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3/7 The energy critical wave equation
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Title  3/7 The energy critical wave equation 
Title of Series  Leçons Hadamard 2016  The energy critical wave equation 
Part Number  3 
Number of Parts  7 
Author 
Kenig, Carlos

License 
CC Attribution 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/20477 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2016 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  The theory of nonlinear dispersive equations has seen a tremendous development in the last 35 years. The initial works studied the behavior of special solutions such as traveling waves and solitons. Then, there was a systematic study of the wellposedness theory (in the sense of Hadamard) using extensively tools from harmonic analysis. This yielded many optimal results on the shorttime wellposedness and small data global wellposedness of many classical problems. The last 25 years have seen a lot of interest in the study, for nonlinear dispersive equations, of the longtime behavior of solutions, for large data. Issues like blowup, global existence, scattering and longtime asymptotic behavior have come to the forefront, especially in critical problems. In these lectures we will concentrate on the energy critical nonlinear wave equation, in the focusing case. The dynamics in the defocusing case were studied extensively in the period 19902000, culminating in the result that all large data in the energy space yield global solutions which scatter. The focusing case is very different since one can have finite time blowup, even for solutions which remain bounded in the energy norm, and solutions which exist and remain bounded in the energy norm for all time, but do not scatter, for instance traveling wave solutions, and other fascinating nonlinear phenomena. In these lectures I will explain the progress in the last 10 years, in the program of obtaining a complete understanding of the dynamics of solutions which remain bounded in the energy space. This has recently led to a proof of soliton resolution, in the nonradial case, along a wellchosen sequence of times. This will be one of the highlights of the lectures. It is hoped that the results obtained for this equation will be a model for what to strive for in the study of other critical nonlinear dispersive equations. 
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00:02
I have been so you have to be the workers so after the preliminary discussion will further properties of this critical elements we see that there is a contradiction comes In this grand scheme from this city filled the theory says that if you have a solution below the energy of of WTO was gradient is below that 1 W which has momentum 0 has the compactness property where the excellent land continues to land dispositive and if the pluses we have this lower abound in this support property the pluses is finished we can assume that the land is bounded from below Texas 0 0 in London viewers the conclusion is that the EU is so this is a complicated description of the 0 function would act again so we see that the critical element we will the winner we were able to manufacture a critical element which had all this property but there also were able to prove that the critical elements had possibly managing so it could be this 0 function so the critical element couldn't have existed and since he couldn't have existed the film at all so that's and there was a scheme to the only question tonight so will not on Florida the so this is the largest city in the moment that news is the founded in May it we're not saying that it is bounded above the consists onesided the 1st is bounded from below implants such about this it's going to go to affinity the has to be no but that of course in the proof we will see the indicated the blast was infinitely we have to have a bomb from both above them and that's part of the probe it's not the assumption could another other question story so it again you know the proof or something like this cannot be done quickly so you have to be patient care so indicated he will we will do 1st decades when the pluses blessing and then when the case when the pluses fine OK "quotation mark so we start with the
03:32
case the blast His blessings and then we need said and it will be at the preparation would take a cut of functional fee which is why reasonable rate 1 In 0 outside the ballroom is still In this case led by our legal feasible and then there's another object that the initiative which is X but X has to be scaled so we multiplied when Felix of so we have to turn context because the groups too much what about Texas needs "quotation mark In now we will call the remainder are of capital are the integral for logics of all of these quantities quantities are all controlled by the energy basically we have in 6 more Wariner 3 controlled by the green 2 and we have to square the wreck square which because when dimensions 3 hard this inequalities controlled by granting script so all of this the quantities have the same directions and now as I had mentioned that at some point I think maybe the 1st day In a lipstick equations there's this city of wellknown identity before that the House approval that certain things have to be seen In this hour hyperbolic and some injured look at today's meaningless and there are more than 1 because we have space and time "quotation mark so the 1st identity this is that there is no you should remember that the city is basically acts and she is basically 1 truncated OK so the 1st identity is that that the Italian of cried BTU he is equal to the minus city have plus 1 and then there's the remain there because we've truncated In the remainder is controlled in this OK not at all the weekend to just write X cried using to you because that's a divergent but the year were convergence then the identity would be that this derivative equals that and there's no way OK so this is just a way to make a divergent integral stores and the 2nd 1 is that you didn't heal is practically EU squared minus values square policies to 6 but the whole of R and the important thing to notice is that this tool right hand side are roughly the same but with different qualifications so that this information is combined because the confessions so this look at Louisiana and then the 3rd 1 in some form we've already seen if you remember we had as proof that the critical elements have to have a moment and you and the reason was because if we performed Lorentz transformation we could decrease the energy and if the momentum was not here and that is basically words reflected in his identity that they accept that at the multiplied the density of energy by X and differentiation you get the moment the negative "quotation mark during the so I have to use identity I know that nobody can remember them but they will trust me that using them correctly came and went to heaven knows who can verify that you trust was not misplaced but so we started with the proof and now we're going to go back to our variation students can irrational estimates playing role I start out with a data which is nonzero but which verifies our size assumptions and then we know that the energy possible then in the hole the role of existence insists the plant is infinitely easier for all the positive this is controlled by the the investors control below by excoriating and therefore if I added square I have that blends CUT so basically for each team it that I have bounced from above and from you for various quantities "quotation mark "quotation mark In the next thing is OK I have to use the compactness of this K unaware that can't show that it is 0 because scientists nonzero solutions so the compactness of the Kerry tells that tales are uniforms this is where the Compaq said the functions in altitude of these tales are all uniformly small show this translates because this is what a scaling parameter is uniformly bounded from below In the saying that given Absalon I can find not such that all of these guys Our smaller epsilon times the energy the energy is a convenient positive numbers to put on the Net so it passed along the side degrees outside the ball around exit the of the there's not much energy we this thing outside the ball there's not much and for that and this is where we use the compactness and affected land based strictly on the from being over possible cost the focus of his other Haitian interim so this is a very full
