may not have financed the not happen anyway because it your solutions captors there won't be a NEW its and then we have signed me of scaling parameters there all much smaller the and my rights as the sum the sold waves plus a small an enemy the coloring is there solutions the exists for all time about note the higher assuming a scenario that I know that no assumptions except that the soup with each will encourage them to move In this case the support each 1 result fine in this case I don't even have to assume that it is a corollary of the case I make no assumptions except that school "quotation mark any other questions "quotation mark you can use really vehicle he was in the back of the head of the the station which from has the dress was of you the translated not no translation because Israeli from the came from so in July will go to the case would have translated and Lawrence boasts of W In all it would take so the rest of way it's all real-time or in already kids on here so that's why General Raines cases and in Dingzhou Australian that's why the grail and numbering case of and on radio cases have to account for a huge number of interests continuum of 1 the solutions of the pro and they're not classified John just and now I'm doing propaganda for the next I have Celtic pickets at the end but then I will provide the notes of the last lectures to but at the end of last so that any other questions there should be more want to see how 1 could prove From what done OK so In the radio carries of concentrating this game and then in the monorail case I will concentrate on the transformed into very thinks so that is the 1st

1:00:37

case then if there is a global in time solution for boss at and the first-phase step is to show that at the at least 1 sequence of times with his pal and this is basically the argument that the show would you if it if we go back To this ground state conjecture that argument in which we said that if the energy what smaller than when the gradient was then it blew up remember there was this differential inequality argument working with the other 2 are if you use that argument carefully here you will see then the man Of the green square is always at most 3 times the energy because otherwise they will blow up in final time and so I won't be the pluses and for the point is that the fact that it exists for all time gives you a constraint on its side In at least for a sequel to his listeners with this so now what we will do is build up on the fact that there is a sequence of times which it is by cost we have much more to do with this was beginning and I was a give this argument because it's basically it almost as a repetition of the 1 I've already the next step is to extract the freeway the radiation term In the infinite time and In order to do that you 1st extracted for a sequence of times on which you know the thing is behind them and then you show that can extracted for a sequence of time you can extract and for all OK so this is the following if you use Iranian solution you have and linear solutions such that provided euro fine distance within the lights cold the new most of their energy on almost so the meaning is that at any distance from the boundary like all any global solution behaves like coming here so all the nonlinear action is strictly contained in now this is a real fear among I'm not approving them because I will prove its version in the radial case in July the London only against thank signed in July I will give the numerator professional the 1 you books 2 minutes from end the of theory the this case is that the because it always gets riskier by this parameter slam them but they're much faster than the time for the tales of all call they have small the move saying that it is outside and just saying that the small In the name of it so you can have provide a London small right so that is an important difference between the final time and of course but that's fine and then blow up and infinite defined time globally we could take away everything outside the light In the infinite time we can take out only most of OK because there's the State but that doesn't bother eventually will deal with that right so the next the next day the next step is to see how our this person estimates implies certain things about the profile and this is the channel energy argument this is an important so I have a globally defined solutions then I explained that there is no sequence of times team with the following property this equals use of land outside of rose and I have a profile the composition that starts where I put all the W's 1st I have the radiation firm then and the other possible profiles and the era analysts assume that were in the nonlinear in the globally behind scattering the but moreover for 1 of

1:06:16

them there wasn't that normally I have for all peoples of the world this channel of energy or if I don't have it in profiles I have it on the remainder but eventually the remaining half a chance and I claim that this isn't possible and we call it the inductive lemon because it's proved but the what let me so we will assume our then that dilemma so there's no decomposition were either from here or from here you have a general and that so you see that this kind of morally forces all of this To all have been done because we know that they will have a channel energy were the only possible and those things and not only that but in this area has some of the city and that's why this dispersal property implies the soliton who could but will see that more formally later but this is what we're the case allegedly in

1:07:41

the proof of so the 1st thing is that you can see that this limit the roles are how far in the EU equals the yuan's compared to ,comma you compared to the time interval the time the energy everything polls this and they have a combined our ability and then the 2nd thing is that the scaling parameters whenever you have a W have to be

