Merken

# 5/7 The energy critical wave equation

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00:04

beyond the the more I think at the technical level of the lectures community completely independent of his OK but they so about I may be just a few words to recall what we did in the previous section so what we consider this energy critical nonlinear wave equation In the focusing case and that the goal is to understand the I think takes a long time a single takes 4 large solutions and thus the general goal then I only I've heard of consumers so I started by a describing them as the work With the franc Meryl in which you proved what we call the new ground state conjecture so that the energy is smaller than the energy of the ground state which we called W which is innovative solutions to keep question In this then mineral energy solutions among Algeria's solutions then they there's a dichotomy is that H 1 or more of the data is smaller than the 1 of W have global existence and scattering both times directions and is this page 1 wrong is bigger you have a final time blowup in both India In the case of equality is ruled out by a variation of Arkansas so that was the 1st result that they described and in doing so were introduced this year concentration compactness rigidity theorem method that allows 1 to reduce the thing to proving that certain solutions with the compactness property the Nunavut exists OK and in that I want to point this out because it will be interesting from the point of view of my lecture Wednesday there's 1 crucial point was that there are no cell similar but blowup solutions which are compact and in the proof of that introduced have similar of course it's who can and we will see that revisited tomorrow but on Wednesday and then after that we started with the general study of type 2 solutions and so type to lob solutions resolutions which is ceased to exist and final time but have uniformly bounded Indian and as I'm saying this is a phenomenon which is very characteristic of energy critical situations and the 1st example were constructed but in a clear shot at the thoroughly in the three-dimensional and then there the Hills constructed solutions in the 4th dimension of the case engine division defies dimensions case for this kind of solution so we're talking about the money in the set of objects and then they reintroduced the soliton resolution conjecture which news a precise description of the and politics of solutions which remain dominant until the final In the energy space and they were description is there present poltical a solution looks like a fine summary scales traveling waves plus a radiation purposes solutions being equated with the term that rose to 0 the as tough time moves to the final tally and then the last thing that we did in the 1st sequence of elections was to prove this result for the Indians broken and approving the rail case boiled things down To a dynamical characterization of the only the radial elliptic solution which is the ground state and that was what we call the general energy argument we showed them all solutions of the nonlinear equations that exists for both positive and negative time creates a channel energy outside delightful less there are police scandal OK and from that we could prove this summer listen to so that's the summary of what we did before it's OK so now we continue so the the subject of this week's lecturers is the memory b worst the world "quotation mark the fact it was more investments OK so In the Redmond radial gains there some new difficulties arise because now we we have to deal with all the traveling with solutions to the crisis and we've proven already Minnesota in the last series that traveling wave solutions are precisely Lorentz transformations often elliptic but when where and on radio case there are many and it takes and there's no classification so it the raid case we have an explicit radio solution there's a formula for 1 0 or 1 excoriated over 6 the whole quantity square root so we have very specific information now there is general solutions to the elliptic equations have many flavors so as we know and they're not classified the so we have a very large and I said the traveling waves the 2nd thing is in an hour dynamic territory characterization of w we used the sort them out their energy equality for range solutions of the three-dimensional way now the corresponding Hassan fails in Hungary so this has it so a different approach this needed so so we believe that an approach that is purely based on dynamical properties of the traveling waves since this is such a ban will be enlarged set of objects is unrealistic so we're going to proceed in a different way focus so state the result for the movement of the describing and so we take a solution won't work that remains bound in in the

07:56

energy space so let's assume that 1st that the pluses fine In it s been a set of singular points so as singular point is a point where there is energy concentrate and we showed that this at the singular points is always more and the and finally in the case when the duo of governments fight so we fix an element In the among the singular said and the result is that the number J star which is bigger than to warrant a little radius our this radius has that and the role that it separates all the singular .period but staying away from the others Inc . 4 pence In the time sequence the analytical released to the plants and this is a well-chosen currency is not an arbitrary time sequence and Syria is the main difference between this result the review In the radial result we were able to prove this for all sequences of broken there's a well-chosen sequence of times so far In scales on the Jr which go to 0 faster than this and similar for special CJD and which are strictly inside the ball greatest peacetime minus the around Lake start for some data which is strictly smaller 1 and the directions LJ which a given as the limit of vector CJD and always peace plus-minus piano and noticed that because of this the limit after passing the 1st such sequences always will be fine as always strictly less than 1 and traveling waves such that within this ball we have in the composition of the solution equals this radiation terrorist fixed H 1 cross to turn plus sum of modulated solid tones so there's isn't traveling wave solutions but Maryland goes to and this traveling waves we don't see each other because of their 7 melody property of EPA OK so this is precisely the soliton resolution results along well-chosen sequence of times in the final book the question the standing on the other some of the the day started the presence of basic the 2 resolutely for this agency started another sequences times what what's more serious so this is the highest based on on history of the final thing and that is that there this whole thing be the hotel so I just separated the different than singular points of it so that they don't see that's the purpose of cocaine has questions "quotation mark now comes the other possibility which is that there an infinite time of existence then there's the radiation term which is a a solution to the linear equation that extracts the most scattering you can have in the so that's what we called the scattering profile and what that does is outside any final distance from but like all the difference between you and linear solution is 0 OK then there's also a numbers J. star now here where allowing the started the 0 and that would be the case with the solution actions In time sequence which is well-chosen going to infinity scales land in jail and which are much smaller than those of similar raids positions which are strictly inside the light directions L J for the traveling waves and traveling waves section the solution equals the linear solutions plus sum of modulated traveling waves class an error ghost an infinity and this traveling waves do not see each other because of their sovereign people open and still there's no Tareq Aziz only 1 infant "quotation mark so this is precisely what 1 would hope to prove course 1 we hope to prove it for all sequences stands end but for now we have it 4 1 sequence this what this wasn't optimistic state for now focus again so yet but let me just on here the last 2 years she ended the hit song is struggling with its labeling non-renewable commentary in the concentrate faster than that there they're dividing I mean that they can open but not as far as the wine and then like on the most infinity because he said he said he losses to have you know but and everything happens in certain societies like OK that's what you're saying right so I will if you remember what I did in the 1st part of the Lexus I proved that the soliton resolution in the Reagan carries 40 plus equals infinity sewing case and you you'll find it can 4 variation but the proof moral the same ones the scattering profile and this is a typo this should be a once we have this Catherine profiles of the 2 crews and more on the side OK so what we're going to begin doing now is in the show how to extract this Catherine profile that's will be our 1st step once we do that then we will the only work in the final Inc because the 2 things will be this but the extraction of scattering profile is a highly nontrivial thing Wilkins so this is work with declare Emeril at the beginning of the year and In maybe I should have said so this terrorist Newton Canada to years myself and so the Jays Howard "quotation mark

