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2/7 The energy critical wave equation

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2/7 The energy critical wave equation
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The theory of nonlinear dispersive equations has seen a tremendous development in the last 35 years. The initial works studied the behavior of special solutions such as traveling waves and solitons. Then, there was a systematic study of the well-posedness theory (in the sense of Hadamard) using extensively tools from harmonic analysis. This yielded many optimal results on the short-time well-posedness and small data global well-posedness of many classical problems. The last 25 years have seen a lot of interest in the study, for nonlinear dispersive equations, of the long-time behavior of solutions, for large data. Issues like blow-up, global existence, scattering and long-time asymptotic behavior have come to the forefront, especially in critical problems. In these lectures we will concentrate on the energy critical nonlinear wave equation, in the focusing case. The dynamics in the defocusing case were studied extensively in the period 1990-2000, culminating in the result that all large data in the energy space yield global solutions which scatter. The focusing case is very different since one can have finite time blow-up, even for solutions which remain bounded in the energy norm, and solutions which exist and remain bounded in the energy norm for all time, but do not scatter, for instance traveling wave solutions, and other fascinating nonlinear phenomena. In these lectures I will explain the progress in the last 10 years, in the program of obtaining a complete understanding of the dynamics of solutions which remain bounded in the energy space. This has recently led to a proof of soliton resolution, in the non-radial case, along a well-chosen sequence of times. This will be one of the highlights of the lectures. It is hoped that the results obtained for this equation will be a model for what to strive for in the study of other critical nonlinear dispersive equations.
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states time Wellengleichung First-Order expansions orthogonal loss argument orders Rolling box position diamond proof area Wellengleichung Blocks building infinity argument expansions maximal scattering sequence hypotheses proof category decomposition theorems normal sum Blocks Errors non-existence 3rd translation Hot limitations approximate scattering sequence profiles operations theorems equations conditions Super Reihe theoretical operations limitations non-existence approximate formalism Computer animation Normierte Räume
functionality time equal ones 3rd translation elliptic inequalities approximate theoretical variance independent PIE variational profiles theorems equations equations position scale graded states expansions meetings elements numbers approximate elliptic category period Computer animation sheaf partielle Integration decomposition theorems equations variations Results
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and this is the kind of thing so as we just continue where we were last time and I review a little bit more about the profile OK just to be so this is still from last time it's the same with the change who we have abounded sequence and for its J. we have solutions of the union wave equations and we have a sequence of parameters so the land the under scaling parameters in the exercise translation parameters and at the end of time translation and the the I'm not entirely consistency in how right sometimes the Jays on top sometimes it on the book in front of the and controllers sometimes adjacent so man From the context said it should be clear focus and we only sequences of triples so course overnight some President say almost always always and if this really have the property that their son goes to infinity so that means that missed 1 of the most Is the sequence of parameters is orthogonal and this is such sequence of solutions of linear wave equation which we then modulate according to this and we find this thing and then we look at the difference it ended linear solutions corresponding to a given sequence and the sums of profiles the model kits then there's error hostile 1st of all remain uniformly but it also he has to go to 0 In this sense of the space-time OK so this season In case we were looking at the this is the latest and he questions the the US uh the UGA muscle strength of the union where quit In the world of adjudication process where I solutions in and you may have a role as used in the the head of the local and that the use of the stuff I viewed the Quran was convinced him to do have aches and a few options here in Europe and the US in the year but you can translate the translating into and then the other thing is that you're Lawrence most unlimited because they keep the bounded sequence In each 1 group should he must face she had PGA history was used to prepare the country for Washington the hole we feel that that's why including the time translation but it is
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the 2 situations In the TJ and oral and the GM has defined and limited then you just solving nonlinear wave equation after finance Minister in the new again if goes tools plus infinitely what you doing to solve the equation at infinity so that's like that what they call the existence of the wave operator and so there is no linear profiles will always exist again by extraction we can always assumed that this limit exists In case when it is finally to make this will think that it is 0 because by the time translations and changing the profile we can take it away and was his plus infinity reminders In the end this formalism all limits have all sequences have limits them done this to you accomplish with the victory How can of frequencies have the feeling that if you say so that the solution yet is likely in the final state is the way over that that's OK and because of this construction if it's a scattering solution the Indians says then you see that limited minus infinity the time of existence this nonlinear provides minus infinity and that a nonlinear solutions captains of mines by construction In the same within just because you constructed price scattering so now we look at the modulated nonunion solutions OK we can still do that and of course there is the maximum let's think positive time existence of this modulated solution can be obtained from the old 1 by this fall nothing this is just the Earthlink Inc as bus Infiniti's 1st facility and so we have this approximation that is a very important to her it tells us that if we have a profile the composition and we control all the circuits norms of the blocks then we can write the solution to some of the box plus a small OK this so where the 1st case of the service easiest state it says that if all the solutions could involuntary nonlinear provides Scott the forward in time then in there Harrison and bullying in solution minus sum Of the nonlinear profiles miners in the news solution associated with the remainder in the profile and the compositional actually goes to 0 in a very strong OK so when I was saying in the 1st lecture the thing we look at each law In In the nonlinear solution justice with we use this kind of the company you may have tuition of what we saw was already gone but not all and the don't know who's not so here is whether we can say that you can go as far as In time as far as all the sucrose norm remains Of the ruling a profile so I suppose we take at times think I am such that it's always winning the diet the forward time of existence of the J and suppose that after that time the stricter norms remains bounded uniformly then you can still so then only solution is assigned to time if they knew the stickers norms remain valid we have solemn and the error moves to 0 but only up to the kind of that you don't see anything even if the yield on the other hand role of the some of the more I have to be smaller than all the OK not otherwise I'd have something that is important that would be to work too the so at area this summer and this is a sufficient condition for this is that they all have very small and 1 grows to long profiles or all the parameters of scattering this is another possibility or this this fake diamonds things strictly no way From the final on the and then in this case you can do that so I have some remarks about the proof 1st this Adelaide norm can also be taken to be a final day of the tomorrow's fungible rooms now the proof of this theorem uses this thing and described them as the longtime vertebrae and he also uses the it is strongly this kind of loss island the property for the L wrong to the 8th all of this modulated nonlinear so this is kind of a peace accord expansion for the look at time
