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Cubic nonlinear wave equation

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as the top and if you can get to 2 thank you for inviting me it's been really really nice to be in Paris this month on a really nice talks to watch as well and I'm just really great except the weather but everything else really great but I am not end into the US it's the weather in a way he notes 100 year experience supposes that it's nice to hear from the in this talk will talk about the but in the 3 so equation that I will be discussing In this talk and it's not it's the CU wave equation in 3 dimensions as this equation is called the it is colonies to one-half critical problem and a lot of people already know this Of course and in fact raw and talked a bit about criticality no 1 during Tuesday but in any event just to arm get into Justice say what is it means that you have a solution to this problem us yeah solution to the equation then in fact you handle an entire family of solutions because you can insert Orlando and and in fact and if you check then you'll see that the norm that's reserved under that has reaches each 1 half normally your initial data In a step demise would have normal here your initial velocity and busses called H 1 have critical problem and and this is this is by no means a on important thing again as as if I remind you of what Ryan was talking about and he's sort of sketched out in a way that on for example his this argument that Kristin colander and Tal used to prove that if you had dated it's that in fact ill posed for dated less regulation 1 so so it's once criticality is it is it really is important now is foresee mentioned some stuff about how would you know native orders and new sort Alice-in-Wonderland type thing or something but I for as bigger than 0 we we know that that's that it in fact that's exactly on that's exactly the right space to think about because we have this local well close the result of and this is by Lynn Blondin inside I think year it's filing by side who also from Johns Hopkins and we have that we have the local well closeness for you know a new 1 for you not in age 1 hand outside the local well below reunited in eastern 1 a new 1 in case .period minus 1 half and just to remind 1 and that is there exists that he of you know what you want greater than 0 says that the unique solution you must insist on my mind as the competition but Europe is being held the local it Bloomberg Politics delegates In the solution of the 2 minutes later on OK so so so this important like a view of the city of you know what you want right because if he had enters depend on the size of did then you could always do some
rescaling to show that in fact the articles and an end to the anyway this is the sort of the thing I don't want to get to talk to much about the functional analysis and all that this but in any event this is sort of the standard from definition of local well posing now there is the group of there is the goal that I was hoping to prove that it's not done yet is to prove that there is a global well pose innocent scat for this problem as it also as global is pretty hopefully pretty clear from their local and mn Warlock Red and then but then scattering is that there exist you not plus the miners in please don't want have you 1 in data from minus 1 such and you have to Minor S & T U not plus ,comma you won't play reached about 1 cross the share my son goes to 0 2 those 2 plus infinity and I think our clients including acid new solution operate too just linear so we have a our solutions and you can the maker of arms so we have a solution
operators emaciated starts to look like a linear equation right and and and now manages for innovation and linear equations on so this is this is the goal of of the D focusing cases focusing we know that that doesn't happen right we know that because of who in the resort talked about the right result talk about some type 1 bold solutions and books on a pyramid and other things like that in From boat but none in India but is that a Dallas for the focusing problems on for the D focusing there is no such Canada exam so this is where I I think a lot of people the the focus probably get an effect that is equivalent to saying that news of the death of 2 so that visitors is equivalent to scatter yeah it's a nice teachers shows but it doesn't take very long I'm not going from 0 cases said then we have then 2 types of of 2 types of ways there are a function of solution can plot than those are appropriately enough called Type 1 and Type 2 for the 1st time this is that you have the H 1 have going off to infinity plus is the you got the maximal interval existence and the pluses is that any 1 of them and were saying that DH one-half Norm goes to infinity I In 1 of
those 2 1 of times
actions so what why's that on why is that automatically it unlikely Alaska get right because you just look at Europe is that a year wave operator and you see that the CSO started in right and can bring back down but I should write a plus minus from the you want as well as the you know OK but not in to bring all back kind of that's why should also but they used it in the New Wave operators the unitary operate find on solar space so so it is if you're too because it is not a lot of things automated games over right that you can't get but but even still it might still felt scatter even even if this type 1 ball
doesn't happen and that although the type to book where you have it's lesson this but nevertheless you still have the elsewhere norms the president yes so that this is the sort of our to types of books solutions now at this point then I want to put this in the context of a couple of the problems that have been solved by a number of people ran down the reason why I want to do that is to sort of I talk about what we have and what we don't have he said basically we have these the reason that that the young the results for the most part at least for that the focusing case on the ignoring that leaving the focusing there is of course the channels of energy which is I think a little bit different than then in the read or I should say it's a little bit it