15:53

such so who is the scattering profile well that the thing that you do is you apply the linear but minus the the solution you then you think the week limit of that and the solution corresponding to this weekend that's what the scattering profile and why this page including the U.S. duality in the previous what they called a detailed they're the same there's only 1 scattered across this school OK so

16:28

that it levity 1st summarized 2 so they know that they keep well amend explained in India the is that we're going to rule out any minute blocks so we know we have this idea of how to the composer solution into blocks which are nonlinear profiles in about the major opera for decompose but I'm not going to go into details of what they are more than that so we know where we're going to see that there is no nonlinear blocks in your solution which remains close to the light called for very large Stein and this system over what should happen if you believe that the only nonlinear objects are these traveling waves because the traveling waves that traveling at work Speed strictly less than 1 so there can't be any nonlinear way that travels at speeds 1 and that's what this promotional OK and how do we showed we assured by combining a family of which is called serial identities which we saw in the previous elections "quotation mark so when tool that we use in In this is actually In part of the theory of the linear where equations in which it was developed by Gerald Friedland In the it is seventies 16 and 50 yes and somehow people working in air the Melrose school "quotation mark of scattering theory kept using the but everybody else moralist forgot about this and this turns out to be an important tool here and we rediscovered that independently but then we found out Friedlander has already them so we will use the notation in the R 4 the other is that there and 1 over our grandson will make up for the contentions that the OK moral standing directions OK so I'll give you this is the linear theory this linear theorem allows to associate the profile infinity to any solution of the new ways of doing it says the following suppose we have is that any solution to the linear where equation with bacon and energy space until then always there were the tangential derivative goes as the host infinite and the higher the terror posters in West Houston there's 6 3rd also this section and there is a profile of the unique function G plus which is a mail to cross as in minds 1 since then the properly scaled teetering 10 at infinity equals these objects and the property the the scaled are derivative at infinity equals furthermore there is an ICE summitry the end of the linear energy Of this solution equals the L to wrong in our process and mine is 1 of these cheap and the map that maps the coaching data to the G plus is by by rejected summitry between the 2 spaces and of course there's a corresponding G minor sustained most of mines in the and this deep Blass and you miners I called the radiation fields associated to the so somehow there's a weight associated a profiler plus and minus infinity which is an asymptotic states for any solution of the new Wave .period as I said the meeting we proved love not knowing that Friedlander of but then we found out that yeah so somehow 1 can use this this objects to patch up together solutions of the wave equation coming from him OK so this is how we're going to use this all year annual there's 1 more thing I want to mention here if there is a well-known property of solutions of the many equation which is what's called it could petition manage but the spatial energy and time energy asymptotic clear the set In fact this shows a much stronger version because this has a miners and has the plus and this goes to them and this goes to therefore there could petition the field of energy follows immediately but it's so much this is a much stronger statement that competition OK come Oh by the way at the end of this series by Friday in it's a afternoon this knowledge will be posted on the CIA tedious websites In the 1st part is already posted In the 1980 s website looking so old and by the way this was for all and bigger than the equal to 3 but everything else is

23:09

also for and bigger than or equal to 3 but I will only be working her OK so here's another title and I don't know if you have noticed but there is 4 this a French notation and that there's a new Western intervention the French notation that the comes 1st and in the U.S. notation that the comes last and I've been using the U.S. military and here I lapsed into French interpretations the polluted by French collaborators and but it should be that the appeals Sec OK 2 something so now it is this the stickers known that we will be working with his elbow L 5 in the balcony next month kind that we will localize it 2 sets of this form by taking 1st this spatial integral and a time in the world and here we have an interval III will call as To the S R 3 5 and then you have an answer no 1 cross-sell too you have a solution you In verifies this circus estimates infants there 1 cross too look at the mysterious and of the president of he here has to measurable let's say nothing in the of well it we will let them we will use it for domains that have dependence market but in the end this definition there's no the need for that but there will always the In the case of a fine independence from who could what it's so that little claimed here is that you have a solution like that then in fact asymptotic elite it looks like a freeze without any and that's the same proof was approved of scattering and in them to and I will use this

25:44

industry OK so sketch the proof of is that of the extraction scattering from so 1st I think that I choose any sequence .period infinity and choose a weakness you analyst because this thing is bound and now that I will call the L. the linear solution corresponding to this week and what I want to show instead this limited Hall no noted that really I want to show this for a but if I'm not for so many of the load for any bigger but it's convenient for me to stay the result as saying that he holds for all any Real's but he was seen as so they say that although they will not consider moved up it covers the outside but moved up look at but and if I know for a nite at the renowned ripples because of continuity of it look