so at a very important consequence of this approximation theory is the fact that this pick the Gordon expansions and held for the linear from funding composed should also hold for this nonlinear profile expansions AP to any time strictly smaller this thing so we have this fact this fact we have the same thing for the it's 6 norm cancel and this is called morally well not morally in actuality you can see that in this converges which it can after extraction and you can make the nonlinear profiling of positions into a linear profile you manufacture the linear prom profile that corresponds to the appropriate linear profile depending on the for 2 "quotation mark OK so I will not torture you anymore with profiles this is kind of a tricky business but there and I put down all the relevant things so now we move on and what we're going to do is 1st there's a number of things the lipstick equations that we need and so start by the developing books so we really need properties of solutions of lunatic equation and some variation and that's what we started out with them so again and this is their associated nonlinear that equation and recalls Sigmund known solutions there's this explicit solutions which is well known from the amount of problem and as I said last time this is up to translation and scaling the only negative and as the result of his new never end if I don't translate this are the only Reagan solutions with plus and minus In this combines the work of original Nuremberg and show ones somehow does it for all these solutions in the radial Bausch I have reduces to the meeting of case and indeed Nunavut gives you some detailed property so we have to put a list of elliptic machinery together but the summary is simple it as I was saying last and sometimes you can summarize in 2 or 3 sentences maybe 25 years of work OK but that's that's OK it's convenient for us wouldn't do the work some OK so as I was saying last time this W is also a call on the ground state so let's recall the Sobel inequality In 3 where in 3 they have a great interest in Miltona function is enhanced and there is the best Konstantin recalled his best constancy you can see 3 is unknown was calculated that I think by buying police already it has a pie is endemic functions so but this is a known quantity show the C 3 is not the way some positive calls and but the known possible concur and then there is the work 1 way in which W is extremely well is that it gives you equality is raising quality and scaling and translations that is the only known 0 function for which the call so that's the theorem of Lyme and now they pay let me go go back to the to the
question multiply this by Q the integration by parts then you see that the key to the 6 they have developed into 6 equals the gradient of Q School Clinton so this is a piece of information that we will so since this is the
solution of the nonlinear equation this whole and we know that this holds the calls this is an extremely so that means that we have a formula for this OK because we have to 2 equations so we can and this is what you get the gradient of W squared this best Constantine reminders to the parliament In principle this adjustment and now where we're going calculated the energy of his lead the formula for the energy is 1 gradient plans 1 6 nominees to the 6 but this members of the same so it's 1 of the and write dubbed squared so we know the grant of a square Tennessee the ministry so this is this and that OK this is what we have the because of the absence and were used heavily used a variation of the authorization of the grounds now because of these things in the W is and solution the number elliptic solution with the least and so understand you the way in which this W is isolating In this set of such solutions to the particular so that investors in the Arab world says that if I have a solution and it's so as to normal living thus enriched white the 1 W it that it can only be done the settlement 2 to be above W have to walk to fix them OK so that's provision out of sketched approved him it's simple proved it when they find plus Anqiu minds and then the 1st observations his captors inequality tells you catalogs inequality tells you that the blood loss another positive part of front-runner is bigger than the equal to the the positive part of public you can and so in this case captors inequality using this you multiply by 2 blasts again and get this In similarly and the corresponding inequality forward to mine and we know that they add up to less than twice the 1 of the new show 1 of the 2 will have serious smaller than that that the important thing is to you can't be innocent so let's assume that the Cumani it is the 1 that small we will show that it has to be easier why is that the calls to mind by integration by parts of the gradient is less than that of 6 inequality uses solid inequality within Wisconsin to go to the gradient now we use that this number is this and we had to and so that we results from here that even this number is 0 or I can't believe I can win divided by this number and get this but this is a contradiction because we started out by assuming that this was less than OK so the assumption that the the ECU miners was not even thinking 0 this absurd and so can miners isn't 0 if you miners is identical 0 2 isn't anything excuse me let them by begin Nuremberg has to be done 2 I think this is a kind of a full-color argument and don't know where it is in print but it is a folkloric "quotation mark associated Press a fact of and take as a corollary is you can use this to say there simply to show that there and if you have a solution whose energy is smaller than the UW there has to be done OK and the the this is as follows from there right to state equations is equals that then the because of this you this but he had this you have to be dealt with and in fact if you precise this calculation you see that