said it seems to me to be a new idea so that's different from however than than some other ideas so because it's it's primarily of analyzing linearly it's analyzing their behavior the exterior so but In general the the and even it's so elements let's let's say elements in common between analysts and away from in in the the elements of proof for for example you have energy critical way so this is you teaching modest Delta you are In forward writing forward even if you do your rescaling you're going to have the concerned that the enormous H 1 cross-sell to many critical this is in force we have to In the they still want critical and LLC which is complete so 4 of these
7 types of problems the elements of proof and generally centered around conservation law and then from that then you
had on ministers stressed energy-intensive along with the so-so for example the the energy critical wave equation but so when you when you sell for example the energy critical invasion might mean that that they have revealed in late so you have your mower testament is bounded by the and then you have an energy and so inundated focusing direction you have the EU combined year-to-year conserved quantity in Europe the water and then and then you just use the I prefer the case and all you do is the a radio cell that Ryan and greater than you get you know 5 1 and then but then for the for the non radio on energy critical wave equation in the H 1 have critical analysts of so these are works by Kennedy and now on you have on and that is the unity has the profile of encompasses sees you find a minimal block element and and then you show this is a element that can't be concentrated because again you have your your more with us and whether that's for the way the creation of the land Strauss Morrow a testament and so the and then and Kristin is the work of the energy provider lesson and master classes well and I hope you know a lot lot different people have worked on this in in the audience and so on and so forth but on I think that the army the interesting thing to me about these problems is in a way they're very and much tied to block result if you pay you if you really boil down to about that is used to call on this is pretty much yeah this month you know what but that was the removal of the whole of the of the world and maybe we'll get some feedback on the interim dividend if you yes that's right we're studying the wave equation during an experiment right now but but but in any event we have this is very interesting to me has been very interesting to me because we had these results of arms for example we have the odds a 3 results but for the energy critical wave equation energy critical analysis mass critical analysts we have we have a conserved quantities right we have his reply whether that be the mass case the assistance you square or the air or the energy for integrated wireless and so automatically phenotype 1 vote doesn't happen at least in the radio in the sense that the focus in case right we we didn't we didn't really do anything other than whoever came up with this conservation laws requested time but but there is no there is no there's no half it doesn't seem to me like to be very that's the sort of given to us but then we have to prove tied to block doesn't have for the mass critical for the energy critical or what-have-you whereas in the case of In the case of the then edge and recently people have been working on now the results for where you don't have these conserved quantities right which is of course the result of the Okanagan Meryl as well just assumes that the H 1 half was uniformly bounded on but the young the interesting thing now is that you know right away that the only thing that those ever prove is tied to not tied to book and some people have gone back and forth in my mind about this now you have to listen to me argue with myself because I am still not completely decided in my mind about this but they didn't showing no to type to blow it's really the same issue right whether it just so happens that these problems have a conserved quantity and that for example each 1 have critical analysts doesn't have doesn't have a conserved a H 1 have norm but but I mean like when I did industrialization do any work to show that the mass was considered right and he now that was easy certainly had figured out like the 1st day I was doing that ,comma use integrated right Europe apart Oh by the way so just as a reference to blighted we're talking about in all these cases we're talking about of who we can approximate was smooth data right so we never have to worry about I like what blood talked about with unity you have a concerted quietly we know we because we're doing mystic by dismissing all these also perturbed if we don't ever have to think about a situation where we might not be eliminated by parties something that we know is it requires much more 1 of yeah and but by the end of that than those of the other thing that's interesting than about problems worried that because we don't have a conserved quantity is at you in a way you have these cancer quite actually give you 2 things if you really think about not just what they give you a conserved norm but they also give you the mechanisms by which their concern so for example this each 1 have critical problem of Khanaqin they say they have a momentum necessary what doesn't control each 1 half an hour but they have concern momentum which is used for the Lynn Strathmore right and that's why I talked about this energy critical radial problem 1st that's really easy because in this problem you have both for the raid radio data it's very clearly about the a concerted H 1 have each 1 and you have the mechanisms you have a positive definite stressed anti-cancer which gives you more testimony is putting together and use the fact that the radio in radial symmetry uses as its new notes and this is the proof of this is the whole purpose of the of the scattering