27:18

so that I'm going to assume that this

27:20

does not hold and willing to reach a constant for the 1st

27:27

claim is that there is an optimal a bar for which the thing of all so there isn't a bar says that if I'm not bothered my the desired property which is that the center space-time enormous find holes In that for all a bigger than a bar this behaves as if it were the new solutions so this should be Europe and that for all a bigger than any bomb but this desired limit holdings and this is a buyer's optimal because this enormous in OK someone to prove it an incident in this statement that is convenient I'm working all are institutions with negative because the

28:18

sketch of proof of this I define a bar as you would normally do you think the team the good thing is that we have to show that the set in London now the set is not empty because of a very large and and this is certainly finance because and my job but said he will have smaller long and then we will coincide outside a light coat with something that scatters and it will be too and that's the reason for for doing OK so this set is certainly not and clearly for any any bigger than a bar this thing is finding by now there's a

29:14

Bach would be real or could be minus infant right now the provisional this the fact that I am assuming that the Theron does not hold forces to and that is how we will reach a contradiction OK because if this is minus infinity

29:37

this would hold for all 8 and that's precisely what I'm trying to prove now let me show

29:49

the 2nd hypothesis the 2nd conclusion so let a be bigger than a lot and now I'm going to concoct and solution event homogeneous problem I will call the and what they will have is you chopped off the X bigger than people think of you to the 5th OK so that this my assumption is that no 1 will too because it is bigger than any and is enormous fun so this is this is that it ain't homogeneous starring 1 grizzled too we have in Linas solution that has this property that was the 1st statement that made that was just likes cats who could now by Finance Bill propagation University "quotation mark V outside the lights on because outside the like global view on this very financing with so than in the previous statement ahead of the 2nd quarter you were In here In this and since you equals and forward linear solutions this tangible 2 0 the same is not true for you now I will show that this the minus the desired L goes to 0 and this will finish the proof to OK I know how I proved that I will just say a look at the associated radiation fields radiation feel associated to the somehow and they went to be and a that they're both linear solutions so they always have vanished and what 1 can check by the definitions and members of the patient is that for any direction Ernesto as long as I stay away from any the 2 registration fees coincide and since the 2 radiation fields coincide and what is approaching its already mentioned feel so is the other 1 the different systems In exactly the region that I 1 so this proves this

32:21

this 1st to property now I I still have to prove

32:37

and you don't have all 3 so I have to show that this this Strykers normally at a bar is not fine so if it were finite then I could make it small by moving the the high right because it's in what is an integral then because this is it well and well-defined object in his book completely linear at this point if this is small I can make it smaller I can still give up a Delta 1 of the 2 so here I get dealt the 1 before then again about the world to then dealt the 1 and then from this is that the warnings in small I can conclude for the nonlinear solution the same thing but this is a contradiction to the definition of a fire because I have a smaller thing for which the think is fine so that's how 1 concludes that the "quotation mark

33:51

Nelson now I've got dilemma that tells me what's happening with this a bar but this is a completely new radio In order to conclude here I have to do now chopped-up into the different direction so I'm going to define this at the regular directions and set OK so let me 1st define what I mean by the regular direction so I have a regular direction if I can go a little bit below a bar provided state In a sufficiently small and so in this direction I can go a little bit and that's what I will call a regular and a direction which is this is the usual angle In the direction which is not regular missing so that the main result about this said that is that the set of regular directions it is not but fine so this is our 1st and then we will obtain a contradiction by showing that has to be empty but if there's a bar I was flying at this at the regular their actions to be compared to the destruction of the Peru so the main tool in brewing the fact that this sector regular that and directions is known in the and finally Is this the there is a doubt that 2 thirds of the 4 major satisfies that for some absolute positive this linear object past smaller than Delta to and what's important here is that I'm allowing the region to go below a by and and keeping the small according to this team and the square root of the Council the curvature of the statement is that if this is small enough but not arbitrarily small Pacific's threshhold the the thing has to be regular OK so that's the very the tool that we use to prove the this singular said Islam and the and fight so let me just say this

36:37

claim it it's not difficult to prove it the compactness of the Sierra the previous proposition and circuits estimates With an elementary geometric arguments and find its piece of property because this will show suppose that it there is no point that the singer than ever .period regular and this satisfies this is satisfied then I cover my sphere finitely many points were these holes in them because of that I can go slightly below a bar so there has to be at least 1 singular .period why it had to do I have to have finally many they caused the if I am bigger than this delta over I'm single-A I have to be bigger than the fixed towers and I have infinitely many points but that was over to get to me OK they get multiplied by large for any harshness so this is the proposition that this proposition is not so difficult to prove that would we will use so now we come to it then the heart of the matter which is to show that this singular citizen and this will give us a contradiction on page so at this point they're pretty much stuck with having to use a profile of the company that tried to avoid discussing very much about profile the composition of the jury that elections I see all eyes glaze over so anyway here we need a profile the composition I recall what the problem composition is I have abounded sequence in each 1 and 2 after passing to the of sequence I can rewrite as the sum of the scaled new solutions to fix the new solutions for less than an hour In this area has small streak as long as I take more and more profile OK now that this only solutions and associated to its linear solution the scale of the translation parameters and the time prior an associated with this objects and linear solution I can't find what we call the nonlinear profile which topical it behaves like linear profile at the sequence of the times over scales a the parameters of the scaling translation and time translation parameters verify also were out the properties that allow us to keep them the profile separated and 1 the consequences of this is that there is also a nullity properties for the profiles in the H 1 prison "quotation mark just keep these things in mind In so the the 1st task is to find properties of the facts Of the singular 2nd until company I would call I will have to provide profit expressions preferably 1 and property he can thought property 1 deals just with the Raytheon situation was not in the other direction to OK it tells me that if I have a profile composition for this solution would be for which the a buyer it it is real then there is no profile With the properties that the translation parameter stay and minus the time parameter which and taking them profoundly composition states further away than the and the same thing when incorporate also the time parameters and the scaling parameters are all shrinking 2 0 and this part here OK know what they will post will have the thankful to be Kansas posted .period but I trust that so few that you will remember look Lawrence cost before OK look so we will