any others solution has to have energy strictly about the changes there is a gap the it is in the nonlinear spectral to call at so now I'm going to use that as some properties of the ground state this variation on characterizations to prove some original estimates and that allowed me to completely understand the situation as far as the global dynamics provided the energy is smaller than the energy OK so this is what the Maryland proved that I guess that about 10 years ago you could wintry in a sense is the beginning of the stuff OK so and now we're going to do a little bit of concessions drew 1 so they say variation 11th says that supposed that my going into small and gradient of not only and his energy is strictly smaller than the image the by among then there is the blocks so we called his energy trapping it tells you that the gradient instead of just being smaller is smaller by a definite them and not only that but there's a certain courses at the moment and in particular the energies to people or look at 1st the proof this as a as I was saying to prove his and the need for the following day 1 variable function one-half minus 1 6 6 3 2 6 white kids you can it's a function of 1 year the 1st thing I noticed that the bodies is the gradient the square this is less than the energy and this is just the fact that C 3 is the best concentration is on the at nothing more than the next day saying I wouldn't do it I'm going to find the positive zeros of this function course that involves solving a caucus simple quadratic equation so we can do that so for 4 positive why this is 0 if and only if it is 0 3 2 the one-half brand news so the zeros are at this a fixed amount higher than grand 3 2 the 1 happens is to continue the war and then the next thing we're gonna find that is what the critical points and so for the critical points we differentiate and defined the zeros and it's exactly 1 see 3 Q which she will remember and that's that and to normally that could square the integral grant them all and added it's an associate is and what is the value of S and it's critical Pauline well it's exactly one-third of C 3 and that is the energy so this W fears for this city of 1 variable function so that is not going up there and they strictly increasing before you hit the 1st critical .period fixes for why small it is clearly increasing because we have power once in the past 2 OK but also this at a critical point is known degenerate because he considered the secondary that this will be strictly positive for 1 positive because can completed its chemical known him so what does that mean it means that it's strictly increasing add up to where get to the critics critical value and on the critical value I have in the value of the energy of the now I am so I'm assuming that I'm a fixed amount below In energy that critical volume that means that half of the great consistency specific fixed amount below and therefore the inverse function is this strictly strict among below the gradient of the square In that proves the fact the 1st part of the recall what 1st
part when the system the monotonicity of the meeting calculated good is that the body and there's this doubt that is is that's a square root and his now where approved the other quality In the other comes in very simply from the 1st called the source here we go to here by the solid investing no we take out discovered in fact a and now we see that C 3 to the 6 is the gradient W. squared and then there's this wall of one-liners dealt the square gradient of those squares so this and this cancel out and we have and this is strictly strictly positive because the bodies spot and that's how you get the cost part it to the next thing that they want to point them Is there and this proof shows that if you are a below the 1st 0 of this function that your energy has to be posted on the yes bounded by the energy and if the leftist positively the energy positive event it this value was a physician so the energy remains possible and it'll be the beyond then the ball or a Granville Square and the amount is 3 to the 1 "quotation mark what is so now I need to go to study some properties of general solutions and their Lawrence transfer remember that the traveling wave solutions they claimed were Lawrence transfer jungle it's true that now so you need a little bit of information about the proposed solutions to the thinking maybe it's been good if I run on the board in the In the end it but at back at AC "quotation mark a to keep wages Alonso a elliptic regularity theory immediately gives you that if you have a solution it will be seen and this is where lower dimensions and better than the high dimensions Due to the 5 is a very nice powers Q to the center of words is not great which is what you get in any dimension applied for it and little but less stick kids to find so curious it seems this and the next thing I know where the city is that Q has the K at infinity and that the case is that the case W W is the 1 that gives you the slowest became How do you see this where there is a simple way of saying it would which use is transfer it is you will remember when you studied how morning functions you can do the Catalan transformation and you still have harmonic folks In 3 The the :colon transform is 1 over a length of X F of X where X square please note there is out of the cabin transform also preserves solutions of the nonlinear the key question so the equation is a variant of the conformal transformations of and killing transformed and does that and so you can pull infinity to abounded .period by the coming transformed the full affinity to the origin and use that solutions of the nonlinear equations small at the origin and that gives the music computer so that's all there is to it so the next thing I'm going to do is and where progress denied they to normal of each partial is a 3rd of all the full grade so all the parties have the same amount Of course for W. this is obvious by the various in the rotation but for any added that the you can prove just by multiplying the equation by this expression you you multiply the question this expression it seems to be a
government of the people will get you multiply the equation by this an integral parts and you see that the terrorists from infiltrating go away because of the meeting the results were in state into the chaos as and when you arrival that union and this is
something we like identity please world now I take it a vector of length less than 1 and take my Q and I view it as if it were a function of X and the Constantine the and added Lawrence transformed that function and then will take in the direction given by what they get is foreign man so don't pay any attention to it the important thing it said is a function of this form X minus comes out so it's translated that is being translated in the direction given by and what is a function Q what do you get by making P equals 0 cancel look and is too much because his opponent and help the truth is that but they work is interesting is that if we if we do our rotation in the X variable we can make this To be individual l times won it's possible and this equations are invariant invitations of the of and little L times seawater when you do that killed days Lawrence transformed in the direction given by Elton G. 1 you have a very specific forms of the game where the X Warren goes to X 1 News Ltd were formed to 100 square and in the team goes to explain the key goes to please t-minus Alex 1 were 1 might of square and the X 2 Annex 3 and the Dutch "quotation mark now this escaped to this Lawrence transformed so that you have some many interesting properties if which of course is nothing more than changing Wilson and chasing everything around it the 1 thing I want to point out many of is that the effect of the Lawrence transform is to make the energy bh yeah a control on the energy gives you control over how how far elders from and it was but said that and remember that this is a positive for all for all solutions of the elliptic equation the energies District almost 0 solution is strictly positive because it's strictly figure than the energy of Q and the energy of Q was 1 3rd of the best constant in the Solomon equality of of 3 OK so we shall all these numbers so let me show now that all traveling waves Our are the form to well for some Q a solution of the nonlinear teach question and this is something that the proved and work with the kind of Meryl but the real proof was then in 2011 but it was not clarified sufficiently in 2014 at the expense part of sir I start with a function therefore breaks the which is a traveling wave function With that in each 1 and suppose that this salsa nonlinear with From Prince have a solid away from the nonunion way so that they claim reason then effort Hester beacuse of for a Q solution to the away more and this is important for us because they were in the soliton resolution and and we wanted decompose things into somewhat belated traveling waves it's a good thing to know what the traveling waves have compassion OK so this is the point of this of this country book so again we can do spatial rotation and his past and his efforts a functional on our 3 dissipation rotation so that L is little alltime 1 so when ethos of this form and now I plug-in what does it mean to solve the nonlinear with me and that's that and you see the 1 minus 0 squared factory in front of the OK let me assume for this proof for simplicity that counties posted look at the case and negative adjusting his team to point "quotation mark so when I have this equation and