1:08:13

smaller than the team and much more so that our preliminary consideration so but the next thing we do is we look at the solution of the nonlinear wave equation which Hassan topical equals the new solutions and we will cancel its the kind of the way it operates then we can assume that this is defined for all costs so of the statement is by if approved by induction on the number of in this profit before for the 1st cases when there is no government In the profile of from the so in this case that you and business solution and so now In the remember we have a profile legal decision in where all of these guys for all and of the solution exists for all time and so far fractionation theory works perfectly here and I can write the nonlinear solution in terms of the other loneliness the nonlinear approach concludes and they it's only in the nonlinear solution corresponding to the state that can be expanded to some often is nonlinear thing corresponding to litigation the blast in only new profiles blast remain there plus the further reading there which is very small so I assume 1st at a a means the total that will hold either for this for this but for positive time and then I will consider the case of "quotation mark if it holds for negative times then this guy's or the have a channel but by author were now the profile these will position this is inherited by now in 1 of these things have channeled this reminded remainder history small so this thing has to have a chance and this is what we conclude but this term outside has fizzled that's just by author reality and assumption that 1 of these guys have a channel or is no uniform and that in disguise uniformly have to "quotation mark so if he would but by Finance speed remember this view and suggest that you truncated outside Milan and so finance speed this is what you going to no I change variables I call this the teeth filled and so exes between it's bigger than the last royal minus the In this is spot this is just from the family members but remember that the and the allied constructed in such a way that this goes to 0 for any fixed the and for any this goes lira and this is a contradiction so I can't have that move now I have to consider the other case the case when the general calls were negative if the child goes for 9 at this time they still use them which

1:12:22

I still use this expansion but now I use it for the equal minus him if I

1:12:32

use it for the team minus the hammer I get this menace that has a lower on the street but remember that by lifeline and speed is equals that outside here so what we get is that but this also gonna hold because P tends to infinity this goal and since the end goes to infinity news in the knowledge outside further and further and as the fixed function so that instead can't be moved I have a little so then conclusion is that I never can have this chance and that's how this so that's how we use the channel Energy Inc In this industry by showing that there can be no profile in the conversation where some of the problems so this was just the kids do when there were no lovely now we have to treat the case which when you have a W inducted and so what you do is take 1 of the most of news and you just truncated for dialog Shh 1 of them and then you could play this very same game again any reduced to the previous because lessened induction process look so now we come

1:14:25

to the main suspect here you are

1:14:28

seeing a lot of the stage for all of you can use it show that is the use of the W cannot give any positive controversial because it's there To the only possible contribution comes from the others the ones of current but if 1 of them has a positive contribution reach a contradiction but analysis quoted Mainz step in the proof but the competition right so how do they do it I started With a sequence of times for which I know the views about and I know that there is 1 and the OK that was the fact that the names in force less than 3 times in and then you have to be focus then where the hell is this litigation turned warrant chosen then after extraction and we have the composition for this sequence so we will leave this sequence of time for this this and the composition for the sequence of times in which we know about then there will be a separate steps that says that this happened in the competition for 1 sequence of time I will have bound in this on all sequences time and so on eventually will will get to that so the logic descended into intricate but it will go on so suppose that this is not true then I have profiled the composition where some of this Harry but some of them are not Red that's what that's what this means that they have or they're all W's and this guidance and moved his union and those other portions so again the limits reminders of past or their 0 and for all and is bad all they're all zeros and ones that NOW but this thing doesn't move this my other there were ruled out by using my inductive my disperses characteristic now this is and then plummeted ,comma condenses the hope so uh we have to get some control on the parameters and then To carry out the proof whispered into their case in each case we use land profile 2 reviews to the situation where X is bigger than Roland with some motorists W's and globally defined profiles that's the important thing we can go outside so that all our provides Croat and globally scattering indicate energy channels in this in in college it was NOW and this is what specific profit the whole give so and this gives the contradiction by this inductively now we can perform this directly and along the given sequence of times at analysts this profile of which is somehow concentrated further from 0 it is a compact support for activation of W with small but I am not so bad case but we cannot say then that we have the time that then we just change the secrets of times by fall by using our proposition that says that we can grow support and then we use the approximation theorem pastoralists sequences time that we started to the new sequence of time and the new sequence of started we get the contradiction so in way we get content the cost of the consists of such an argument .period long but this is the basic a principal approved for so evening less than you have faith is simple the company so this proves it for the 1st sequences times in which it was by so how they will conclude so that these arguments are mounted multimedia but there are more honest so but let let me go through and so the 1st step in the 1st step it that once I had the legal position for 1 sequences of times for which this holds I 1st show 2 facts 1st that the gradient converted to J. cut J. times the gradient of the Dublin 2nd that he did with his gold if I have the I immediately get the boundedness the because if this has limited its foundation in this as a limited spot and that's all I have to say so from this I can already go to other sequences but it also fixes how many W's each style I get to proposition for sequence of Stalin have because Jr is the ratio of this and that and this depends on you and this enough 2 adjacent is fixed so the number of bubbles is 6 and abound in the polls professor there seems to be this as being improved so has had to improve so let's 1st show that this is true OK Soon then this isn't then because