41:38

I but let me sketched the proof which is usual it we will allow you a contradiction that's assumed at the time translates thing it's always you remember that when we do a profile the composition we can reduce to a situation where the PGA and ii the 0 or TJ on national Lonnie overland Indiana the distinction Britain that those are the 3 cases otherwise by changing the profile the Federal was reduced risk to the fact that in the last 3 a series of lectures I explained how there is a certain lack of uniqueness and the profile that's tied up to the possibility of making this change but that's the only lack of uniqueness in the pro OK OK remember that this profiles are obtained this week minutes and so the fact that this is the profile of the property company the composition means that this week limit gives me exactly with the energy of the linear energy of the profile OK and the definition of the scale profile the modulated profile is this big cost TJ any 0 otherwise we would have minus DJ and on the Jedi so now I'm going to this calculation I do the grain you Jane and I know what this goes to most of which is positive because it isn't a new approach now because so this profile is 1 which is concentrated In this but

43:35

and directed to prove that there is no profit from from 0 so I assume that the barriers and then there is a contradiction and because of this

43:47

but the because of his property this is integral concentrated in that region now in this region because this is than actually here this is a property 1 this is beginning to us to actually has an natural here outside the bar I can replace you buy this video we already proved that so I knew that I still have a little 1 but here and now I can replace this region by the whole thing with the L again by this property it's concentrate forget now I can use that the asymptotic properties of given by the radiation monitor radiation field In that will tell me that this integral actually goes to this is a new and and using the radiation field I will get that this is 0 so that this object on the 1 hand goes to this and the other hand goes to general that's a contradiction Beijing has so this is another way in which you use this radiation fields he has citizens time tending to infinity and many have Anderson drive there is the case where the limits of the day and overland Indians infinity and this is worked out here that hearing you use that not all the radiation field associated to the and then you get a contradiction in a similar way OK now we come to the

45:48

profile property too Rockwell properties to deals with the singular direction and brought property to tell you the following suppose they have there is that there is a fixed number Delta 3 says that if he and goes to infinity and all major is a singular .period after extraction I have a profit in the solicitation sets that 1 of the problems which I'm calling you 1 is substantial and it's concentrating around in the direction of Wyoming and there additionally so this is the spatial concentration and it's in the direction of a looming at the end of this and the scaling goes to 0 in that time translation the time concentrations also going to and this is because otherwise if there wasn't any concentration on the strictest norms in in near this directional major I will have a regular report so this is a singular point there has to be some then approved it is the use of 1st propositions and geometric arguments and you have to obtain that result OK so these are the preliminaries no we're going to get to the heart of the matter the heart of the matter is this proposition suppose you on a borrower before an absolute again then I can find a sequence of times going to infinity a a number of positive such that I can make the X-1 unpleasant the key as the next 2 lastly the extreme concentrated along the E-1 1 directions up to level out from within apps campaign some and the interesting thing is that OK if I had just been X wondered it stood in the X 3 and the detainee this would be much better but I can't hope for that much but if I can add to the DAX 100 To OK so the 1st thing that I'm going to this show you From this proposition I can show that there is no single element there's no singular director and what I will do it I will show that this implies that he won and that that's tied up to this day 1 entered the direction the won in here I will show that the 1 it cannot be a singular address but of course since the 1 is an arbitrary rejection of the rotation that shows that single sentence and that will be the contradiction please so playing here this limit approved it will be approved without using 1 insisting the prove this for the direction the water and then true that this implies that anyone can be I think that you the social life of Europe yesterday nite key Adkins if I proved that I can prove that none is singing yeah I can prove it for any other judges 1 room but it could be approved for any of for the direction in which the possible but I will show that I have is that this sequence of and a sequence of stunning gives me but he won any legislation on this and this is where the right and using that sequence I can show that that the action could not be singing this year residue that they knew OK so let's use this so that in that it's OK that's part of the logic here on the logic and so let's show that this implies that 1 cannot be a singer OK this is enough and then of course I have to prove the purposes and approval of the proposition is the main party flocked to know what we've been doing is peeling layers so as to get to the heart and this is the heart what so I now going to show how Proposition 2 implies is 1 does not belong to so I I now have to do a little digression into get so I introduced a new normally energy so I think it's a fun at the energy states I defined the ease of 1 or To plus the Warren and to open the gates to open the the OK let me 1st convince you that this is a you at 1st you couldn't Of course the triangle inequality is obvious but what is not obvious at least without thinking for a 2nd is that if the norm of energy 0 energy has to be easier OK celestial that suppose that the normal veggies 0 then asked 2 partials that there 0 right and it's in H 1 then the 3rd partial has to be easier but they can't be NH 1 function that depends on only 1 OK so I mean it's 1 is 0 but then because of this being 0 G 0 so this is Wilkinson OK now that the interesting thing is that for the linear equation we have to conservation right the 1st conservation of the energy but we also have a conservation of the moment and if we combine those 2 you can see that this quantity is independent because what you're adding it is To the the 1 the and that's the momentum in the direction of the war so this is independent of the because of that and now we can plug this into the profile composition and we're going to look at the Gloria expansion sorry with we have profoundly some position and we have this pedigree expense because improving there's all users there so we're not out of the parameters and invariants under the law Prince so this is the I disagree expansion for anyone

53:34

what and the next thing that we need the following claims suppose somebody gives me a break and then there is an absence such that if I have something In energy space With bounded by M. energy norm that very small he won district norm can be made as well as OK so somehow being small In this norm forces the circus and more so how approve well I proved by contradiction if not I have a sequence which is uniformly about whose normal doses 0 and Corsica's mountains Tuesday right this is what I need to that's the same the show that this is true let's take a profit on the composition for this now because of the a Pythagorean expansion all the profiles have to be because there's no standing to sue but each profile will have 0 E 1 and the 1 remaining 0 I it means that the thing is 0 therefore I am only left with this person In the present work in their profile the composition and the dispersant carved has Cosmo so that's the end of story proves that OK this is clear this statement falls because this goes to 0 I have no problem and therefore on OK and I