for the radio wave equation as a critical and so said then the question is can we extend these results to other situations where we don't have a quieting and and and that say it to me that's an interesting question because end so that the type to block results by way removing the harder of the 2 obstacles but it's still an addition an obstacle removed because you've removed away the the aim you don't have a mechanism by which conserved quantities conserved and that's you have to on you have to do some more work I think too to build up to it on you know even even the mass critical unless you have the conserved conservation of the L 2 norm which is extremely useful for the interaction right it's extremely useful so anyway I hope I haven't said anything controversy over but if not I will move on and spoke of a halogen just move on to so from now on I'm talking about the results in the radio please instead spot and the other in the radio case and there this is the result of myself and Lowry that proved no 2 please this is a radio data so so we know that there is do we know this type 1 block of focusing we don't know about the focusing we know about the focusing this type 1 loss in focusing but we show that in either case there is no time to block and In the 2nd result is we have the human while Qatar's so if I cannot say should make a brief mention again I'm very happy about killer going before me because now I can just mentioned so so we have the stated it's the 1 half plus epsilon but now look this accident to Epsilon that's that's equivalent to saying it's in it's like saying it's an a 2 . 1 have myself right because you have more than one-half process on Meister said so we're just slightly around the on the critical regularly both lower and higher right so so so that's good I mean we we want to and we don't want to be it will be because there were scaling we want to have a norm that somehow interpolate speech 1 half and interpolated to the to one-half norm but then we also but but if it so that we have just an absolute right so if if any if if it is son goes 0 that the great writers I would mean that on the injured the whole result right but of course as probably most people know just removing an epsilon not always as simple right often the because now it's so hard for but in the event of you know and so on are still at the center let me discuss how these results the group said the truth that the 1st result is is approved with Lowry is provided concentration compactness so we have a quantity and this is just a L 2 inner product USA Today the cost piano this is the result of that agency's you've got this this more this gives you a nice concentration of Kennedy additional regularity shows that year minimal below solution satisfies the you have to you will see them so you and maybe this is even the wrong words of 1st you show that in fact you're minimal books Sinise in each 1 habits in each 1 and then from there then you know Kennedy's always conserved and then you've got this nice Moritz inequality which since has set out anything in a New Year's take a time out here because the derivative so you take a time average an estimate the endpoints in and will the that shows actually energy has to be 0 and then turn the Davos a case already you know then that means you 0 in the photos encased in a result of the stonewalling killed finish on and say Well if its energy is 0 it has to blow up in both time directions but you've you've carefully chosen the minimal will you take in the middle book solution that doesn't block in both time so as I liked it as an analogy I like to take analogy to the Sir Walter Scott who said our a tangled web we weave when 1st we practice to deceive and nuts falsely attributed to Shakespeare a lot of points it's actually through all those guys what it means is that the of the idea of the country's compact associate OK let's assume that it doesn't blow that it does book solutions will then you've got this all this on this than that actually means you got a whole family of blood stations because of your scaling and things like that and you can't end the Compaq C intake limits and get another solution and eventually this bloke solution is caught in its web of lies that is created and then you realize it that is not true and incident stunning usually the more people know about plastic bags the funnier they think it is at least that's what maybe it's the heat doesn't know but I try to tell myself that is it's a good idea it's a good analogy but but this is this is that this is the standard this is a standard recipe right I mean you use you but that was what I want to point out is that for this type of thing we have to get additional regular right because we don't we don't have any more we tested the skills of these 2 1 half level In the foot in the way crisis so we have to To do this work but but then the the other a bit of a difference the argument because this actually he uses the I'm quite heavily armed but he did so and and along with the hyperbolic "quotation mark so this is a a work that Italy's because the the doing and how long corners in the 2nd Back to tutorial proves in nice estimates for data that started in a compactly supported region so let's let's take our data that we have the in In India norm over here while we know that it on Israel and so we know that onto it too tired it was he said well OK so His alike and let's say we start with data in here in this part region here and involve forward in time and we know it's going to stay inside it's like and not sits inside this on it when you if he draws the hyperbole good lines and to Torres wasn't a radial mismatches result but it's alliance this hyperbole then and so if you calculate then the wave equation hyperbolic on anticipated you a nice estimates on on the way request but where would we do for this this result over here we don't know that it's in its compactly supported but below what we do know is we know like on about the fire propagation fees and so we know that outside of a ball a radius Our data is going to be