10:47
slide let's try to with the existing that the 1st Lehmann said therefore excellence all this CD is there a constant is an absolute calls so it actually small an hour is very big I can find that the the said that the 0 is controlled by seek times are and foreigners 0 lessons here less than 2 0 0 this is below are mine and Donald of such and at the 0 it is exactly OK so how do we think about this remember that 1 of our normalization of the effects of 0 0 Eleven those 1 OK that was in a state of shock T equals 0 growth this ratio said least than this now I gave following the remember now that acts of steel and continues that was 1 of my assumption so I and like hit are minors are 0 actually the 1st stop and that's the easy way OK in the statement of the land mines that I hit the the 1st time below sea times are for some constants you could hit that 1st thought that the threshold of our mothers are 0 eventually bye seeking this has to be approved but that's the at least with the limb assessed OK now let me just say that if we were dealing with the radio suppose that we were in the running and case that X of the wouldbe always 0 and then there this already the OK but we're not in the range of tastes so we have to do 1 extras In the extra thing has to involve somehow the moment because that's what course the momentum of any radio function is 0 addition remember that's independent any equation is a radial function pairing of functions in multiplied in the gradient of 1 times the other agency look at the 2nd Lamont says that for actions small are large then this the 0 In fact has rebounded from below by wondering which are Weirich OK so what is the 1st one Davis about from above the 2nd uses abound from below and another from below and above her above are not compatible for additional 0 and that's why we get the contradiction is why no such function can exist OK right now 1 erection is not small and it's eh so is all in this together some of these 2 lemons the first one is a little bit easier than the 2nd 1 cup that neither 1 of them is at the heart of provided the knowhow to combine this really OK so as to the in the 1st one the question that isn't true and then I never up to see times are I never reached the value our minds you can't and we will see but that's important but if my memory not true I never reached this value In that required to OK so now I look at z Zia which is the sum of the 2 quantities in the 1st 2 and now it is some is what works well in 3 the In hired the you have to put coefficients In this is because of how the different terms I waited in the Maryland 2 so it was sound and let's see what the derivatives it turns out that it is combined To me and then the air OK plants with a combined now what do we know about this spot well what we know from our allies regional estimates is that these quantities courses so that his certain reported so this whole thing it is basically minus the image that's what we know from our radiation and but we have to deal with the no let's understand what happens for x larger than are because this error Call it gives us an integration over X bigger than us OK index is based R and P is between 0 IN CAR this thing is bounded from below by our Xerox along by the triangle and pull right because this is bigger than are minors are minors are 0 my conceptual so you get Argerich so this whole thing will will be bigger than ours Europol vegetables olive X is bigger than are my a regional integration is contained on where this is bigger than that are 0 Absalon but the compactness gave me that they're integral over that whole region of integration is small so this error it bounded by West so that the error cannot cancel what we have on the means that's the conclusion so the theory that there is negative as that of the size of seasons because this guy cannot reach to Council was this it gives because actually small now I'm
18:06
going to give up .period westbound on Wednesday but down from about and then I combine the 2 things and get them what is the point was bound from above is OK In here let let's look at the largest that the 1st 3rd the large thing is X right excess but bounded by our and the rest is just simply bounded by the United but because of coefficients and the 2nd term it's the same because I multiply and divide by the length of X and anew over the length X by hardly is again bound by the end and so I also get hard currency and the 2nd so I all in all I get the disease bounded by OK now I integration between 0 and PCR are and I use the fundamental thing so I get this for integrating Zia prime and From the bounds at equals 0 and an and the equals so I did this is bounded by that but if I chose are I Josie so large that c Tennessee filled that is bigger than to see 1 E I get the contract so I had to have reached see for this large so that's an Islamist group the other 1 it it's a bit more complicated and I remember that this is the 1 that's crucial in the numbering case because it crucial in and radio case that momentum has to come and so they we're going to look at 4 0 less than the 0 the function why why it's so far out of the which is the 1 where when you take the derivative used at the moment all minus the moment OK that's where you use this so let let's look at what it said we know that 4 p less than the 0 the Overland of the is less than our minds are 0 Mappsville alone so if X is bigger than this whole Qantas is bigger than Avenue 2 days as the same as before now the momentum is 0 so the main terminal derivative of this thing museums but if you recall the derivative of this thing is minus the moment if the momentum 0 then that's 0 and so we just get the error In the error because that and for Expedia bigger than on his it is big is controlled by R R and R is controlled by Ipsos eh so therefore this whole thing is less than the length of the interim which is keen epsilon so that's money bound and this difference and now we're seeing them this bound gives problems so the 1st thing in the rebound is why you OK so why are of 0 all I'm going to do it splits the region of integration In X less than are 0 and that's bigger than ours In that part of X less than a 0 the size of our bounded by a 0 and the rest is bounded by the get this point In the part where X is bigger than are sighs of R X is bounded by X 4 it's less than 2 and 0 otherwise so I replaced the size alive are and what they get is an integral outside our 0 action prepare so the demand of this I keep and this outside our 0 epsilon by the definition of little are is Bob and compactness is bounded by E. actually but this is my for 1 hour so I get above from above not gonna get a bounce from below 0 for why are of the 0 and there will be the balance of whom were "quotation mark so for why are of the 0 I stood the reader of integration in this region and is called In this region will necessarily access to be bigger than than are because forex bigger than our this is bigger "quotation mark so I can replace our rights by X because that's what this is for excellence so I have this now in this region the PCI is always bothered by and by the compactness I get absolutely so that's my bound for for the taking and I've got this part inside so this is what I'm dealing with note in the part in science I add and subtract minus and 0 2 0 in London and very added and subtracted remember that this is the thing that's exactly in size our minds are 0 so I know exactly what the size of from below "quotation mark this now I'm just left with the density of energy but I don't quite get the energy because I'm only a integrating on his region so I added that the parked outside to get the full energy so the Bild is that this is bigger than this from the contribution of this and the whole interval that my
25:21