56:10

let me go to the proved that he won London To now I need to recall for you prophet

56:17

property the profit

56:21

property says that if I had at the end voluntary already announced their 4 sons of sequels this property also this property will seize property holds and there is a Delta 3 which is such that and the comes from this claim

56:51

but they have a uniform bound in this is smaller enough then this system is smaller now than they OK so

57:04

I haven't given by the Supreme at traders debate that we dealt 3 over to wear about the 3 is the thing that gives me the substantially all of the of the 1st profile In profit property OK by the proposition I can't find the sequence the primarily outflow since and this leads small this gives me the sequence from this sequence I take a properly composition verifying the properties of profile property then I'm going to show that the the this implies the rights smallness of anyone look at so since this is an inner product associated to this normal and we have the thing glory expansion for this norm we also have this week for the profits remember we use this before for the a 1 cross to normally not for the E 1 we know by profile properties 2 that this did this profile is concentrated where this is less than Alfonso this part is small now I look at this in a problem and splintered into this region and this region in this region is too small and in this region I get about lapses therefore I get this whole thing is bound by optional something goes with but this gives me that you what you want Norm square and that means that this thing as a 1 on less than that is the 1 is less than absolute move given the way I chose my data this means that there as normal everyone is less than that of the 3 over to because data was that the too but the property to send it had to be beaten them 3 and so that's the contradiction so this tells me that's all I have to do is approved Proposition 2 and that is the heart of the matter OK In the next hour will spent improving proposition there is a logic at least clear Of course some of the calculations we have to do it at home with the fish had but this is their plan prompted opposition to this is that it the important thing given Absalon we can find the sequence the prime minister and announced that such and listening show business look at this is our task that was proposition tell him didn't trust me that this was proper OK for proved and this is the thing that is proved using very aligned this is the heart of the matter shows that things cannot be traveling its people near the the like ,comma OK but so now we start the

1:01:02

difference chapter so the 1st thing where there had been little era produced some kind the defect OK so let's balance many sequence both times spending I will call the roll of ecstasy To me you to the saic's lesser gradient this isn't being 1 of 2 the 1 Square in the heart of the it so what would it works this role representing this I'm scaling the at the singular .period and I'm doing it beyond a buy hardware things OK and then I have I put in the linear and the energy than the new energy but I also throwing the 6 Norman Harding threw for movement so that this statement is that after were given sequence after extraction I have a non-negative rather than measures such as that Ross said at the time the head cause weekly to this non-negative measure until but that supporters the Taylor after the next 1 name How can To because and that's where they the fact that all my homies a singular that at a bar Is there makes it appear that way OK the 1st the statement is obvious right because there's abound but the measure has such sequences about the sequence of measures as a subsequent the conversions with nothing more to be said so the important thing is the support OK so In

1:03:28

Proposition 3 we showed that if I went and epsilon outside the EU convergence to media if I go an excellent building outside the a bar the you can be replaced by the and biannual using that the planes outside the tensions here the 1st statements to end this 2 statements were also true because this to quantities so we just wanted to use to show that the support but since the EU equals ADL in this major if I choke if I choose that this is going to 0 for in compact supported think this system is supported in a X 1 less than or equal to optional for every actually and therefore needs 1 us equal show that and this is not a linear solutions so what am I gonna use pointed at times are going to infinity so whether I believe the regulations openness when you usually using so this is what I have this will radiation field I 1st change variables and now I change this by the radiation field and then make a smaller no change variables here and now because this has compact support of major is different from the 1 this goes to 0 and therefore by dominated conversions singles you can have 2 different directions and no 1 is willing to infinity by exit support open all right now we

1:05:41

have defect major utilities and anger and the composer's into the delta mass and the origin busses remained In the remainder has the property that the charges the origin 0 we have no way of knowing that this is an absolutely continues measure and there were no I'm not going to claim OK so I just think that the parts corresponding to the origin and take the rest so the next limits as the following but the infinity and actually being here then after extraction there exist too non-negative rather lamented the euro and you and tuna at numbers said the 1 don't charge June the role at the head converters weekly to merely replacing the word of the year the row I P and miners Alpha for attend was weekly to anyone seen 1 of the the supports the meeting days it's 1 less than or equal to 0 and here is that the punch line there's also were announced and me 0 0 In the ball rate is less than elsewhere smaller than abstinence anyone in the ball radius 7 out of 10 more please each 1 of these 2 things is obvious from the because me 1 and museum or charge 0 2 the origin scientist small ball that they charge a small the only .period it is that there is this uniformity that they can use the propose something proportional to the ALF In both cases OK and so what we're used to uniform think there's only 1 thing you can do and as his finance been providing it this here were working with the wave equation and sold by Finance

1:07:57

being propagation and small they theory you can see did you can make the same provided there it's because but where

1:08:14

only Alpha over 10 a part in the allowing of plain excess who could so that's where the the weight equation of playing it's been propagate not that