1:21:58

alone Desalegn have the the right to limit because of this the

1:22:03

composition from peace Selanne clearly this equals change time's up because difference Costa 0 and normal this is the sum of the Norris by the loss of another the per OK so I know what I want alone with the sequence the salon for which I have the so I assume that the this doesn't hold for all of then there's the sequence for which the TE and 1 for which this limit is this place for some actual which is non-zero but which I can choose Weiss said this if they intermediate values theory In the continuity of the slot they can go In between and choose any intermediate value that I want between the limit not existing and fixed limit that time this assessed intermediate value theorem continued to fall that allows me to do now let's fix that maps to to the determined in advance this actual would not be going to 0 and with the some fixed things because of this the yield on the 10 year US but by if it is infinity in between being finer than being inferior that they sold about credit can always move current Celeste intimated using the allows for that at so this guy guide is bounded because of this because realities about therefore is bound 106 6 by the not remember that the energy is concerned and the energy is that gradient squared plus the the squared minus used to the 6 squared and that's a constant the gradient square than they used to the 6 5 on the energy is constant so that means that the DDP always calls so this is about so now I have in this sequence I can do that the composition of the Steinbach by right so I do that the composition and therefore there is limited it has to be some integer times ground square that's what the composition gives me an integer and so on and now I choose an absolute so that this is never an integer times grant and that's a contradiction but they have accountable number of possible values and was missing listen so that's why the limit has to exist equal Jason's and proves that this limit is 0 it's the same because the longer sequence the surrender Ltd 0 because I get W ,comma 0 and sold a similar arguments analysts you need I always have this place so now I've proven I know how