nose the part that I added which is in the region outside our 0 Rachal and so is controlled by absolutely unique and this guy is still our minds as you can In the end of this guy In here the absolute ideal this is less than hour 0 bye the way chose the regional integration and the rest is the density of energy and so I guess and you know the negative signs don't matter because we know that the energy anyway it's about from above and bounded from below yes I usually inherited was not no I'm not I'm not but that bounded from above because I have a minus sign and so they may get all the time I got too "quotation mark show my todo control of this from below is all of that and if you look at the different make are larger and Epson's small so this doesn't hurt because of the actions or small this doesn't hurt because the new thing the are large the actual small tells me that this term does hurt me and they are large tells me that this doesn't hurt me and so from below by are 4 where was he that you have to express its usual around the world in which case you see it hit the ground the you flight would add about from above I'm going to get the band from his actual money from above and since the moment from delivering a negative side the so I put the absolute value and the integral I assume that it it's useful to moon because they are absolute values outside now I put it in science because the interval that's a lot of the integral is less than or equal to the interval of the absolute value than the absolute value of this is bounded by In the absolute value of each 1 of these terms is controlled by the INS has said that he should and I again and again certainly didn't think that but it's good to keep militants have created that applicant so go on and on and seriously by taking I allotted Ansel small is very easy to see that I can in the bone from up by EEI before and now I put all my balance together and the point is that on the right they had the absolute because in the momentum was 0 the southern derivative have the bound by and therefore Pinault is less than a traditional over he are over the age of at and that's a statement look at so that takes care of the cave are of people plus an equal and then we regard that as an easy game In our angle to the case the plot final and then we know about the support and renewal of the lower bound and in all of the movement OK Of course if the plasticizer they can risk to make it 1 this 1 risking so it's easier to write things with the plus equal OK so the 1st step is to prove that land that not only is bigger than constantly over 1 wins the but is also small the constant over 1 the latter forced to reconstruct b of the size of 1 of the women and this is a big step the fact that you can proved that and this is the last time we will use the very enlightened once we get to that we've spent all our real money with him so we have to find something else so all we assume not and then you have a sequence sending to 1 such that distancing thing and we will see that this is absurd cancel the contract you know I look at 0 p is the same object that I had before but they don't need to 20 why don't they need to truncate because of the support I have combat support and so X is bound to I can make "quotation mark so what's the prime then it's clean for it's the same as before when others will remain so by the original estimates again is a prime now is bounded by minus CE for all then the next thing I would say it is at the limit of the of the passed the tends the 1 is 0 why is that well let's look at the 1st part exit smaller than 1 minus the seoul exit the disk gives a woman's to him and the rest is bounded by the energy I don't have any and so that tends to see the 2nd part of the multiplied in the light absolute value effects as Absolut Alexis still bounded by 1 modernist eh and then you over X you to buy hardly controlled him so this is true and their derivatives is bounded from below sold by the fundamental theorem I get this the city of tape is bigger than concentrations 1 and I would see no then days in this final compact and that's where the contradictions during I do
32:39
that so we shown Aziz of 1 minus the intends to 0 and this is a contradiction with the suggestion that had to be Q so now going to explode the moment 0 remember in the definition of the but for free I can put explicity it's a 10 over land can because the momentum is "quotation mark and I have the division by 1 might instead have the division of women so now they're sick a few steps but they're not too bad but the sipping that somebody give me an epsilon and now I'm going to look at the end of this terms in the region where X plus acted in overland appeared it's less than epsilon times 1 minus 2 so for the 1st time fired and this is absent unusual minus the Oregon this back for the next spots and multiply and divide by experts axle in Overland and I can apply hardly at any oranges and that's where the term used in the same but for fixed the I can apply hardly index where would the origin wherever I choose it to be I choose it to be at minus excellent him 10 so then what we have is back so this part of the integral divided by 1 minus PEN give me so it's what I ate arming the good situation now I have to say that the other party is not too large 10 so that is so in order to do that I 1st have to control X of piano Rolando T. OK and I claim they have to be less than twice 1 month really twice sending to not seek kind to 2 look at why is that In this is not true they intersections of the intersection of this 2 walls is empty trying and therefore this interval Over here because this is supported in here and says this doesn't meet the support Bentonville is now what happens on the other party the other part this and that's the compliment In here I can just be a grayscale things and change variables and I get but remember this object is In an integrating outside something and my assumption is that land at the end times 1 minus10 goes to infinity so what happens instance it goes to 0 show all of all the Gradient square integrates to something goes to 0 this cannot be because they're my solution would be 0 and I'm assuming that he has a finally 10 books it's not zerosum therefore X Land of piano Overland of the is indeed smaller than 2 times 1 OK so I got the size of X of the universe let stay so now I'm I'm going to this is the quantity have than and of course the X is smaller than 1 women's team and X 0 the Rolando stands less than 2 times 1 minus the and so this hope Qantas is less than 3 when a device that 1 might and they have them and now this again doses 0 by the by the fact that this goes to infinity and the compact and other terrorist handle and then I get back to the contradiction than 0 the handover oneliners the when I saw that it was bound from so at the end of the assumption that Lambeth Hinton's 1 runs the intended to infinity I could not hold so and that the end is bounded by constant over 1 ministry and landed and is of the size of 1 or 1 minus Hamilton so at this stage of the game then we
38:08
have found that landed 15 is the size of 1 or 1 minister and believe me at this point you ran out of the usage of ruling so you have to do something else so the 1st step is to show that because of that by some general principle where in the excess tea can be gotten rid of and you can replace the land precisely by 1 minus the and this is a compact option "quotation mark OK and this is a very simple argument I don't think I want to go through it it's just a few lines and not much to do to do so please believe me that now we can assume that X of the 0 in the land of the equals 1 Over 1 minus 2 I am now so now we are in the south we have a family of solutions which have the combat out we have a solution which has a compact and profittaking but this scaling parameter is 1 where 1 and they have to show that this cannot happen and this is somehow a crucial point in the future it's fact crucial all the way up to the soliton and this is always a very important step In critical problems to be able to rule out some similar objects but here we are not ruling a solution which is exactly some similar but 1 which is so similar up compact 10 note so now what do we do but this stage 1 may be stuck and the pope this thing here is as as I said in the 1st lecture and there's a lot of elliptic theory now we go to a part which is really