 Thank you for inviting me. It's been really nice to be in Paris this month. Really nice talks to watch as well and just really great except the weather but everything else really great. But no, I mean to be honest it's the weather in a way you know it's 100 year experience supposedly so that's nice too I guess. Okay so in this talk I'll talk about the in the three so this is the equation that I will be discussing in this talk and it's the cubic wave equation in three dimensions and so this equation is called the is called an h to the one half critical problem and a lot of people already know this of course and in fact Ron talked a bit about criticality on I think Tuesday but in any event just to get into just to say what it is it means that if you have a solution to this problem or yeah solution to the equation then in fact you have an entire family of solutions because you can insert a lambda and in fact then if you check then you'll see that the the norm that's preserved under that transfer H is the h one half norm of your initial data and h dot minus one half norm of your your initial velocity and thus it's called h one half critical problem and and this is this is by no means a unimportant thing as again as as if I remind you of what Ron was talking about he sort of sketched out a way that for example his this argument that Christ and Callender and tau use to prove that if you have data that's in fact ill posed for data less regular than h to one half so so h to the one so this criticality is is really is important now of course he mentioned some stuff about how you know in the negative to negative orders and you sort of a Alice in Wonderland type thing or something but for as bigger than zero we know that that's that it in fact that's exactly that's exactly the right space to think about because we have this local well posed in this result of and this is by Linblad and and Saug I think yeah it's by Linblad and Saug who are also from Johns Hopkins and we have that we have local well posed in this for you not and you won for you not in h dot one half oh sorry so we have local well posed for you not in h dot one half and you won in h dot minus one half and just to remind what that is there exists a t of you not you won greater than zero such that a unique solution minus t comma t um u is in l t low lx low lx just lx and the solution depends continuously on initial okay so so so this is important right because you have this t of you not you won right because if it has it can't just depend on the size because if it did then you could always do some rescaling to show that in fact you had a global solution right um and and so then anyway this is the um sort of the I don't want to get to talk too much about the functional analysis and all that of this but in any event this is sort of the standard um definition of local well poseness now there's um good um there's the the goal that I was hoping to prove and it's not done yet is to prove that there's global well poseness and scattering for this problem that's the goal so uh global is pretty hopefully pretty clear from you know local and then make it all are right and then but then scattering is that there exist you not plus or minus in h dot one half and you won in h dot minus one half such that um u of t minus s of t u not plus comma u one plus h dot one half uh cross h dot minus what goes to zero as t goes to plus infinity and then the same minus infinity s of t is the solution operator two just the linear so we have our solution so we have a solution operator it's amazing it starts to look like a linear equation right and and the non-linear just perturbation of the linear equation so this is this is the goal of of the defocusing case so focusing we know that that doesn't happen right we know that because um well I mean we saw a talk about that right we saw a talk about some type one both solutions and blobs on a pyramid and other things like that and but but um in the but this is what that was for the focusing problem um for the defocusing there's no such um counter example so this is what um I think a lot of people believe the defocusing problem scottics and in fact that's equivalent to um saying that u is in l t x four so so this this is equivalent to scattering um yeah it's a nice exercise to try to show that um but it doesn't take very long I'm not going to do it um okay so so then then we have um then two types of of blow up two types of ways that our our function our solution can blow up and those are appropriately enough called type one and type two blow up so the first type is is that you have the h one half norm going off to infinity t plus is the you know you got some maximal interval of existence and t plus is is the endpoint of that and we're saying that the h one half norm goes to infinity um in one of those two um one of those two time directions so so what why is that um why is that automatically it automatically I rule that scattering right because you just look at your um you just look at your wave operator and you see that um oh yeah so sorry I didn't write it I'm not going to bring it back down but I should have wrote a plus or minus in front of the u one as well as the u not okay but I'm not going to bring it all the way back down just to do that so okay that's what I should have done um but but you see that I mean your wave operator is a unitary operator on on sub less spaces so um so if if if u of t goes if this norm goes off to infinity then automatically the game is over right this you can't scatter but but even still it might still fail to scatter even if even if this type one blow up doesn't happen and then that's called the the type two blow up where you have it's less than infinity of the soup but but nevertheless you still have the l4 norm equals infinity yeah so this is these are sort of our two types of of blow up solutions now at this point then I want to put this then in the context of a couple of other problems that have been solved by a number of