1:26:18

many bubbles that have and you know that I have done in this world and have the rest is basically continue the know what hampers the continuity argument at 1st Is that your land the land injuries that you can now change for any sequence but they may not depend very nicely in the sequencing may not depend continues and if they don't depend continues there can you continuity so the 1st thing to do is we replace the land by another 1 that killed let them go into constructive community which is contained but I will tell you how the scaling parameters can be construct so that the next step Hungary the choice of the sky so I I define them this number the subjected I can for individuals in the U.S. the contribution of Jay Maine is 1 bubbles classical tradition of their 1st bubble restricted to the union ball you can't always a number last number and now I defined landed Javed team which will be my scaling parameter the Of all the land was such that this is bigger than the equal to the and if you think about that this defined slammed the jailing of the 2 countries but so I have a continuous functions so what what does it have to do with everything so the next claim is that if think Guyanese is any sequence going into infinity after a extraction I can't have a position but were now my Landers I was given by the previews finish originally I had some land which ended in control no I can't use this land this this was proof of that because this is recalibrating the we know that there is some sequence that Beijing 1 for which this system you know I will show that the land that 1 ends are comparable to land jail and therefore I controls 1 for look if I was in response that the finally the decomposition this equals that this is because I have had the conversation with the 1 now I I look at the KC 5 0 is less than 1 then this by the definition of this system is strictly smaller for larger this were bigger than warned this would have to be bigger for so if you think about that carefully what this means it said this in the limit part of the statement said this is the ability of the Rangers warned against what 1 by other and the price of the error which still boosted and this gives the claimed that now for any sequence can move the land In this moving areas continues in who pays so I'm building up now what's by next spring my next problem is the science writer was the last thing we have workout because like I can't have 1 combination of science for 1 team and another colorful for for 1 thing and then another combination for a different 1 disaster the always the same but this is somehow easier for Eisner is somehow not hard because the state's only 2 values so by continuing the ones I have 1 value they should always be able to keep the same set of values otherwise something drastic will happen and I'll get out of and I'll be the next OK so that's another continuity books so this is the choice of science the end of the "quotation mark so I'm going to consider In end couples up J. couples of signs for 1 2 of the a and for any Delta considered functions which are madam Close to this has doubled rescaling W. unions With this signs given James as I said the Lambert James Have Yourself tonality property mentioned by them OK In this I'll call a supply OK so what is the point the point is is that if there is doubt that is very small and they have 1 of these J. couples and different income and 1 is in 1 of these guys in the other 1 is in the other there OK because what's the point if you look at W plus era scales of plus or minus and W minus every their fear among the people there can be cancelation that's why this things in their closing round In that gets preserves and opportunity and that's the reason for this to be true and once you have it it tells you that is used to chose to stop signs intelligent origin the and you start falling and you're always close to a continuous thing which is your original thing you can't change the sun you're applying it to humans and so the choice of size remains for once you start being close enough and that's what gives it for all In sequences of times this because so and In that concludes that Prop the cultures of the 1st of the year you have to let him tell me anything I know is the 1 on which it is bound other than that there's anything that you accept it that his firm that you get out of here that you won't have the known so many lives back through personality sequence the union there were some subsequent he added that the composition OK then he approved that the norm is bounded using because the gradient minus the radiation the solution minus radiation Square has limits which is James times the gradient of Doc which is something you pray OK and then there the the team of the solution miners most you know you have that now you prove bounded for all time whenever secrets now you take another sequel systems are returned Our 1st there said that whenever you have a sequence of time per cent sequencing at the compass and I knew that they could you get the legal position for some subsequent and then you prove once you introduce scaling factors that the scale links can be changed to this ones that are contained the fastest sequences so as to keep track of the Normandy don't keep the land you have all the land injuries you replace them by ones because there are comparable in size to get position here this year because that that the ratio the ratio tends to 1 you prove that this region tends to 1 and that's when you can and then you have a continuity on the land and then you get to the fact that you can avoid changing the sign when I had the limit is the same for whatever sequence of it's the 40 secret himself sequence for which you have the right answer but is always the same and so the subsequent sentiment here because of the you know that he can get the world strong cauldrons were discovered but it's convergence where that's all that matters there is no other questions for many elements comments I think the thing to remember about comments now With this they recommend this to build up for the northern Arabian case so in the radio keys we have this whole huge collection what possible traveling waves so when really have any solution of the elliptic equation and you do it any Lorentz transformation that you have a possible traveling way so to attempt to get the at this person fiber or a channel type characters station for all those things seems completely impossible so in the normal range in case you have to have a different strategy you cannot use this kind of art that's the 1st thing the other thing is this out the energy called this that with falls and it's not that the with of the prove that just falls so you need to change the way in which he obtained Utah channels and that's who what will happen in July is the explanation of how you overcome still he said some of the proceeds from the sale of the coalition of course you have to find in propagation and feud another that's all thank you anyway so what I'm saying is that this is considerably complicated you go on and on and on you start by something you can never hope to 2 of my lady here and the hope thank you and to have been now