parable In here so we're think of the cell similar case as if we were dealing with the parabolic stick to it and when you're dealing with parabolic situations that you want to understand the blowup introduced of similar event that's the method that was pioneered by Indian corn and it plays a role he can't so we go to that the somehow introduce a similar event and this was introduced by gains call in the following the case Maryland I used it in the in the new wave equation case but in a subcritical regime and the effects of their their work the worse in the power is smaller than the conformally invariant power which is strictly small the energy critic Path and somehow all this this formalism words differently the lower above this critical part so what you do is you continue function W so now a new variables will be Y and N S why is X over 1 minus the and as is the longest of 1 woman and you introduce W S 1 minus the to the onehalf you of Eckstein In in the Y as their roles it's like that and is the final for s between 0 and infinity can the support is always in the bowling greens 1 because X or 1 ministry was less than 1 and support and now for technical reasons we have to introduce a parameter does and the technical reasons as sources of you buying things were multiplied when you want so introduces regularizing parameter downtown which corresponds to looking at the solution this shift there in time so now why is actually 1 blows dealt the ministry and S is long over 1 or 1 because of the minus the NW while as Delta is this the onehalf and it wouldn't be a bad and now this stubbornly is not defined up to plus infinity is apt to define at 2 local 1 without and when Delta Blues 0 that gets bigger and bigger and the support of this thing if you do the calculation is not always within 1 month that the city never get to Delta equal to 1 and that allows us to do calculations near Y equals 1 there would not be allowed otherwise 2 the next thing as well as the equation
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for both W and W Delta value calculated the queen and it turns out that it's some kind of a nonlinear wave equation but I not so surprising there's a power 5 vessels surprising you get some extra terms In the DVDs that's the look the harmful 1st but in the end they helped and and lipstick pirates very where overall was limited role the right "quotation mark it is coefficient role then generated wide equals 1 so this equation no is an agenda to the point so this is the price you pay now we get degenerative take equation instead of the ordinary the plant soon and we get this this out this 1 is a law on terms and this 1 isn't lower or total will think too much about them that we had extra term awarded to have see With the best what does that do not that few comments about about this and that the wages if you if like and if you analyze it you see that can be written Raul 1 overall burdens and then the identity is ranked 1 matrix widens wine please so if why is strictly smaller than 1 I miners that rent 1 matrix is amended me because I don't reach the 1 but s widens to water and losing 1 direction some of this so this is an elliptic equation is smooth coefficients for wider and 1 about the gender that way to OK so what we're going to do now is our universe is W and we're going try to compute what are the formulas in the W union union and see if we're going to be able to kill this double hope and that's how we're going to kill ourselves in that's the plan for the 1st this summer if there is anything remarks W 0 at the boundary because you will hear about the remember you when x equals 1 minus the is 0 so X over 1 minus these why y equals 1 W's and that it is extremely important if you can do it well you wish you were here because we had the support property in the inverted cone remember that guy From the compactness you get that and they were restrict motion when the NBH 1 so that's why was only 0 so the . 1 sense is in the sense of tracing sensible space so so that's this is the correct technical In the gradient squares control because when you change variables in the widely used in that's why you have the factor 1 minus the to the 1 half of the keeps the home he keeps scale the W 6 the same thing and this 1 is a combination of things and if you check it it's in the system you go to see the any of the no uniformly in whenever there is a Delta independence will show within in the picture In this is uniform and in and now there another called the that you get which is for free which is W squared is inseparable again is this highpower away and that's the heart of the inequality where instead said instead of being at the point it's at the site OK so this is another hardly so that all of these bounds a uniform and now we introduced the core and they had the right for this equation OK
49:35
so this this expression and now you see what the purpose of the bill the belt becoming means there sins why is supported in wide less than 1 miner's Delta this expression converge because word that this wakes becomes the generous my solution "quotation mark program so this is on its quantity I'm allowed to write at least for them port How do I decided that this is the energy and you multiplied by Esther riveted W In integrated their parts in this kind times that the role and this is you In the same way you reduce the energy for the way quit 10 now there in the thing is that this is not an energy that's because of the extra terms in St but OK with only worry this is live better for there until the energy it's a positive Kwan OK so this this is a miracle the director of the energy is the sense the energies increase by the way ended when the powers that are smaller than the conformal power which is 3 in this case the energy is decreasing Inc but here is increased and again OK I have is now enormous power of 3 have just 1 but that they still converges because and that the double only tests 0 or near where near equal 1 and therefore it's as the at the it's Sicilian it also so I can write so it was my 1st for me my 2nd formula that calculates so this is the the route India the 2nd formula calculates the primitive of and it is just to tell you how to get there you look at the equation instead of multiplying by W Times role you multiply by W and you integrate but computer and so this is our formula and of course there's the formulas may look awful and at 1st but they they grow and how can you get used to it In fact the route and that the final thing it that that about I'm approaching the final time existence which is global water without them and I look at my my energy In the limit I get exactly the usual continued you must and this is just a calculation is nothing more than a cult but now we combine 1 and 2 and 3 and we obtain immediately In this energy is bounded from above right because at the the last time it is boundless so and it's increasing so before it's and this is a very nontrivial by the original CDs with the naked eye From the independence movement the obstacles such as ways to deal with the USled Hutu you viewers to who was on his way to the wait somewhere you so everything called they compensate they compensated with a shrinking so it's a calculation that you termites yet it is surprising at 1st but is that remembers the wait outside is the 1 minor stick to the 1 in the the 2 good books and you remember what you did in realization you you have to do that now look at the other question is OK so the Internet and now I'm going to tell you our 1st improved 1st improvement in this if it were a very many you will get a abound 1 over them From the way supporters Maine because the support it ended wall reduces oneliners Delta so that's how far you can go in white 1 oneliners square and what I'm saying here is that that's a bad advice there is an improvement and is an improvement that's so strong that you get along not just the fact that the OK How do you get there in the improvements welcome source here the really this is a truly parabolic thinking you're going to have to find the right this function To put into the equation to and in the most series of parabolic equation is that Nash's theory logarithms OK and so on it is workers at identities so what this is is that I passed my eyes equation by with W times longer you can't said and then you get there is not a very nice formula it is seen here and in the pot inside his bracket there's no way there's only the long way Here we have this way so the long waiting here gives you the law where the by and so when we integrate the goes away and we had 1 about an hour here when we integrated remember this is a bound