this proposition says that suppose ashes and Nunzio functional areas in our and equation holds then necessarily elsewhere in this lesson learned and there a Q solution of elliptic equations such that it's the law and stressed OK so the proof costs in their 3 different 1st cases elsewhere in his begins if elsewhere is bigger than 1 there's really cannot happen because of finest piece of property so you just have to produce the argument that the shelves and there the argument that it is as follows I call you ecstasy this thing where am doing I'm uncertain doing the Ltd here so this is a global and I'm solution is great in the news that a mysterious thing is down the it's 1 so because in each 1 I can't find any a large and given the bachelor I find larger firms such as this is small for the tail segment of at times but if details at times you were small by Finance because of propagation I get that this propagating into the future and so if they give up a little bit on the M I need to give up a little bit to use some kind of functions to be moved to while use it's been the propagation many I get that form all-time team this morning constantly and the constant comes from this most cation so once I have that I I looked at the compacts is set next to an extreme and 2 numbers a and B. and the 1st observation is that these things large and I have X 1 in this interim and X 2 in extreme this the then I know that the length of X has to be vigilant the that's because alleys bigger 1 could the moving only by the this is best for all 1 way of using the alleys begins so once we know that if I look at this of course I have this fact from the finest be the propagation but I have my formula for you and the changes of variables that I that this thing equals but so this thing for fix K and fix a and B is small in Seattle for every epsilon so this thing has to be 0 and so are hostilities now that tricky cases when 1 because the design human justice not and that's where you know the way you get out of it as you go back to the takes but in the year when use elliptic theory not to the so because remember that their factory in front of the 6 1 is one-liners and square when it is 1 that factories here so you're equations set your function verifies an equation into materials instead 3 so my ass verifies this equation In this 2 variables and for almost every 1 Hassan regularity so we fix such at 1 and wouldn't have been there is and will show that the 0 by using apologize Oregon the idea involve argument is that from the equation you you get this identification which even checked by flooded its rule I'm from this side and that the EU integration by parts integrated parts both sides of the divide integrated this side again 0 with an intimidating and Virgin In any event integrating this the only thing that happens is that this falls on the and that gives me for over 6 Is it too from his evidence and 2 from the formula and 6 from there and so the conclusion is that after the 6 the after 6 0 2 0 and so that's the case in this last L is equal to look at and to justify the integration with parties users from truncation that I buy with that no finally assume that elsewhere is less than 1 and now I a fixed them the 1st variable problem by doing this if I do this now I find it very finding usually From this equation that with the one-liners us elsewhere and that the elliptic equations so s verifies the sticky question and elsewhere has to be this was result from home so let's start now With the approval of ground stations conjecture so this use is essentially concentrate on compactness and rigidity theorem so let me start by reminding you of the precise statement of this In so precise statement is supposed to have a solution it is energy is smaller than the energy of and you will be the corresponding solution of the nonlinear from care then if the gradient is smaller than the gradient of W then the solution exists for all times in both time directional signs caftans and actions and if it is strictly bigger then it goes up in both times the book direction new many say OK you're missing in a case which is the but it turns out as the case doesn't exist there is no such folks the energy constraints and this in combat there is no such thing this is the work of usually however you want to see a witness to the needs the strict inequality Indian focus In fact the case for energy of his equals the energy of the museum is a very interesting case that was sold in the paper but you can the classified with Athens and there you have it if you need scattering in both directions or In in in this case you have to use captain in 1 direction and will do some objects called W minds in the other direction and in this case is that you go off in both directions are evolving 1 of the time directions and the other 1 you go to another object called OK but I'm not going to enter into that today Red so so that is a major thing with King but it is also understood broken the list in flawless look at that so that's 3 things to Tokyo analysts 3 things to prove their last 2 unless there simple consequences of the variation analysts innocently and the real content this property 1 so remember that from the local theory of the pollution problem this condition is equivalent to scattering so this is it what I would say it the hard part "quotation mark their lending disposals in 2 parts 1st and then we will start developing the theory that goes "quotation mark so too do
deal with the 3rd point I would recall another regionalist which is the quite he is false it says that the the agreement is less than or equal to the of development and the energy is less than or equal to the energy of W then a degree in squares is so it a way to say is that the ratio of the green squares is less than the region the so that's prove and we have a pet that's the 1st thing that I'm going to do is I'm using his claim to show that there cannot be something with energy Slessman energy of W ingredient equal to the grave why is that because if there is a great audience are equal and the range all the energy of Dublin and and the energy the energy of being divided by the energy of the lose bigger than equal to what From this inequality but it's also so strictly less than 1 my assumption that's the constant so this claim immediately gives them that 3rd cases the insect insisted the ratio of the gradient of the and ingredient of W's 1 so that that this ratio of energy has to be bigger than are equal to 1 but we know it is less than OK so now that lets proved that again we use the same function of 1 variable that we used before this polynomial the urethra and we noted that after a 0 0 0 OK FOR grabbed randomly Square is the energy will release and after immigrating into smaller than the energy by the solid the arrangements in this effort is concave because the 2nd derivatives negative Wiesen for a spot of the widening and they get them because it's my musical a constant sometimes the greasy so we can take that is cocaine so since its concave and from 0 0 and half of 1 it so I looked at half of Esperanto square that's a concave function and add 1 in equals the energy of W squared and 0 2 0 silicon ,comma gives me so efforts cracked down square is bigger than the equal to ask that the energy of the common 0 and now it chooses s currently give me precisely the claim it is this arguments over a simple calculus but of course you have to know what to play what so this takes care of 3 "quotation mark now I will prove too and there's an extra assumption continues using them so too says that if the entities smaller of gradient is bigger and will so I will do it for functions in Alto it can pass the general case .period truncation using fine it's beautiful appropriate so let me just do this you can say that so the