00:00

Auflösung <Mathematik>

Term

Gradient

Eins

Übergang

Richtung

Dynamisches System

Arithmetischer Ausdruck

Spannweite <Stochastik>

Diskreter Bewertungsring

Theorem

Wellengleichung

Vorlesung/Konferenz

Parametersystem

Güte der Anpassung

p-Block

Rechnen

Fokalpunkt

Unendlichkeit

Integral

Linearisierung

Konstante

Energiedichte

Konzentrizität

Quadratzahl

Betrag <Mathematik>

Kompakter Raum

Beweistheorie

Numerisches Modell

Aggregatzustand

05:06

Wellengleichung

Lineares Funktional

Parametersystem

Punkt

Prozess <Physik>

Ortsoperator

Kategorie <Mathematik>

Aussage <Mathematik>

Ähnlichkeitsgeometrie

Gleichungssystem

Auflösung <Mathematik>

Zahlensystem

Frequenz

Soliton

Computeranimation

Energiedichte

Negative Zahl

Zahlensystem

Einheit <Mathematik>

Sortierte Logik

Beweistheorie

Mereologie

Inverser Limes

Aussage <Mathematik>

09:10

Theorem

Subtraktion

Term

Computeranimation

Gebundener Zustand

Eins

Richtung

Arithmetischer Ausdruck

Ungleichung

Theorem

Wellengleichung

Funktion <Mathematik>

Aussage <Mathematik>

Wellengleichung

Fundamentalsatz der Algebra

Multifunktion

Rechnen

Linearisierung

Unendlichkeit

Energiedichte

Quadratzahl

Lemma <Logik>

Differenzkern

Flächeninhalt

Ungleichung

Rechter Winkel

Beweistheorie

Mereologie

Beweistheorie

14:21

Stellenring

Subtraktion

Dreiecksungleichung

Hochdruck

Störungstheorie

Rechnen

Term

Computeranimation

Konzentrizität

Quadratzahl

Einheit <Mathematik>

Ungleichung

Physikalische Theorie

Inverser Limes

Perkolationstheorie

Gammafunktion

Leistung <Physik>

18:36

Folge <Mathematik>

Subtraktion

Momentenproblem

Gruppenoperation

Abgeschlossene Menge

Zahlenbereich

Physikalische Theorie

Computeranimation

Gebundener Zustand

Eins

Ungleichung

Existenzsatz

Schätzung

Kompakter Raum

Inverser Limes

Leistung <Physik>

Aussage <Mathematik>

Schätzwert

Wellengleichung

Reihe

Physikalisches System

Summengleichung

Lemma <Logik>

Rechter Winkel

Beweistheorie

Konditionszahl

Ellipse

Pendelschwingung

Beweistheorie

24:41

Stellenring

Subtraktion

Streuung

Klasse <Mathematik>

Term

Computeranimation

Ausdruck <Logik>

Ungleichung

Vorzeichen <Mathematik>

Schätzung

Kompakter Raum

Inverser Limes

Zusammenhängender Graph

Analogieschluss

Auswahlaxiom

Gleichungssystem

Aussage <Mathematik>

Wellengleichung

Zentrische Streckung

Aussage <Mathematik>

Vorzeichen <Mathematik>

Physikalisches System

Vektorpotenzial

Unendlichkeit

Inverser Limes

Quadratzahl

Lemma <Logik>

Physikalische Theorie

Richtung

Hill-Differentialgleichung

Beweistheorie

30:30

Vektorpotenzial

Stellenring

Punkt

Gruppenkeim

Gleichungssystem

Element <Mathematik>

Zählen

Gesetz <Physik>

Computeranimation

Übergang

Gebundener Zustand

Kompakter Raum

Wellengleichung

Einflussgröße

Parametersystem

Lineares Funktional

Physikalischer Effekt

Reihe

Vorzeichen <Mathematik>

Störungstheorie

Teilbarkeit

Linearisierung

Arithmetisches Mittel

Menge

Kompakter Raum

Sortierte Logik

Beweistheorie

Physikalische Theorie

Beweistheorie

Aggregatzustand

Theorem

Subtraktion