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for the interview and you to 1 it's not for ETS is through the Internet by integrating you make things better and so this is the
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1st time this is not good and now we look at the others In this 1 is going this 1 this is good decisions by and in this 1 adjusted use caution and then you contempt for the area controlled by look and I start cranking a machine island improvements now it along with nowhere near the square meters along for Warner this was the same OK so let me explain why this is true In
58:15
that sits in this formula let me put this terror on the other side this there doesn't because the energy is bounded from above nevertheless the monotonicity of the energy and the fact that they know what it's like so this Thurman OK this what do I do with them whether W term I have no trouble because of my hardly equal the number W. squared supports up to 1 miner's y squared square so this 1 is hard and here well I have a dangerous storm but they use kosher shortened use with previous bomb and I guess in a square root of of officials cohesion and then and now I'm going to this terms my and my first one widget somehow lose all Hassan the right side so I can ignore it I just throw it away as the roadside and the other ones are fine by Gucci Schwartz and what they just prior and the fact that on WE can put a who can answer to this is just such a utility that something different to do something to practice as well as to which is going to vote you so this is that they got their notorious structure of some similar yeah so is precisely that and you get information so you have to know what is of course so the 1st thing you get and the 2nd thing the 2nd there follows from this 1 because if you remember the only negative terming the energy it s the 1 win and the energy Is the bounded from above and so it is integrate before the Thurmond the energy that's bad that's negative is the 1 that comes from again now I now
1:00:56
going to go and improve this boundary what is interesting here is there and the length of the it's log 1 but my bounds is about onehalf of the 1 register so I've been able to show that this is going to 0 in some way From this battle so how do that well I just think express this that's the difference of the energy at and this time and at this time the balance from above at this time I just away and then develop from below I use the miners along and the previous steps OK 2 months old and
1:01:55
convincing but now this is a very powerful unions because Of the course the correlated Coraline that deserve an answer buyers of Delta which is between 1 and law 1 over downtown to the threefourths such that on something of line 1 8 again bound by anyone over warning 12 so this lemma follows from this 1 just by pigeonhole argument I splits the the long intervals into smaller and the roles and come home when you can have and then they the Suns have computations here I'd do
1:02:52
it in distance and the roles of this letting the number is there's life lines alliances onehalf and I again I'll tell you where I'm having I'm having to try to prove that govern these independent of us they can do some manipulation to make this independent unless if I make that DES through 0 that's how you make it independent West and that's what I'm trying to help him and tried to make the hideous 0 OK new stage that sense in this and the role I guess that very good but I don't where I think the average until I get there the bound because interval long therefore I being defined sequence in which discourses the use of relations between 1 AM and for those of you the eye and the number of the the important thing is that the "quotation mark along in 3rd while I get the ball on the coast so now you know a little bit of we're going to eventually have to take them out there willing to 0 for all of this to work and so I have to get the a convergence and for that they use the compactness because so I get all my things are compact and so when take Delta's Jay's going to 0 I can get compactness so I can find the sequence Delta Jerry going to 0 says that this guy's converters the W. star and they're W. star is in fact independently but the next thing about that is that it is OK is independent left but it cannot be 0 because if it is 0 it's a limiting point of 1 of the things in the compact set my original compacts and that means that you is a case can be taken very small by modulation but then by the small date the theory a cannibal often find OK so this W. stock in all these years so the W fire words will have to so the same equation except that there are no answers is in because as independent and so what I get is that the generative that the key questions for the 1st time and I know the Government stars considerable NOW starting 0 on white equal to 1 nonaggression
1:06:04
crucial next .period is that there's some extra bounce for my governing style In addition to being a spy and is that this to objects of fine OK so this limiting object this 0 and the way in the air at priori I have no idea why would his interval before and that is the point is that because of the way we chose With don't test W why as Delta Jays and we had this uniform bounds on because of our formula what was the pier called the variable for so we get this thing that have uniformly bounded in In and therefore in the limited the W stars have this dominance but the devastated defendant so I can take away the integral to just get the bounding it's OK can this year capital and have contributed to the is
1:07:21
that the 2nd and they're all where there is convergence it's I'm in the wrong from the local existence focus a I
1:07:41
got this things and now I I conclude by using a unique continuing so this is where there's a connection between what presenting world 1 of the connections and other market because of Morris the 1st person to insist that 1 should study the continuation for equations without them and the qualifications and he did this in connection with and with the way the queen and so he and that the 1st person to actually be able to do something like what proposed was a column in in his work in the 30's and 40's and this Parliament's work that gives basically that this W. stories you I can't after after a lot of things so what we have to show now that if we have a W star soul into the general elliptic equation with 2 additional bombs it has to be easier for while less than 1 minus 8 0 0 I have a linear what with smooth coefficients With critical not by a wellknown argument used to Trollinger journalistic theory we can show that W. stars inbound was published diaries bound I can forget about the nominee editing and considered that a little to the 5 to be down to the 4th time that wet and called W 2 the 4th the and say I have abounded potential so now I have an elliptic equation were kind of analytic equation with abundant In columns theory that is that the if advantage in an open that they have to to be but so if I can show the W star is 0 nearer Y equals to 1 then and that propagates inside solarcell action his new wife so if they're giving you the actual profile of model the problem in slightly easier according to the flattening and so if a slap in the circle and I look at what migration looks like it looks like where are you now is the perpendicular direction and then have the general operate there and get some sign here that is wrong this is minus the OK but so as a type of miners have in the tangential variables disease I don't have any bitterness they have the real losses than I have seen and the star and then have started now I have
1:11:17