1st thing is that this argument in this function that last which we 1st bid when the gradient was smaller the agreement of the new also has a version when you pass the critical point but when when it passed the critical point they are forced increasingly get to the critical falling that I started will die OK so that means that is his style with a brilliant being bigger you will stay a definite amount it's the same argument but on the other side so this variation unless tells you that if the energy is smaller by bundles not and it is bigger than is a certain amount so using this this is a static argument now you can plug it into the hole until then flow Of the full preserves in the energy so if I start out with the energy being smaller and continue this being small and in the gradient was bigger it's now bigger by a certain amount so for a little bit of time he was still remain bigger than the agreed but then I reapply the static argument that I get a fixed amount take them by continuity of the flow of the flows continues it will remain strictly Tokyo closer continuity so for all teams in the maximum interval this is a certain amount so the argument that we use for the it is an argument that goes back to work on Harold line I in and with the anyway this is from the early 70 there is this argument that's been used for brought problems there a similar to arguments of his car over glasses for the change good so now I'm where I am saying that the energy is smaller than the energy W and so there's a certain amount of Delta for which it is still bigger this is strictly bigger than you have room to put an end it and now I sold for Use it militarily into the 6 in this inequality and this is what they get they put this on the side of the 2 6 and the science and this is like NO I multiplied by 6 and this is what they get that I will actually in the next slide multiplied his by 2 if can survive multiplied by 2 here I have 6 here I have 6 and you have 12 so this remember that precariously on its goal so now we go to the actual proof I called y of the equal you squared and that's why I needed to assumed the new wasn't something that makes sense and primaries 2 attended attained this is clear was more interesting is the calculation why don't for and this is 1 of these New England and the state that was talking about the last time and there's this sentence I know I I can leave you to check that this nothing more than fighting in the equation integrating parts OK so now I use now inequality from below to you to this sea that was in the previous life and so I still get this to you the square now again the 6 and here I have a 6 on this but to give plus 4 and then have been minus 12 and you know we're now I have some other positive collapsed you can just plug that's all of them in the eye add to the 6 gives me a and I remember the 3 times the energies From the so then 104 -minus because 12 times is 3 thousand 4 telling that and I remember that ingredient of squared remains strictly bigger than a grain of the wood so I'll throw it away get this inequality so I assume that nite and the role of existence goes all the way to and I'm going to see that cannot be that is this function is strictly increasing while the prime minister made upcoming white prime restricting Wendell Primus strictly so why Prime Minister to increasing so eventually would become His exists for a long time and it will become part there and then it will increase from that point on the remaining ports so that's certainly a team not for which white Prime is a spot so for the bigger than the knowledge I multiply this inequality by y which is you square and forget the dealt when I get there but this we recognized From coaching shorts that is begins why Prime's where was white prime is said to you the key OK
and so where look at this I get wide double prime overweight Prime is bigger than to wild prime over what I can do this because why Prime never manages to officials of the most recent losses for is but I know how to use all the means of the this tells me that while Prime is bigger than some and that is why swearing but that was the news show this cannot be the 4th focus so that's the argument so if the energy is smaller than the energy of W and the gradient is bigger and he has the ball and then the positive time direction within the negatives think messing with me work so now the 1st thing is the heart of the matter so the 1st statement is that if the energy is smaller than the energy W and the gradient is smaller than the grain development the solution exists globally and scarf it's so now we learn and develop the machine to prove that this is not a few lines from will take consistently next time you know just to claim that it was call I can't but they must be patient and set it up the 1st thing I would do is I'm going to use my variation OK I went to my original estimates the device that would data which is strictly smaller and the gradient is smaller then for all time of existence this remains small and this remains being but there was the variations but noted that this statement in particular implies that the energies because the energy has one-half and 160 and this has one-on-one who can moreover because of this lower bound you can actually say that the energy is comparable today to encourage to each time this is age 1 there is certainly bounded by us by buying this book this lower bound is bounded from below the team just sits there and this is comparable to the initial age 1 grows old to water because constant so this energy is the energy and time 0 and time 0 this equivalence holes what is the situation there the situation is my norm is on the but that does not mean but my solution of this global his captors because we have the examples of priggish likened the term we have this property but ceased to exist in so we have to exclude that this is of course the country this it happened but concentration so we have to to see that our sizable strained excludes them look at office and now I'm going to use the concentration compactness procedure to use the concentration compliance procedure introduced the following in state and uncle and scattering its C for all you not with H 1 cross-sell to such the gradient into smaller than they've been in the value and the energy a small and the energy of value than the corresponding solutions these go well and scatter that's a statement that I would like to be proved as subsidy as the my theory that seems too some of so now I would say that the fixed function with verifies this hypothesis the science research constraints we say a of museum anyone holds it for this solution on its maximum literal this L 8 enormous enormously so that's the same as saying that the maximal until is Little is hard and the solutions captains both kinds OK we saw that from the beginning scattering criterion international space-time normally stickers only side so now we're going to start slowly "quotation mark if you will hear my allegiance to the flag of the no I is the maximum OK so I'm always using them the maximal enterprise I will so now I'm going to start and I started by saying that if they're there they small certainly assets the views eerie 1 holds by the local right now we have small they thought I showed or claimed but the solution exists global in scope but now I haven't translated into the the energy looking and data whose energy is smaller than the energy lovely on greater smaller than a grain of W if the energy being small and the H 1 Brazil to normal being small are looking because remember we