1:08:30

important factor here this is an extra piece of information that this page tells you which is it with defined role of X PTI the same expression that you had before where before you had P and enable team then for all key between Kinmen is also retained a key and this will be smaller than to actually provided are always charging this annual OK and that's so yes this is this is the place he said that seems to be all because otherwise might and might not might might prove 1 not you I need them to be called look at this because of the geometry poems and racing history indeed any other questions 12 you have do so and From this this could be the result of the curious thing you exactly where they had been designed not to fix what he wanted he said the best the exact yeah that's the best thing that's the only .period so this property rose star shows he is uniformly In this range of the enzyme in this range of things are small OK so that's what we should take from OK now we're going to review material identity that we've seen in in the 1st part of the class but the cost but will so and the wine is gone so that's a so little R is always opinion this Payless she a cut function which is 1 4 x less than 1 quarter and 0 4 experience 1 and fuselages scaled OK then we have this collection of identity various are very elated the first one tells us what X .period Grandview the DQ key you that's where we think it is we get this constant minus-3 have plus a half grant you square magnitude 6 and we have to take it a cut of function because X grows and then there is an error and the area is bounded by OK the 2nd 1 so don't copy the 2nd 1 the blast should be reminds us that whenever the gradient square than the into the 6th appear they have to appear with opposite sides because when the focus Inc OK so this a minor so In the other variable is where we have used the BTU and then we have the same type of thing but with different numbers here 1 his mind's 1 and he is my to have his way In this Mary is really severely his identity is related to the 1 variants of the movement and says that if you take the energy density the new multiplied by X then there is nothing of that isn't and we have to chop it all and then we get there and we will also be using that to conservation look at that and the derivative of energy 0 under the areas of the moment of and of course we will always be using also truncated versions so if put the fuselage here we get a little capital of and how they prove this things improve them by brute force and you multiply the equation for her by an appropriate objects in unions agreed but Wilkinson's usual multiplied "quotation mark so now I'm going to take a sequence the 7th inning to infinity and fees allowed for what before I was calling fees of knowing announced an illegal logging into the fine is the land of X to be in this city translated and scale you so I definitions and our previous Lemmer give us information about these U.S. then converted to make it to the this ,comma his weekly and then there were the smaller than alpha thing OK and now I'm going to introduce 4 1 2 aso is that the truncated by the health the missile is the great squared see celebrities Newton 6 and the so then is the X 1 times the which is the 1st component of the moment just and they're all

1:14:46

truncated by but and then I will be averaging this quantities for the between T-Online itself and the OK and I will be proving things about the averages In from proving something about the averages is how I will manufacture another sequence the prime that comes from the area OK and then there are the terms but the averaged held upper body OK now that the rephrase the virial identities and my information in terms of the but in the 1st 1 on the 1st material wasn't we have the few we have and there is nowhere to go these old maps along comes from that property stock I can remember the star was the thing that was uniform between Kinmen itself over and the OK so that was the 1st material where this season find attendance space there that it is in and this 1 assists in the 2nd period when I think about it but there and then the coefficients all for minus 1 the 3rd 1 was that the derivative of the X 1 times the energy density which gave me the momentum here is X what mine the momentum here imperiousness Indians once I get there and he was and governments but all but this is the the constancy of the momentum truncated and this is the constancy of the energy truncated the that so that's what this this things are in the capital actually it's uniform in the it's been right so what I will do it is force of the bright relationships between these that comes From my assumptions who so things were taking an average and this is roughly constant it is the difference between his average is smaller than absolute which is the point was different but then since I'm averaging again and half it the same for the energy the government that is being put on the market this misapplied against thinking yeah the thinking yeah that's a plus good so far now I wouldn't approve 3 the facts In the end about so this tells me With the 1st factor is that in this situation wherein the energy equals the moment With minus that's what the 1st ones the 2nd World 1 says that this this combination is almost 0 In the last 1 is that this combination is the momentum parcels so let me assure you that the proof of this I will only show you a but you will see once and through a hole the reason OK so it made relates to the their energy into the moment and why and how do I really the energy to the moment remember there isn't any a relationship that the derivative of X 1 times the energy density equals the moment that's what that would "quotation mark note so integrated that relationship between the man and the environment of and the average I get mine is the but the average of the and here I get rounds this is the X 1 times the energy density at time t and my nose at time t and minds of but this is a fundamental thing In this going around 5 the length of the give me this is just the fundamental fear should I go back to the identity that used to get through the what instead it I think

1:21:08

interval between 10 and 10 winds of Over 10 and on average they get this and then I get this at the end points by the fundamentals who fool with "quotation mark

1:21:29

so here what will happen what at nite I'll explain in words with this guy will be small because I have heard of him In it this is a consequence of that Starr about In here I haven't at PM miners off over there then I have to change the space a scaling but I have to make a space translation to match when they make a space translation I will get why 1 class of home repair another white 1 term will be small and the outflow repented is the 1 that will give me their OK so let me know the calculator the 1st it is this look at and

1:22:37

backwards I have to understand this quantity I have the 1 In this is exactly the outflow wrecks the musical acts In the limits no it's important that I have 6 1 here because X 1 vanishes as the origin and that's why I didn't get the dealt mass and the region in the that car and this is precisely less than human and that was 1 of the properties that measured the new 0 hat so that's what way again where the other 1 I have to do this to interview so what I have is that it is an hour do that to you and of course 6 1 has a wide 1 Byron and then apartment no 1 1 which is this 1 and this 1 is the 1 that gives me precisely what I think the average the the of office endit In this 1 now it's everything is centered the right way will give me absent so I have this of course I have the energy but I have edited at the at the wrong time but the is roughly constant so I can change so have it here so I get this identity but I can change the energy and together and this is precisely the 1st 1 of those things listen when I called