are no estimates on his W. star In this are crucial estimates it translated in this way and now doesn't a hattrick hero which is that this that and the the question can be this singular by changing are to be a square because as the square and if I do that and I called VW's terribly squares in the NBA Is that so I equation now becomes known generator and I know that the thing is 0 era but of course that doesn't allow me to say that the solution is there any extra information and the whole coaching the but my extra information so this In my extra information gives me this this thing is fine and then the generosity of the equation forces the in the sink the singularise equates to vanish on the mound and now I have the whole cautioned the vanishing and now I can use unique continuation cleaning this in the coach probably like to show that the W stories In that finishes approach the conservative can the long journey but that finishes the provisions look at so let me conclude with a this part of the of the cost With formulation that's a slightly different of this city alternative so supposedly averages less than the energy of government N they hold H 1 presented to enormous small then the solutions exist forever and scatters and if it is big it blows up in both directions and the dignity before our formulation was without the new 1 but there the formulation would be 1 and without the 1 I equivalent and this is very all In the 1st claims shows you why this is purely regional if the energy is less than the energy value than the gradient is less than the gradient of W in final the other woman happens and the same for about on this is you know playing with the variation it's because of this energy constraint OK so will leave this means that and now we're going to perceive yes he refused you he was awarded the rest of the year the communities around him no that up and this hypothesis this to us this to work with but he shot you begin your view you can you when you want it you get the she got from a human deals with this movement of the history of the existence of the rest of the world would be him but it would be wrong the I'm not saying that they that there are equal numbers but 1 a smaller if if the other ones start so now where we're headed next prove this soliton resolution In the Reagan tax cuts are next to task and that will be what we will do what until the end of month and then the 2nd part of the course is to prove soliton resolution in the for sequence you can be sure of that no so far I avoided being real but then took us through this more difficult thing you 1st have to do it in the so so for the 1st step is to start early give a general study of solutions which are tied to the launch of nations so that means that there they cease to exist in finding time by the norm remains pop and as I mentioned in the 1st lecture there's that many examples by now such hurricane and this is where the happens by concentrating so now I'm going to OK and I recall also that tied 1 rule it means that the norm actually goes to infinity but a priori there could be something that needed tape 1 replaced 2 if it's bounded on 1 sequence and bound but then immediately .period typed and we will show that in the ring case that cannot there isn't such a thing Donald mixes and Texas now the 1st thing I want to clarify is what this means to be applied to all and for that I introduced the notion of regular and single people conclusion it .period is call regular if for every Epsilon there is an hour that's independent of the that's the this thing becomes more so far this is the usual Laurent I'm adding the hardly known for safety so what that it means is that there is no commentary and display put and the block time near this point at this point can and we called In X not not regular if it's not it's not regular call it sink and now we S the set the singular .period "quotation mark now
1:18:29
let's 1st stated talking to the bed about it In this notion the theorem says this is the fair which indicate Emeril that there's always that at least 1 singular point if I'm tied to so that's what happened there was at least 1 singular .period but that there only finally many such and finally many depends on its energy home in the same energy about moreover you always have a week limit as you approach this is the final thought it is insisting instantly plus an interview stay away from each 1 of these finally many singular .period you're actually approaching strongly "quotation mark this is the real yeah I will explain the bit about the proof of this and in a few minutes to know that there's going to be a definition so Of course you for any sequence the end ,comma his weekly to some limited after subsequent because of the boundedness assumption the point of this 1st statement is the limit is independent of the "quotation mark but of course if you believe that there's only finally minis .period and that outside those points the limit is strong and that gives you the uniqueness of the week look at this as "quotation mark so the really important points are that there's only finally many of these points and away from the singular points this limited stock cooking so now let you will be as in fear leaving the solution which of the final time has this date that means you 1 so we called is need a regular part of you with the blast and the difference between you and me this singular part now the thing is that it's very easy to see the because of this strongly made away from its singular points the support of you miners a singular part it's in this inverted cones centered around each of the singular .period so let me give you a to draw pictures but so the support has to be there because everywhere that the bank cut here from here on the 2 solutions that are very very close so by finders fees of propagation there can't be anything except in this inverted cone in which they "quotation mark during so now that approved what we have had to use in off like shown the region to frontlines times where the but the redshirt Nova noted that in the William setting things up because I stop but the 1st time there is a singular point of the view real well you won't see run I still I mean that what is possible if 1 develops instead of the Koshy horizons of consumption but that in this new way of looking at things were stopping the 1st time there's a singular and then we just rest so that never happens because so the main point and approval for areas of following the yes there is a fixed number of Delta was such that if at this time the 0 this small OK that the pluses 1 Indians and the cluster the water and they have the cuts To this neighborhood this has to have and the reason for this basically is a small data I think the Delta wants to be there constant in this small lead the theory and then I make it a solution that equal fees and GeoCities the of TCU at time PC and that and then this small data theory gives me that that has a limited as you your approached equal to 1 because they were possible data and still fight it's been appropriation will immediately the this the point is that it is the varies it is very simple In the next point is a corresponding .period attitude look at way I do it outside in Estonia .period but a
1:24:44
corollary is the following if I have a singular point in this thing Hester remain bounded from below because if it were smaller I could use the previous level and and the combatants in H 1 result told means that there will be a regular now this already gives me this finally many singular points because it infinitely many I can approach like this because smaller and smaller as the ghost once and then I will get the if I get had to singular points I would like to get the bond of Cape Town's Delta 1 for this age 1 person to raw but the age when Crothall's enormous uniformly bounded from the the number of points King has to about To conclude this already tells me that this finally menacing and as you see here is the proof of this
1:26:00