Ortsoperator

Streuung

Bilinearform

Term

Inverser Limes

Störungstheorie

Drei

Gleichungssystem

Minkowski-Metrik

Aussage <Mathematik>

Wellengleichung

Dispersion <Welle>

Durchmesser

Logischer Schluss

Stochastische Abhängigkeit

Aussage <Mathematik>

Lineare Gleichung

Vektorpotenzial

Energiedichte

Flächeninhalt

Mereologie

Parametersystem

Energiedichte

Richtung

Normalvektor

44:08

Parametersystem

Theorem

Verschlingung

Durchmesser

Streuung

Mathematik

Gleichungssystem

Vorzeichen <Mathematik>

Computeranimation

Energiedichte

Forcing

Nichtunterscheidbarkeit

Störungstheorie

Aussage <Mathematik>

46:49

Wellengleichung

Stellenring

Massestrom

Term

Obere Schranke

Computeranimation

Energiedichte

Ungleichung

Iteration

Kompakter Raum

Physikalische Theorie

Beweistheorie

Innerer Punkt

Gleichungssystem

48:16

Resultante

Ortsoperator

Streuung

Auflösung <Mathematik>

Zahlenbereich

Drehung

Auflösung <Mathematik>

Soliton

Gesetz <Physik>

Term

Mathematische Logik

Computeranimation

Iteration

Unterring

Translation <Mathematik>

Vorlesung/Konferenz

Indexberechnung

Folge <Mathematik>

Soliton

Grothendieck-Topologie

Durchmesser

Kategorie <Mathematik>

Indexberechnung

Profil <Aerodynamik>

Aussage <Mathematik>

Vorzeichen <Mathematik>

Störungstheorie

Ordnungsreduktion

Integral

Objekt <Kategorie>

Arithmetisches Mittel

Energiedichte

Randwert

Lemma <Logik>

Betrag <Mathematik>

Kompakter Raum

Kategorie <Mathematik>

Term

Beweistheorie

53:51

Resultante

Gewichtete Summe

Punkt

Randwert

Mathematik

Fortsetzung <Mathematik>

Kardinalzahl

Extrempunkt

Soliton

Computeranimation

Gradient

Temperaturstrahlung

Vorzeichen <Mathematik>

Wellengleichung

Folge <Mathematik>

Inklusion <Mathematik>

Parametersystem

Kategorie <Mathematik>

Profil <Aerodynamik>

Temperaturstrahlung

Arithmetisches Mittel

Randwert

Lemma <Logik>

Einheit <Mathematik>

Rechter Winkel

Beweistheorie

Aggregatzustand

Lineare Abbildung

Nebenbedingung

Folge <Mathematik>

Subtraktion

Theorem

Gewicht <Mathematik>

Sterbeziffer

Streuung

Klasse <Mathematik>

Gruppenoperation

Zahlenbereich

Auflösung <Mathematik>

Solitärspiel

Term

Physikalische Theorie

Differential

Ungleichung

Spieltheorie

Reelle Zahl

Inverser Limes

Verband <Mathematik>

Abstand

Optimierung

Aussage <Mathematik>

Drucksondierung

Wellengleichung

Erweiterung

Kontinuumshypothese

Modul

Unendlichkeit

Energiedichte

Quadratzahl

Parametersystem

Differentialgleichungssystem

Energiedichte

Helmholtz-Zerlegung

Richtung

Kantenfärbung

Normalvektor

Siebmethode

Term

1:06:14

Parametersystem

Statistische Schlussweise

Streuung

Kategorie <Mathematik>

Reihe

Profil <Aerodynamik>

Soliton

Computeranimation

Gefangenendilemma

Helmholtz-Zerlegung

Energiedichte

Existenzsatz

Flächeninhalt

Helmholtz-Zerlegung

Inverser Limes

Richtung

Beweistheorie

1:08:12

Folge <Mathematik>

Benz-Ebene

Bruchrechnung

Theorem

Total <Mathematik>

Statistische Schlussweise

Orthogonale Funktionen

Reihe

Familie <Mathematik>

Profil <Aerodynamik>

Term

Ähnlichkeitsgeometrie

Physikalische Theorie