people and the reason why I want to do that is to sort of um talk about what we have and what we don't have here so basically we have these these that the um the results for the most part at least for the defocusing case the ignoring that leaving the focusing there's of course the channels of energy which is I think a little bit different than than um the or I should say it's a little bit um um it's a it seems to me to be a new idea so that's different from um how I've um than than some other ideas so um because it's it's primarily analyzing linear behavior right it's analyzing linear behavior at the exterior so um but in general the the in any way it's um so let's say elements and let's let's say elements in common between the nls and and nonlinear wave equations let's say um in in the the elements of proof for for example you have the energy critical wave so this is utt minus delta u in 40 right in 40 then if you do your rescaling you're going to have the concern the the norm is h1 cross l2 um and this is in 40 this is 2d and then the h dot one half critical nls which is in 3d so for for these um types of problems the elements of proof have generally centered around conservation laws and then from that then you have a um you know sort of a stress energy tensor along with that so so for example the the energy critical wave equation um so when you when you saw for example the energy critical wave equation right i mean like let's say you have radial data right so you have your Morowitz estimate is bounded by the and then you have an energy so in the defocusing direction you have you know you combine your your conserved quantity and your your Morowitz estimate and then and then you just use the for the for the radial case and all you would do is just use the uh radial cell blood embedding right and then you'd integrate it and you'd get you know an l5 bound um and then but then for the um for the non-radial um energy critical wave equation in the h1 half critical nls uh so these are works by uh kennig and mel um you have um that you know do you have the profile decomposition right so you find a minimal blow-up element and um um and then you show well this is a minimal blow-up element that can't be concentrated because again you have your your Morowitz estimate whether that's for the wave equation or the the lindstrasse Morowitz estimate um so yeah and then and then of course then there's the work of the energy critical nls and and mass critical nls as well and um you know a lot a lot of different people have have worked on this in in the audience and so on and so forth but um i think that the um the interesting thing to me about these problems is in a way they're very um much type two blow-up results if you really excuse me if you really boil down to it oh i have to move okay it's just your cloth it's a great thing against the microphone okay well maybe we'll get some feedback on that and see if yes that's right we're studying the wave equation doing an experiment right now um but but in any event we have this is very interesting to has been very interesting to me because we have these results of um for example we have these or let's just say these three results um for the energy critical wave equation energy critical nls mass critical nls um we have we have a conserved quantity right we have a conserved quantity whether that be the mass in the case of the mass critical nls let's just use squared or the or the energy for the energy critical nls and so automatically we know type one blow-up doesn't happen at least in the rate in least in the defocusing case right we didn't we didn't really do anything other i mean whoever came up with these conservation laws of course did something but but there's no there's no there's no it doesn't seem to me like to be very that that's sort of given to us but then we have to prove type two blow up doesn't happen for the mass critical or for the energy critical or what have you whereas in the case of in the case of um but then and then recently people have been working on now results for where you don't have these conserved quantities right which is of course the result of uh kennig and merrill as well they just assumed that the h1 half norm was uniformly bounded um but the um the interesting thing now is that you know in a way the the only thing that those ever prove is type two no type two blow up and some people and i've gone back and forth in my mind about this now you have to listen to me argue with myself because i'm still not completely decided in my mind about this but they're showing no two type two blow up it's really the same issue right whether it just so happens that these problems have a conserved quantity and that for example the h1 half critical n ls doesn't have a doesn't have a conserved h1 half norm but but i mean like when i did the n mass critical n ls i didn't do any work to show that the mass was conserved right i mean that that was easy um so i mean i figured out like first day i was doing that from right you just integrate it right integrate by part oh by the way so just as a reference to vlad we're talking about in all these cases we're talking about a we can approximate with smooth data right so we never have to worry about uh like what vlad talked about with you know do you have a conserved quantity we know we we because we're dealing with strict arts estimates and all these results are perturbative we don't ever have to think about a situation where we might not be able to integrate by parts or something we can always integrate a part as much as we want um yeah and but but and but then the the other thing that's interesting then about problems where you know problems where you don't have a conserved quantity is that you in a way you have these conserved quantities actually give you two things if you really think about it not just one they give you a conserved norm but they also give you the mechanism by which they're conserved so for example this h one half critical