so now I'm I'm going to start my preparation the proof was assaulted resolution thing in the region and so we do a little bit of gearshift and we're going to study some properties of linear gradient so for a solution of the linear way equation the linear energy in onehalf of plus gradient perceive square and so the density of the energy is then and recalled his identity the infighting at the interior of the energy does the same and as the spacetime versions of this 1 OK but this is of course just differentiation and this is what gives you the fact then the energies right because he we integrate this annex that's the narrative of the energy and they entered offered spatial materials is 0 the development the very clearly energy integral 0 so energyconservation falls so now I'm going to introduce for all no negative numbers saying they will take all the Algerian Energy which is that the integrated outside the light call displaced by this is terminal so the claim is that for all a bigger than revolted 0 this is the decreasing function of the port the possibility that increasing function of the 4th and that's the way you picture it is if I integrate a small steps I should get something smaller Of course this isn't really the justification for this claim because the functional and integrates changes with each team that's the memo monarch device to remember whether the increases at the no 1 can so how do we have to prove that this this Israeli troops we integrate this derivative of the energy on X bigger than the Jose say and have to show that this is bigger than required there on less than a and we invigorated by this formula integration by parts so what we get is that this difference at the time as 0 and 20 0 you can now do the integration by the parents ceremony time and what you get is this kind of thing and this quantity here is called the flux on this Terry will be important for us when we do this all the time the solution in the normal rate case but here it's important to us that appeals hearing gives us that 1 so because of his monotonicity I always had the following elements that exist and this is the kind of an interesting formerly from quantify how much it increases decreases In terms of the flux so they have the
1:30:27
approval out energy inequality because I have in the very 1st it says that it is a real solution linear wave equation in 3 and a isn't earning his number and then for all peoples of the world the negative this is bigger than that the roughly speaking it says that the outer energy even at 40 . 2 plus infinity of for the going to minus infinity there's not to see has a limit which is fix possibly number which is the know this isn't quite true because at but they are inside it's still to and we will add discussed what the differences in if you but let me prove this fact and as I mentioned it the 1st time this is something really dumb number could have have and the cost this 1 so how the suppose it they can as read solution of the linear waving our 3 then if you look at of To the RV of the any extend this to the oddly only 4 the less than 0 as is well known and it's also ordinary wave equation in 1 the some grievances and 1 fact and you can check it was nothing nothing but to OK and I will call on 0 items in 0 and F 1 the ones which she the qualities now they input they another important factor is that inequality In Remembrance says that squared the lower our squares is integral in 3 D but this surface of the measure is our square the are so what you get is that this thing is fine and because this thing is finding this says 0 is actually in each 1 because of course it would very gingerly 0 this season each 1 because the zeros in each 1 but it difference in the ah will get just the 0 of his original to ways OK so we have a data in each 1 of 2 for the wave equation in 1 now I look at the quantities in 0 1 and where I look at the the the and the the R minus the the thing and this parenthesis belongs b so Of course the sum of the squares of this too give me that "quotation mark nothing so because the sum of the squares give me this at least but until at least 1 of them is bigger than 1 4 0 OK and this is what we choose whether it happens with deported to hold fatigue so let's assume that this side equal to 1 that passes the importance of this 1 this and then I will prove that theory for the peoples were teenage now the wave
1:34:39
equation in London remember the 1 has keep plus the so the king minus the of the warnings we feel that so that way equation factors us to transport equations and 1 that's all I mean and because of that 4 thousand lower than are available at the town derivative of function I have 0 this function is constant so I have this guy's constant alone characters I am not going to do it for the lesson 0 and I started facing this it is the same as that by this the quality now I change variables they give to her and I think that this is less than or equal to that nation and using this inequality and this was bigger than the initial data have a natural In all use was that I had won the heat equation and therefore I have a constant thing characteristics and I could decide whether is for the force of new 14 at according to which of the 2 parts consumption dominates so it's the composition and tingling outgoing waves then have to decide whether England waves of more powerful than the that tells you to what the time direction the consortia concealment is not much interest what so now since we now we will use the following calculation the very often remember that I had this quantity here are age of and I little development I use the product rule I get that and developed the square and then there are the other 2 terms combined and then this 1 of course they can integrate when I get back so that it now you see what's the difference between this and precisely so this term it always smaller than that when this country so as a corollary on the lefthand side we can replace what we had by the outer because that's bigger the best and we always have that now you could ask for well why would you do something OK and that's the reason I I do this is because I don't really care about 4th because 40 going to infinity this guy will go to you With for solutions of the way questions will will make no country why is that because we have a businessperson bound to oppose the reasons infinity and insults wave equation then for large stymie of XT is bounded by 1 over the 3 pride and minus 100 tool is the decay rate England is 1 of the 2 is 1 of and so that the extra time he doesn't play role now at time 0 0 you cannot replace this and this is easy to see once you think of it because he chose your data the 0 2 the the Newtonian potential for are bigger than any it's a constant the wiring 4 up 0 1 OK and we want to be so what is the solution of the wave equation with this data for R & B given a plastic it is precisely 1 overall by unique because inventory potential source of the but plastic away from the know what happens to this guy when you when you calculate In on 1 or are limit you get 0 the out of on the lefthand side we would begin year Of course this quantity is known to but if you replace them by that it is 0 because our times on is 1 so OK so that's why we have to formulated in this way which is the way you prove it many can't prove something falls but you could eventually hit and would be a good idea now so you would ask what do you do in higher dimensions and there are corresponding inequalities in 102 and to do that you can you think about what you're doing here about what you're doing here in the following ways in which you are doing is doing the orthogonal projections To the complement of this onedimensional subspecies of each 1 cross too "quotation mark we are bigger than any given by 1 over our common so you take away the orthogonal projection to the banned substance onedimensional subsidies and then that's what you have not seen equal and then once you see this way higher dimensions there mortar to subtract and so in every all the mention there's an equality of the stifle attract more and more terms depending on the bench and then there is always a fine dimensional space that you subtract but the dimension increases when the dimension of the spacing and that inequalities still true with constant just using the but I also know around them you have to use 1 or administered but than other derivatives but notice that a few extra things that they want to do today the 1st it is an extra property the way queen so this is a proposed howling and solutions and not necessarily assuming that Israel in this In this I think I have parameters and the scale things In then I have that away from the surface of the light ,comma mm that approve this year 3 1st that we can assume that Islam is scaling parameters are all won by me scale now solution by go back to supported the which you can always do when and then the next thing you do is use stronger Corrigan's principle and that tells me that this thing is supported and so there integral is not just pretending to Europe this and that's the end of this program so you can do this in any other now how about this fact in even image you cannot prove like this but it is still true in winter soon will see proof maybe it was 6 so the meaning of this result is that morally for large time the energy of your solution is concentrating on the boundary you might call this would is going His away from the boundary line which is this does not conclusion look so now I'm