Computeranimation

Approximation

Entscheidungstheorie

Variable

Lemma <Logik>

Wärmeausdehnung

Parametersystem

Wellengleichung

Aggregatzustand

1:12:20

Folge <Mathematik>

Wellengleichung

Theorem

Prozess <Physik>

Statistische Schlussweise

Approximationstheorie

Orthogonale Funktionen

Computeranimation

Approximation

Lemma <Logik>

Wärmeausdehnung

Spieltheorie

Parametersystem

Wärmeausdehnung

1:14:22

Transfinite Induktion

Theorem

Folge <Mathematik>

Gewicht <Mathematik>

Ortsoperator

Annulator

Zahlenbereich

Mathematische Logik

Ähnlichkeitsgeometrie

Computeranimation

Gradient

Eins

Theorem

Inverser Limes

Normalvektor

Aussage <Mathematik>

Analysis

Folge <Mathematik>

Drucksondierung

Wellengleichung

Parametersystem

Torus

Approximation

Dispersion <Welle>

Kondensation <Mathematik>

Aussage <Mathematik>

Profil <Aerodynamik>

Temperaturstrahlung

Fokalpunkt

Approximation

Energiedichte

Lemma <Logik>

Existenzsatz

Forcing

Kompakter Raum

Beweistheorie

Parametersystem

Energiedichte

Kategorie <Mathematik>

Helmholtz-Zerlegung

Term

Beweistheorie

1:21:55

Folge <Mathematik>

Zwischenwertsatz

Einfügungsdämpfung

Subtraktion

Theorem

Gewichtete Summe

Kontinuumshypothese

Zahlenbereich

Physikalische Theorie

Ähnlichkeitsgeometrie

Computeranimation

Gradient

Stetige Abbildung

Inverser Limes

Analytische Fortsetzung

Normalvektor

Aussage <Mathematik>

Parametersystem

Mathematik

Machsches Prinzip

Ähnlichkeitsgeometrie

Temperaturstrahlung

Inverser Limes

Energiedichte

Auswahlaxiom

Quadratzahl

Existenzsatz

Einheit <Mathematik>

Ganze Zahl

Rechter Winkel

Parametersystem

Zentrische Streckung

Term

1:26:13

Quelle <Physik>

Punkt

Fortsetzung <Mathematik>

Element <Mathematik>

Ähnlichkeitsgeometrie

Computeranimation

Gradient

Eins

Temperaturstrahlung

Stetige Abbildung

Vorzeichen <Mathematik>

Wellengleichung

Urbild <Mathematik>

Analytische Fortsetzung

Auswahlaxiom

Elliptische Kurve

Zentrische Streckung

Parametersystem

Lineares Funktional

Verschlingung

Kategorie <Mathematik>

Zirkel <Instrument>

Gebäude <Mathematik>

Klassische Physik

Vorzeichen <Mathematik>

Teilbarkeit

Koalitionstheorie

Helmholtz-Zerlegung

Menge

Beweistheorie

Strategisches Spiel

Beweistheorie

Aggregatzustand

Folge <Mathematik>

Subtraktion

Ortsoperator

Abgeschlossene Menge

Zahlenbereich

Transformation <Mathematik>

Stichprobenfehler

Spannweite <Stochastik>

Weg <Topologie>

Endogene Variable

Inverser Limes

Aussage <Mathematik>

Kombinator

Physikalisches System

Stetige Abbildung

Unendlichkeit

Energiedichte

Auswahlaxiom

Quadratzahl

Flächeninhalt

Parametersystem

Mereologie

Normalvektor

### Metadaten

#### Formale Metadaten

Titel | 4/7 The energy critical wave equation |

Serientitel | Leçons Hadamard 2016 - The energy critical wave equation |

Teil | 04 |

Anzahl der Teile | 07 |

Autor | Kenig, Carlos |

Lizenz |
CC-Namensnennung 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 | 10.5446/20499 |

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 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. |