problem of kenning and merrill they they have a momentum that's conserved well that doesn't control the h one half norm but they have conserved momentum which is used for the lin Strauss Morowitz estimate right and that that's why I talked about this energy critical radial problem first that's really easy because in this problem you have both for the radio in the radial data it's very clear that you have both you have conserved h one half h one norm and you have the mechanism you have a positive definite stress energy tensor which gives you a Morowitz estimate you just put them together and use the fact that the radial you know radial symmetry you use it and it's it's you know it's I mean this is the proof this is the whole proof of the of the of the scattering for the radial wave equation energy critical and so so then the question is can we extend these results to other situations where we don't have a conserved quantity anymore and and and that's a to me that's an interesting question because and so the the type two block results are in a way removing the harder of the two obstacles but it's still in addition an obstacle removed because you've removed away the you don't have a mechanism by which your conserved quantity is conserved and thus you have to you have to do some more work I think to to build up to it you know even even the mass critical n l s you have the conserved conservation of the l two norm which is extremely useful for the interaction Morowitz estimate right it's extremely useful um so um anyway I hope I haven't said anything controversial but if not I will move on um I suppose if I have I should just move on too um so from now on I'm talking about results in the radial case um spot again um in the radial case and there's a result of myself and Lowry that proved no type two block and this is for radial data so so we know that there's we know there's type one block for the focusing problem we don't know about the defocusing we know about the focusing we know there's type one block in the focusing case but we show that in either case there's no type two block and the second result is a we have the cubic object scatters if so if I can uh make a brief mention again I'm very happy about Caleb going before me because now I can just mention so so we have data data in h to the one half plus epsilon but now look this x to the two epsilon that's that's equivalent to saying it's in that's like saying it's in h to the dot one half minus epsilon right because you have one half plus epsilon minus two epsilon right so so we're we're just slightly around the um the critical regularity both lower lower and higher right so so so that's good I mean we we want to um we don't want to be because of our scaling we want to have a norm that somehow interpolates the h one half norm um or interpolates to the h to one half norm but then we also um but but if so we have just an epsilon right so if if any um if if if epsilon could go to zero that'd be great right because that would mean that um that would be the whole result right but of course as probably most people know just removing an epsilon not always so simple right it's often the hard part um I guess now it's the hard part but and even um you know so um yeah so let's see so let me discuss how these results are proved so for the the first result is is proved with Lowry is proved by concentration compactness so we have a quantity and this is just a l two inner product use of t equals the energy right this is the result of energy so you you've got this this more this gives you a nice plus conservation of energy plus additional regularity it shows that your minimal blow up solution satisfies e u of t equals zero so so maybe this is even the wrong order so first you show that in fact your minimal blow up solution not only is an h one half it's an h one and then from there then you know okay that energy is always conserved and then you've got this nice uh more of its inequality which says that um you know you can and then you you know take a time out you know because it's a derivative so you take a time average and estimate the end points and then say oh well then that shows actually the energy has to be zero and then so in the defaults in case already you know then that means you is zero in the focusing case and you have a result of uh Stovall and Killip and Vishan that say well if its energy is zero it has to blow up in both time directions but you've you've carefully chosen the minimal blow you've taken the minimal blow up solution that doesn't blow up in both time direction so as I like to as an analogy I like to um take an analogy to the Sir Walter Scott who said ah what a tangled web we weave when first we practice to deceive and now it's falsely attributed to Shakespeare a lot but I looked it up once and it's actually Sir Walter Scott so what it means is that so basically the idea of these concentration compactness you'll say okay let's assume that it doesn't blow that it does blow up there's a blow up solution well then you've got this all this um this then that actually means you've got a whole family of blow up solutions because of your scaling and things like that and you can end they're compact so you can take limits and get another solution and eventually this blow up solution is caught in its web of lies that it's created and then you realize that it's not true and and so then it's done and usually the more people know about concentrated compactness the funnier they think it is or at least that's what I maybe it's the heat I don't know but I try to tell myself that it's it's a good idea it's a good analogy but but this is this is the this is the standard this is a standard recipe right I mean you you but but what's what I want to point out is that for this type of thing we have to get additional regularity right because we don't we don't have any Morowitz estimate that scales at the h to one half level um in in the in the the wave equation so we have to to do this work um but