have Issaquah so I know
that there is some later 0 positive said that if you 0 verifies our hypothesis and his energy is really is small then scattering holds for this year's cereal there is nothing more than a combination of original estimates and small data so what what is my strategy my strategy to to stop increasing the energy as start increasing the energy slowly and in the 1st time that something was wrong and my goal is that that 1st time is the level of energy that the /slash And that's the strategy this the concentration compactness procedure of course that cannot be just gonna like that for what has actually was last right so and recalled if the the smallest level of energy so a subsidiary will be a constant such that it is 0 E 1 verifies the that hypothesis this and the energy strictly less than and see their scattering and I will look at the optimal use of C with this don't and we know that this piece of after restrict less than 16 bigger than some number anything on there only considering members smaller than the smaller or equal to the energy can so my goal and my fear is the statement is of equals the average In the subsidy equals image of dollar might interest so I preceded by contradiction has shown the use of that the strictly small then the energy stocks and that it would patients will reach a countries but the contradiction comes after a very long time consuming there with so this is a stretch so we assume that the subsisted stricken smaller and we reach a constant so the contradiction is reached because of that all 3 different events so the 1st event it is that there is an element in each 1 cross-sell too With energy is a critical level of energy and therefore quality of its gradient is smaller than the gradient of the and the solution corresponding to it actually happens doesn't get so this the critical level of energy is realized by some folks the critical data so that's the 1st thing to prove the 2nd thing to prove it is that this critical functions so this is a very bad functions is the main gate of citizens with a very bad functions so the next thing I know is that this functions have a redeeming property they're not all that bad it was the from the redeeming properties a certain England in In the column possibilities of his functions as functions cannot be torn apart and that is precisely expressed in terms of the following mathematical studies then you can find is scaling parameter land In Leyshon parameter acts such that for such a critical function that's saying that the this thing goes back to the positive side it goes back to the positive side lost the negative side so consistently goes by the the positive side horses will go about for that is then informed the positive side we can find these modulations that said that this thing behaves as if were a single function sort of Singleton so that there a specific formula for and that means that the trajectory of this is compact Of course it's not Singleton than the next best thing to being a single plan is being a Compaq set this so this is a redeeming feature and this is a very normal this person's property because it's easy to see solutions of there is an online newspaper because solutions of the linear equations if they have this compact it's properties have to music compared with this simple fact and then the 3rd part of the rigidity theorem part so this is kind of like a new now that the new rules himself suppose you have a solution like this very fine size the only possibility is that the busy who can so the can be such objects With this nonlinear became but this guy cannot be 0 because its level of energy was a critical level of energy and the critical level of energy was forced bigger than this cannot and that's the thing so you have to wait quite a while to to OK OK so in my notes I wrote a very detailed from but they don't realize that they will he bore everybody to tears if you just need to prove so instead of a given you detailed proof from an interview with the idea of when his sister OK so let's start with the 1st such a device that will let me qualify that the proof of Proposition 80 and Proposition B I T it's absurd to say that and the lengthy and so on and the use heavily the profile of the company but so I sketch the idea from the proposals and then give a complete proof of the rigidity that that there is no such object which is not and that I think is kind of the most important part of the probe the other parts are easier to understand conception so let me explain why Proposition 84 history OK so how would we trooped proposition and in what would be the most native waved from proposition so this is the critical level of energy so it slightly higher than its things will go very bad so for each the number there would be a function of pair of size of whose energies E C plus 1 such that the s normal relations by definition being the critical level energy and the fact that the strictly below the energy of W so I can go a little bit about but still not reached the images with so something has to go back if it goes badly that this long is it and then what you would do if your name is would say OK now with this uniformly bounded functions and its past the leg Of course you cannot take a strong element but week what happens if you try to do that the Indians way is that you face because you at the critical level and you cannot exchange we conversions and millenium when your critical so what you have to do is use a profit the composition too substitute for this failure the most compact events so this makes complete sense so what do you do how did you use a profit in OK so now we think the profiling composition of the sequence of 4 the 1st thing that too we will observe is that for all this profile the energy force why possibly because of their effect Oriana or 7 the conditions the energy and the gradient will be smaller than the ones of W and we know that if we system among smaller than W then the energy remains box so all
of these guys will have positive energy as well as the remains and the remainder will also have to for the next thing that show Is that there cannot be more than 1 the non-zero profile why is that but suppose that there were 2 that is there too 40 each 1 of the profile and I have to say however that the energy is strictly smaller than the sea because in the 5th and Warren expansion and all the images appear the energy of the of the date of the U 0 and you want and is going to be easy tending to his and I have to whose energy is known 0 I can always subtract 1 of them and then the other 1 which remains on the right-hand side will have the district is smaller than the right and I also have the ingredients being strictly smaller than the gradient of W because but my variation considerations if the great into smaller and the energy is smaller than grains smaller by fixed them Of course it's a slight problem with this argument here in that I cannot use the author malady the expansion for just the space but this we overcome by a small fixing off at the time and so don't get stuck on this city's is too long but then each profile has energy strictly smaller than anything ingredients strictly smaller than in the