small registers from reading offer this week we know it's small it's small you saw what happened outside the so in particular Al here the Alps for North all of our data is of our solution is is so then were just left with back in the situation and that you know to talk for example had to deal with where we're just you know we're just taking is inside the slight come and then at that point then there is the into good observation of stuff Lonnie and repairing Shan who realize that on that in fact if you were to shifter hyperbolic ordnance so you have the yellow the new telecom ,comma test because you have since as of far as tens you if you can since then needed telecast tests on tentatively accepted this and 1 assault on 1 what do you want to get your data on tally pose a threat because in the eastern town these themselves -minus the chief it's going to solve Jurica like equation of and but then out here again are and and Iger the picture wrong because these these ventures should go toward so 1 of 1 as you as you go along but I'd enjoy right because I don't draw well but you you can but nevertheless it's going to be pretty well approximated by by the way workers right by linear wave equation evolution right just because of the fact that you're starting with small data and settled in the remainder is going to be no small iteration of that and so then the 2nd thing bed and then there a there's a result Irish chap a further result by repaying Shen observed that In fact look look at this so on you have the arms you have the you'd we have their answers the tower ,comma and equal to one-half you know the the tower -minus said that you tell my service plus one-half you're not Eatontown clusters even tower clusters plus one-half you tell myself and you close you want of strong this property as the plus and then for a visit to give retirement SBA rights to work out you as they sought out in this region so approximately without a linear evolution costs on in here where women have more little more difficulty than we we can handle that but then in fact this wave equation as an energy being V the channel 2 the Agency for unspecified assistance so that I might have forgotten as greedy somewhere but that's that's pork on a general this is all but they aren't we have this energy but now if you were to calculate the H 1 norm all this of this energy right so you Capitanich 1 on to study you take the derivatives of Aristide's reinvestment rate from 0 to infinity but then you have another turn left over from the unit commuting here derivative in space with the asked the same thing it all it all works out in fact if you if does directly calculate the derivative of Europe Sts it's bounded by integral of you not and then do a change of variable is gonna you not squared are cute do you are you not primed square art for art's script his IQ Dr and then plus rollover too to you 1 of all where do you and and so In fact this is precisely Europe could you ride out in In just taxes this is enormous skills like age the 1 hat write to the energy is controlled by enormous skills like reached the energy of the years can at least out at the at the wings is controlled by enormous skills like this 1 have now and and then if you tried to and end if you played with the fall in the energy you could be scaled costs on its so own so if you have you you have a finite entered and true if you have a finite energy you they also satisfies and then you have a Uglow result scatters because he has his energy and he also has a more with tested but it but it's at the right scale and right it at the scaling of age to that of it's it's it's more testament that bound by age to the 1 but there's still leases that were the energies delegates 1 half and you have role of each of the 4 4 times great DS are Esquerre times since the the arrests are a cinch yeah the detail is grounded by your energy so OK so that's that's how skills but now have but you can do the exact same thing and show that this norm gives you good control over the ASA 1 half plus Absalon norm on Tariq was 0 as well will now arrive at that point what do you do you got this 1 half Norm control it was absent uncontrolled you want put uses more ways estimate which clearly controls the Act doing both sides this over the attitude change of variables this controls the ELF 1 normal you inside this come so is 1 last thing you have to do you have to show that your energy age the 1 hapless fund arm is somehow propagated through and yet that we have to cut off in frequencies and show that in Dannemora and show that that energy of Europe cut off but both 1 gives you a good Moritz estimate unit controls the area's pretty well and to also the energy is bounded but how do you that will that's where the time stakeouts estimates ,comma to say day and nite and and make it did to good control on the areas and then you get a nice bootstrap calling you can't control you can change all this for a 48 truncated peace and then the longest against us Mississippi will then the linear behavior in the period is basically linear at higher frequencies To put it together can you don't have any that strong it news I I have to him you don't have and I think you have would but I would guess is probably like the at least 1 of rap's onto the power of some but but could be even worse here you I had to get back to you on that you don't want to questions for me you few and now
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Metadaten

Formale Metadaten

Titel Cubic nonlinear wave equation
Serientitel Trimestre Ondes Non linéaires - June Conference
Teil 19
Anzahl der Teile 23
Autor Dodson, Benjamin
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/20810
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2016
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract In this talk we will discuss some recent results regarding global well -posedness and scattering for the cubic, radial wave equation in three dimensions. Both type one and type two blowup will be discussed.

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