1:25:02

cooking so now I'm going to add a OK when I have a and

1:25:23

"quotation mark it wouldn't Bloomberg a and C. I get the environment is a embark -minus two-thirds see embodied equals capital now I have been to it if you remember has BMI my nose and hands and I think it's plus here so they entering and cancels the big airlines In late and that is with the fact that Barton is small it overlaps so that blood this back to the the 1st thing and this is small can put with the old and so I have 1 have on the 1 hand me a lot because the Obama is overlaps but now let's remember would be a lot beyond alien have a handle had been the square b and has the green squares and the other factor would have and the embodies precisely the 1 but the pain you know I complete that into the square and you see that that's precisely the E 1 the 1 long has been the 1 was leaving the square the lesson 2 square opposite the exchange so I just come in going and that's what I want to show a small that's my conclusion when this kind of expression being small but it's averaging time such as the averaging time is it has to be 1 sequence of times forwards this thing is small OK good you for every year of 1st time this is how you choose the prime and from passing from the average 2 particularly set the price but as we saw in the proof that that ruled out that was a singular .period any sequence was enough it didn't matter any sequence can "quotation mark served now all that you have to do it is city where the supports are and see and this and that creates here also but instead of itself at the end this election the seal relapses but that concludes the proposed then the statement said an elaborate proof you can please sit down and do it look at the end of the day I don't know I just say we have the option of continuing or if you're tired and you can we can stop for them and respect them this would seem who were located on the will have to change subjects so maybe this is the 3rd of the international questions on the end this is a big chance to have swelled in the summer of phones have you tend to fall below In conclusion show show in the state of the U.S. has been ruled by absolute Minerva II have yes yes I I have no desire to go to war but OK so let's let's go back 1st this statement of the proposition the

1:30:05

proposition is antibodies would we had that there is a given then there is the sequence and that this this small OK so I think maybe 1 the good is to review that this implies that won it's not a singular .period I think that there will be do I saw a hand I cannot understand the legislators said that you have seen seasonally the the music was transferred to the coming you have to make some decisions that will be record new wine but we can say it will immediately kill all the provides the the only sources of overseas New year because 1 of his presidency it the as soon the issuance sequence that it's fall on kind 1 he was sufficiently rich and there is no Hill or only once but we have you have to work 1 direction at the time of this story is is that the value of I think but in the thing is that the killings that direction so you you have some direction on your sights that you want to eliminate and to eliminate did we find the sequence of time that has all these properties but then that in places that could but let's go through the prove that this proposition implies that the 1 is not a singular directions "quotation mark this review so we introduces no and while introduced this Norwell go back to proposition to it appears here and why doesn't appear here as we saw in the approval of the underwriter contend this this is the thing that appears naturally when you how's this burial identities and you see that the energy equals the momentum until then you can combine do when you combine them you get precisely that type to was the 1 one-half dignity squared plus one-half great squared plus the 1 the team is exactly there will be 1 of few merger OK but of course those functions are not allowed because they can be but so so we

1:33:01

introduce this thing is a natural objects for our troops the 1st observation is again you have to go online and environment and because they're both independent of time this quantity is independent time so this is a constant thing that using the square and because this is time independent and we have the also were not at the properties of the parameters this is immediately implies a peace agreement for the profanity cope and the proof is the same as the 1 for the usual if you recall the proof from the official added still missing you need introducing the area OK look at then you're going to really being small in this 1 wrong to not having profile to having that this circus norms OK this statement tells you that provided they have a uniform if this enormous small then this norm has so to prove that what you trying to show is that provided this is uniformly bounded if this goes to 0 this ghost OK so if this holds and the ghost 0 then this festival to see if this is what we need to shore to show that we just take so we have this to property and where show this to properties means means there is no 0 profiling the property OK why is that because there is a piece of foreign expansion so each profile squared is controlled in the E 1 is controlled by this 1 and we know that there's something has anyone on the 0 it this year that was the 1st thing that we show that this this is and that's so therefore all the proposals they from this tracks and the Pythagorean expense when she had it all the profiles and 0 that means that only the remainder is in the probe of the composition and the remainder has this problem so that proves this claim you want to know what they see and you know you you know that you can have a of initial she was sequence of new yeah because the limited identified and that it had received this this sort of way pollution you cost may be the some of the but the 1st instance what no other

1:36:38

Prudy 1 cannot be it by using proposition toe on this property to the profile the proposition to governance I'm in this situation and the 1st profile is substantial With substantial the Delta she said that investment in it is right here this is not a willful don't transparent forget so now I'm going to contradict this and using his OK so I have a sequence the primarily used for such sequence have profited from physicians and the property composition lessons from now I will use it the weakened amid property that tells me that the inner product when you give me the then on the property and this is just a consequence of the Basic Law and expansion worth of another Press note from our assumptions this 2 assumptions it tells you where the profile you In lives in this region for he because the other parts are no not where the union is profiles always are concentrated near the lake for once you up into the sky and so this is realistic look at what you can use a radiation field or the concentration in the nite call either way OK now I look at this in a product and splintered into this part and the other part of the by corresponding to this it gives mineral goes to 0 and then the rest that use the course which works with Interpol and I give this to message is bounded by that no this coverted to this quantity squared so the confusion this and this it is less than half of but my claim about me that this is less than Epsilon there s enormous lesson and data and data was dealt 3 2 so this is that the 3 were too but the profile property to told me that he has to have to be billions of so what do you think is that if somebody gives me a sequence of 4 subsequent comprehensively compass and then just work with them and profit property word for Indian and cares is this more clearly "quotation mark I will not say that this is not a trick improvement is the tricky part the so let me let me just say that was what I would do next time it is now I will abandon the infinite time because a final time and infinite time behave the same way once we have this part this is the part that's the main difference between the fine but we've done so on that and I'm going to go to the final time case and prudent so that will be the last 2 that was the 1st thing to and say something about how to find the sequence all of those areas you think continues the some there by the new strategy is a good thing we have several times but then I'm not willing to say more than 10 minutes Garrison I think it is a very difficult quest to pass that was already difficult to prove for sequence of times but the pass to the general case you need to understand conditions of solitary wicket and this is a very tricky focus and that's what now in the range of cases of course you have only 1 sold everywhere this discursive properties of solutions that allow me to understand the the influence of the future and