gradient of the wonders that mean about the profile by the definition of the sea then then only a solution exists global in scale then I can use my approximations approximation of himself in all each profile scattering the out bits of the of the sequence capped but that's a contradiction because the fixed to have infinite In space that restrictions so there can only be 1 Ground profile the next thing is once we have only 1 known your profile I will show that this remained the this W's that we know voters enrollment this person nor have to go to 0 in each 1 per cent the argument for that is the same because of those didn't go to 0 in each 1 proposal to I could subtract their in and then did the 1 provided I have with have discussed there that can be for those guys are now willing to 0 and now my critical elements is nothing more than lonely profile the public OK so that was produced in the critical now to show that this critical elements now that I have the compactness that's the 2nd thing all I have to show is that they have a sequence of times the and converging for the final time I find scaling parameters and translation parameters such that when I my solutions the corresponding thing convergence in each 1 OK that's what they need to the House do that well I suppose that this doesn't happen and they produce a sequence of kinds in which it doesn't happen now I look at length critical element and that level of times and then look at its profile evil the profile legal position again can only have 1 on your profile the same argument as before and the remainder hostile now if you think about 1 1 0 profile I and the modulation and loans 0 profile and put it on the function and that's that's converging and that is moving you can so it happened this is how it works so you can consider proposition that AMP proof Arkansas I'm sure you know what the House action approved pooled it takes a while can see "quotation mark so now we come to the region and the descent of want the World Inc more full details so the rigidity theorem that if I have a compact the solution of this compactness property and the energy is smaller than the energy of W integrated into smaller than the gradient of W then that the only possible with this disease OK this is what we have to do by the way are sweet continue With this will see that this solutions with the Compaq prop with its compactness property are essential In the they if you want to understand singularity formations show solutions which remain valid in a 1 critical too they're the bubbles 1 can prove that any such solution bubbles software compatible and this is a very widely spread phenomena and assisting in a statement that Hold for basically any dispersing equation for which you have a profile OK so there is this so the basic bubbling objects and then and eventually you want at the time of rigidity theorem that says that this compact objects that have to be traveling with and here what we're doing is worse than where being below the level of energy of the smallest lipstick solutions so the the state is this a lot of it's true the only possible visit the Compaq of OK this is a tiny bit of confirmation of this the so this puts this in the framework of the larger thing that would turn to Of course this was done 10 years ago so he you have to go up to now what do we have to see this but this was the beginning to but the worked so the 1st thing that you need to to think about this is a very subtle thing you know when you have this compact solution solutions with the compact property he just assumed that there exist a function and the Tainan a function X 2 with this certain property no what tells you that this functions are the same measure tough you have no knowledge of any property of this function except that their function so it is the 1st step if the work a little bit to show that if you haven't being some function it can regularize and to be actually smooth functions before the final plan exists and to do that you work a little bit to make them piecewise constant functions and then you can regularize the peace which they don't want to say more note so that you can always assume once you have the functions that may be in a can of crazy functions it they're always transformed them into continues regular
federal why it keeps I think it doesn't like me to their look at the
next level it's also approved the company why this system can the limit tells me there if I have a critical element I can always manufacture another critical element for which the land it is strictly Paul OK why is this true I'll explain it in words and gestures suppose that they I'm given 1 critical element which is banned and the land that goes to 0 I the approach the ice I approached find that you have this the land of opportunity in the catalog from central Canada the chemicals and I respect which would you know it can spread the in between we had difficulty something because McAlinden and Alice-in-Wonderland must have learned you don't different papers presses Islamabad other than the 1 Overland we always confuse themselves when you get back toward them saying it also applies to 1 land than being bound from to so you can always assume 1 or the other as you may find it convenient but not both at the same time it you well August so that was the proof of this suppose this fate then you have a site your critical element the land that is going to 0 as you approach to find OK but you have a compactness property and so your solution after you modulated Hassan limit as you approach the fine and now you reverse the time from the final near land that that was going to Lira no I don't arise as you go in the opposite direction and that's how it's going to be the if it was going down as I went this way when I go this way it'll and that's all there is to it and all this is work by the composer Brokers so we we just assumed then you can see it in the notice there's this very carefully now I'm going to go to a hour and I'm not going to stop stop waiting the hands and more specific proof the 1st thing is I suppose I have a critical element and the time is fine were in the final .period case and of course after scaling Wilkinson it's 1 the final this land not only becomes being but in fact it goes too far as a lower bound by 1 over one-liners and this is a very general property than just has to do With the fact that there compactness property holds and the homogeneity of the problem if you were working women enshrining the of waved it without square this is 1 miners who were working with Katie leaves have the 3rd 1 and so Turner may show the proof so you only need to consider P and going to 1 and we look at you and 0 0 to this objects which is our modulates it's broken now we know the UN this Yuan's of 0 former Compaq said and if you remember the corollary of the perturbation theory warns that London started with the complex said again uniform time of exists so the solutions corresponding to this day data have uniform time exists comparing exist and until and at times he 0 which is uniform in now and this why 1 should be there it's a typo John Niland take a critical element In write it in this way so that I can tell you what the solution corresponding to this things In terms of the solution corresponding to that so as long as I'm before the final time of existence we get there the UN modulated in the reciprocal way give me a critical Of course this is just the uniqueness in the coach because it it will be a solution and at times 0 it's exactly this and this equals that and the solution with that Baker will "quotation mark the system that's all the time