1:43:51
going to notice that the following things which is an anomaly released it is this personal property at nonzero solutions although linear wave equation unless the fold for all of the bigger than 0 or less than a week with you that there is an hour 8 consistent for all Altedia memory with 2 0 4 less than a week with the there's always outside their image "quotation mark are this isn't always true for the for the way it is true for for all infinite time :colon father we proved that we have the tools to prove it's very simple this is nonzero debate In this state this thing equals that because remember that my integration by parts this this went out at 0 minus the EU's with in 0 square so I 0 that versions of the same so this quantity is on 0 but since is no 0 no we can't give up a little bit and integral and find that it is bounded from below once we have that this thing is bounded from below our and corollary to our art out the energy faulty proves that for either team positive 414 negative this is bounded by and that's the stupid OK so too do the soliton resolution in the radial came the key tool is to extend this to solutions will not New Wave radio solutions and the normally equation in 3 which I'm not the solid deputy welcome to this as our next task this will last until the end problem 4 W. this way before was W if you think about it behaves like the Newtonian potential for large banks but this 1 or 1 plastic squared to the 1 that for large 1 Rex and wider than the Newtonian potential play a role in handling a case if there was something you're the closed at the time equal to 0 to the solution of to competency In the then find are by where you stop being dependent on imports still some trade unions here the 2 have to be aware "quotation mark I so the 1st task is to show that In China to do
1:47:29
this we will introduce of following this is a useful litigation with using the money it all the time suppose somebody gives me in his area 1 in each 1 Crevalle so radio and a number of armed to sports I'm going to call you 0 still the U 1 filled depending on our system are to be the solution where the you 1 until they just chop off by 0 that's just as to show I'm allowed to do that and the other 1 just by using you and the point of this engagement is the sequel the outer wrong is exactly the normal his truncated right and now we will have to use these things In the 1st proposition have not been approved but I just want to explain what this means he said if you didn't go well in time solution and says that for some are for both reported that the and negative the this limit 0 instead of being positive then there's not much that they can say about this then the salute the date that is in the compact to support them and of course if its compact and supported such a thing we will happen by the thinking are much larger than the support for it's minus the scales doubling his compact support and those are the 2 options so when Justice personal property doesn't holding in science it tells you something very specific about the In the of course this is a nontrivial proposition but they keep cool improving this nontrivial propositions is out there and you don't so we're continue on Monday thank you if by them week's analyst stop the the at the since time the review the book the cost of this sort of person you know something so that it is to go on and on and on and the union have so here we have a lot of money and you think of land to be 1 and here's what goes to infinity key North 1 over tea and so this should be planned in Genova 11 OK so that the world is a source of you that you know that the end is going to infinity and London and is 1 look at so that was a type other questions the 1 with him minus have more the losses through the result of a week in which the were on the and so on but this is the the president wants to problems him in the flower of they have to find another proof but it is and this is an important use plants is perfect the question and
00:00
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03:27
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32:38
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38:06
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49:31
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57:18
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1:02:50
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1:07:20
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1:11:13
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Theory
Number
Hypothesis
Frequency
Average
Uniqueness quantification
Theorem
Equation
Analytic continuation
Subtraction
Conservation of energy
Uniqueness quantification
Lemma (mathematics)
Neighbourhood (graph theory)
Horizon
Ring (mathematics)
Limit (category theory)
Existence
Estimator
Exclusive or
Computer animation
Film editing
Units of measurement
Limit of a function
Extension (kinesiology)
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Point (geometry)
Differential (mechanical device)
Interior (topology)
INTEGRAL
Multiplication sign
Image resolution
Mathematical singularity
Materialization (paranormal)
Food energy
Number
Summation
Derivation (linguistics)
Mathematics
Population density
Term (mathematics)
Wellformed formula
Negative number
Subtraction
Proof theory
Wave equation
Logical constant
Image resolution
Lemma (mathematics)
Point (geometry)
Gradient
Element (mathematics)
Food energy
Mortality rate
Functional (mathematics)
Proof theory
Category of being
Radical (chemistry)
Computer animation
Estimation
Function (mathematics)
Integration by parts
Mathematical singularity
Linearization
Equation
Theorem
Normal (geometry)
Conservation law
Identical particles
Resultant
Flux
Friction
1:30:24
Logical constant
Scaling (geometry)
Wave equation
Multiplication sign
Equaliser (mathematics)
Direction (geometry)
1 (number)
Parameter (computer programming)
Mereology
Food energy
Dimensional analysis
19 (number)
Derivation (linguistics)
Square number
Moving average
Proof theory
Wave equation
Product (category theory)
Real number
Infinity
Food energy
Variable (mathematics)
Measurement
Functional (mathematics)
Maxima and minima
Proof theory
Category of being
Arithmetic mean
Hausdorff dimension
Equation
Linearization
Resultant
Directed graph
Computer programming
Divisor
Real number
Characteristic polynomial
Calculation
Inequality (mathematics)
Rule of inference
Theory
Number
Term (mathematics)
Boundary value problem
Subtraction
Scaling (geometry)
Surface
Lemma (mathematics)
Projective plane
Dimensional analysis
Line (geometry)
Euler angles
Limit (category theory)
Wärmeleitungsgleichung
Inequality (mathematics)
Vector potential
Calculation
Summation
Computer animation
Estimation
Boundary value problem
Units of measurement
1:43:43
Point (geometry)
Sequel
State of matter
Scaling (geometry)
Wave equation
Nichtnewtonsche Flüssigkeit
Image resolution
Multiplication sign
Propositional formula
Insertion loss
Food energy
Number
Manysorted logic
Square number
Soliton
Physical system
Proof theory
Area
Scaling (geometry)
Image resolution
Lemma (mathematics)
Propositional formula
Infinity
Food energy
Limit (category theory)
Vector potential
Category of being
Proof theory
Computer animation
Integration by parts
Equation
Linearization
Right angle
Boundary value problem
Resultant
Directed graph