00:00

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07:54

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TVD-Verfahren

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16:25

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25:40

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27:19

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29:11

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32:16

Unendlichkeit

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33:48

Resultante

Folge <Mathematik>

Punkt

Gewichtete Summe

Landau-Theorie

Gruppenoperation

Regulärer Graph

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Eigentliche Abbildung

Translation <Mathematik>

Turm <Mathematik>

Wurzel <Mathematik>

Ordnung <Mathematik>

Aussage <Mathematik>

Ortsoperator

Schätzwert

Tropfen

Parametersystem

Zentrische Streckung

Krümmung

Kategorie <Mathematik>

Winkel

Singularität <Mathematik>

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41:35

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Mathematik

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43:46

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Sphäre

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Drehung

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Mereologie

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53:34

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56:16

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Mathematische Logik

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Wärmeausdehnung

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Eigentliche Abbildung

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sinc-Funktion

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Innerer Punkt

1:00:57

Ebene

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Computeranimation

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Vorzeichen <Mathematik>

Prozessfähigkeit <Qualitätsmanagement>

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Mathematik

Physikalischer Effekt

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1:05:36

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Physikalische Theorie

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1:08:10

Resultante

Folge <Mathematik>

Subtraktion

Gewicht <Mathematik>

Euler-Winkel

Momentenproblem

Klasse <Mathematik>

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Nichtunterscheidbarkeit

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Objekt <Kategorie>

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Lemma <Logik>

Quadratzahl

Forcing

Flächeninhalt

Mereologie

Größenordnung

Energieerhaltung

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Geometrie

1:14:44

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Unrundheit

Derivation <Algebra>

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Raum-Zeit

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Fundamentalsatz der Algebra

Kategorie <Mathematik>

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Energiedichte

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Beweistheorie

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1:21:22

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Mathematik

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Nichtunterscheidbarkeit

Translation <Mathematik>

Inverser Limes

Term

Gammafunktion

1:24:57

Folge <Mathematik>

Thermodynamisches System

Aussage <Mathematik>

Frequenz

Teilbarkeit

Computeranimation

Sinusfunktion

Arithmetischer Ausdruck

Quadratzahl

Betrag <Mathematik>

Gewicht <Mathematik>

Mittelwert

Beweistheorie

Aggregatzustand

1:29:54

Impuls

Lineare Abbildung

Folge <Mathematik>

Impuls

Gerichteter Graph

Computeranimation

Richtung

Wärmeausdehnung

Gruppe <Mathematik>

Nichtunterscheidbarkeit

Vorlesung/Konferenz

Biprodukt

Ordnung <Mathematik>

Ideal <Mathematik>

Aussage <Mathematik>

Gammafunktion

Normalvektor

Lineares Funktional

Kategorie <Mathematik>

Güte der Anpassung

Singularität <Mathematik>

Aussage <Mathematik>

Temperaturstrahlung

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Entscheidungstheorie

Inverser Limes

Energiedichte

Einfügungsdämpfung

Innerer Punkt

1:32:59

Lineare Abbildung

Subtraktion

Folge <Mathematik>

Impuls

Extrempunkt

Gesetz <Physik>

Computeranimation

Temperaturstrahlung

Weg <Topologie>

Spannweite <Stochastik>

Wärmeausdehnung

Gewicht <Mathematik>

Eigentliche Abbildung

TOE

Stochastische Abhängigkeit

Normalvektor

Aussage <Mathematik>

Parametersystem

Thermodynamisches System

Kategorie <Mathematik>

Stochastische Abhängigkeit

Singularität <Mathematik>

Profil <Aerodynamik>

Aussage <Mathematik>

Temperaturstrahlung

Skalarproduktraum

Fokalpunkt

Unendlichkeit

Objekt <Kategorie>

Divergente Reihe

Konzentrizität

Flächeninhalt

Sortierte Logik

Konditionszahl

Beweistheorie

Mereologie

Strategisches Spiel

Körper <Physik>

Wärmeausdehnung

Normalvektor

### Metadaten

#### Formale Metadaten

Titel | 5/7 The energy critical wave equation |

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

Teil | 05 |

Anzahl der Teile | 07 |

Autor | Kenig, Carlos |

Lizenz |
CC-Namensnennung 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. |

DOI | 10.5446/20478 |

Herausgeber | Institut des Hautes Études Scientifiques (IHÉS) |

Erscheinungsjahr | 2016 |

Sprache | Englisch |

#### Inhaltliche Metadaten

Fachgebiet | Mathematik |

Abstract | The theory of nonlinear dispersive equations has seen a tremendous development in the last 35 years. The initial works studied the behavior of special solutions such as traveling waves and solitons. Then, there was a systematic study of the well-posedness theory (in the sense of Hadamard) using extensively tools from harmonic analysis. This yielded many optimal results on the short-time well-posedness and small data global well-posedness of many classical problems. The last 25 years have seen a lot of interest in the study, for nonlinear dispersive equations, of the long-time behavior of solutions, for large data. Issues like blow-up, global existence, scattering and long-time asymptotic behavior have come to the forefront, especially in critical problems. In these lectures we will concentrate on the energy critical nonlinear wave equation, in the focusing case. The dynamics in the defocusing case were studied extensively in the period 1990-2000, culminating in the result that all large data in the energy space yield global solutions which scatter. The focusing case is very different since one can have finite time blow-up, even for solutions which remain bounded in the energy norm, and solutions which exist and remain bounded in the energy norm for all time, but do not scatter, for instance traveling wave solutions, and other fascinating nonlinear phenomena. In these lectures I will explain the progress in the last 10 years, in the program of obtaining a complete understanding of the dynamics of solutions which remain bounded in the energy space. This has recently led to a proof of soliton resolution, in the non-radial case, along a well-chosen sequence of times. This will be one of the highlights of the lectures. It is hoped that the results obtained for this equation will be a model for what to strive for in the study of other critical nonlinear dispersive equations. |