of existence bigger than senior but at least I'm sorry it is bigger than the not so as long as this is smaller than not this exists so this exists so this must be less than 1 because 1 is the last time for which the solution so therefore this is less than 1 whenever this is less than half and plugging the To me the ratio seen 0 Rolandvue don't and I sort of thing and what is the compactness gives us the uniform at the time of existence and land you just this is how the program works so we have a law about the next thing that we prove is that the compactness shows also that we have fine and time of existence we have the the inverted called picture but but but but but but but but but the solution is there I decided to simply that's what we're going to because of the compact hatchback if it lives in 2005 finally has to concentrate at OK so highly proves this the 1st thing he in which many humans is a property of finance piece of property so we have data which is controlled In and end of we know that at times 0 the tale is a certain amount small for all further declines this thing is small and the way it did you do it this EU truncates your solution to look like the other 1 tale and you will coincide with the outside by flying at speeds of propagation and the 1 the like this you make it completely small my feeling it inside the hall here in the studio question the killer was clearly like square near his home this from hardly equal what so it is always control the the people you knew want to take care of the capital the women differentiate the kind of that's all I'm doing but it's important to note that I have this is hardly turn always at my disposal because that's controlled by the H 1 . bigger than recalled to 3 "quotation mark this I will be used for be using it constantly so good thing to keep so we always have that in this can be made small and therefore this can all so now to to carry out this proved the 1st step 1st step is to show them this limited testing the goes to 1 it here "quotation mark so how do you do that this is nothing more than the compact all of this city model modulations of this In collection of functions together with the fact that the land that tends to infinity assistance to what we saw that the land that is to invent this status 1 because is bigger than 1 were 1 state and that ghost infinity assistance to and what happens when you work look at the tail lying idle land of pushes the tale outside and because of the compactness the thing goes to look at the date the unit from small from compact it's a lesson this limit is always now I have to see
what do I do with this thing the X of the overland this is what I have to 1st control to be able to get the point is to expire so because this expression at time s was small I can use fine instead of propagation going backwards in time and I get that this thing is small for any and that's the start something small the backwards in time and then you will have to be small and then there the point is that this implies that this guy you always have to be made this is they don't remember founded this thing will go to infinity and and the 1st sequence but then eventually this will be contained in that and goes to 1 this is going to infinity outside of infinite then get any belong that they want and then I let SB 0 here In the past the limit in and and get that for any ball this but they can't feed because life the solution was crosses the critical elements of it's space-time norms in champions so that already shows me that acts of the oral the it's bound and then for some sequence standing to 1 I have an export so from the center despite the simple properties of find speed of no once they found the center for an large this is going to music expire and for any not fixed for an enlarged Is this inclusion holds for any
at all for for an hour not
that may depend anything and with this I see by the inclusion of the for any as this thing is for any international force when that is support of looking at the level as the has to be contained in the ballroom is 1 minds expired so the picture is really it in the final and blocked the next thing is a fundamental property of critical now the energy critical wave equation has 1 more constant motion and that's the moment OK In less than this fact for any the missing quality and how he checked out by the difference between the using 0 by the equally OK In and the next property and this is a crucial property for he said if you have a critical element and you prepare its proper you make the land the continues even the classified ads are effort is infinitely make inbounded the ball focuses for them and then you you can prove that the momentum has to be easier so that means that in a way that compact object is not moving OK so this is something to be proud I think have enough time to sketch the proof the proof of this is that is not short I'm completely and so we will exploit the Lorenson Aryans because Lorentz transformations played a role so I assume that momentum is non-zero then 1 component of the moment amusement 0 we might as well assume that this is "quotation mark the 1st component is but if it's negative real change in a sign of X 1 2 minors so we can always assumed that his fostered lessors consider the the blast and let's say at the bus is 1 which is without loss of generality and we reset the dissolution so that the 6 bar is now the origin they expire that we found in the previous OK now it's more convenient here too consider you will 1 plus the and the team equals 0 tip of that hat ,comma so let's make it this peak with but but but OK and now because it's the 1st component of the moment and as long as there we do the Lawrence transformed our solution in the In 1st according to direct With the parameter L that whether it used to be very small so we have our new solution and then it's easy to see that this bill solves a nonlinear wave equation for a time that's bigger than mine mishaps and that this support he still inside the car look at this is just a simple geometric act now what's the point of view the point is that when we look at this solution let's say at times is minus 1 the phone before Ellis small world that learn to look at elevation so now we're going to look at the derivative of the energy as a function well as a ghost the the point is that there is a derivative of the energy is the negative the moment look at this a general strike in the Lawrence transform is furnishing that in the Florence parameters and again the negative of the moon and so but now this is negative and then there the other thing that happens it is that the the gradient for some Ltd 0 is remains small now what why don't we just have to pick up some teasing you because of we calculated by during the integration in space and time in which we can and do the Lorentz transformation space time intervals you can undo within the industry ancient anyone was small the other 1 is small and then you can pick a time at which it is small because if the integral small or the average of into the small there is a time in which to small you can't but then this has the energy smaller than the energy of W. gradients smaller than the gradient of those for L small and therefore exist globally as captain but that contradicts his support contradicts word lives and that's the 1 we gained by the energy being strictly you see the energy for falls from the energy becomes less than the critical level In the momentum is 0 then you can decrease the energy by doing Lawrence transfer and therefore you reach a so the only thing that could happen is the momentum is To the property now in the infinite time the case that calculation is that a bit more complicated but the principle is the same so I'll skip the infinite but you see that we're building properties of this critical elements so that we would prove the rigidity theorem we have more things at Canada's Paul that's the the story OK so I think for today where we can stop here and continue on Friday into the future and my