but then the the other proof is a bit of a different argument because this actually uses the i method quite heavily um but again so and and along with the hyperbolic coordinates so this is a a work that at least goes doing it in a hyperbolic coordinates at least goes back to Tataru who proved some nice estimates for data that started in a compactly supported region so let's let's take our data that we have that's in in in this norm over here well we know that it um it's radio so we know that um so what Tataru did was he said well okay so here's our light cone and let's say we start with data in here in in this part region here and then evolve forward in time and we know it's going to stay inside this light cone and thus it's inside this um if when if he draws a hyperbo it's going to lie and Tataru's wasn't a radial least symmetric result but it's going to lie inside this hyperbolo identity and so if you calculate then the wave equation in hyperbolic coordinates this is going to get you some nice uh estimates on on the wave equation but what what we for this this result over here we don't know that it's in it's it's compactly supported but but what we do know is we know like um about the finite propagation speed and so we know that outside of a ball of radius r our data is going to be small right just just from reading off of this we know it's small small h to one half norm outside that ball so in particular out here the l4 norm of of our data is of our solution is is bounded right so then we're just left with back in the situation that you know Tataru for example had to deal with where we're just you know we're just taking we're just inside this light cone now and then at that point then there was a good observation of Staffelani and Rupeng Shen who realized that on that in fact if you were to shift to a hyperbolic coordinates so you have a v of v of tau comma s equals e to the tau sinh s over s times u of e to the tau sinh s e to the tau cosh s um then if you if shift to this and then you want to solve on um on you want to get your data on tau equals zero right because then v sub tau v is going to solve minus s over v cube so it's going to solve a cubic like equation and um but then out here again our and and I drew the picture wrong because these these branches should go towards slope one right they should have slope one as you as you go along but I didn't draw it right because I don't draw very well um but you you can but nevertheless it's it's going to be pretty well approximated by by the wave equation right by the linear wave equation evolution right just because of the fact that you're starting with small data and so the you know your remainder is going to be you know a small iteration of that um so then the second thing that and then there's a there's a result by Chaffelani and then a further result by Rupeng Shen who observed that um um in fact look look at this so um you have the um you have the u of we have that s of v of tau comma s equals one half u naught e to the tau minus s times e to the tau minus s plus one half u naught e to the tau plus s times e to the tau plus s plus one half e to the tau minus s times e to the plus s u one of s prime s prime d s prime c plus and then for and again we're talking about s big right so we're out s big so we're talking about out in this region so we're approximating with our our linear evolution of course um in here we're going to have more a little bit more difficulty but we can handle that um then in fact this wave equation has an energy e v of tau v to the four times s square d s so so i might have forgotten an s square d s somewhere but that's that's the polar coordinate integral right so that's just all that that's okay um we have this energy but now if you were to calculate the h one norm of this of this energy right so you you calculate the h one norm of this energy so you you would take the derivatives of s times v of s and integrate it from zero to infinity um and then you'd have another term left over from you know commuting your derivative in space with the s but same thing i mean it all it all works out in fact if you if you just directly calculate the derivative of your s v s it's bounded by integral of u naught and then do a change of variables it's bounded by u naught squared r cubed d or u naught prime square r r squared plus times r cubed dr and then plus integral of um r cubed u one of r squared dr and and so in fact this is precisely your if you write it out in in in just x is this is a norm that scales like h to the one half right so the energy is controlled by a norm that scales like h the energy of v is can at least out at the at the wings is controlled by a norm that scales like h to one half now and then and then if you tried to and then if you played with the ball you know the energy you could rescale of course um and so so if you have u you have a finite energy what shen proved is if you have a finite energy u that also satisfies this then you have a global result that scatters because he has this energy and then he also has a Morowitz estimate but it but it's at the right scaling right it's at the scaling of h to the of of it's it's it's a Morowitz estimate that's bounded by h to the one that scales like h that where the energy scale like h to one half and you have integral of v to the fourth times s squared ds or s squared times cinch or cosh of s over cinch s ds d tau is bounded by your energy so okay so that's that's how it scales but now but you can do the exact same thing and show that this norm gives you good control over the h to the one half plus epsilon norm on tau equals zero as well well now at that point what do you do you've got the h to one half norm controlled h one half plus epsilon norm controlled you want to put use this Morowitz estimate which clearly controls the elf after you do a oh sorry sorry um s over after you do a change of variables this controls the l4 norm of u inside this cone so then there's one last thing you have to do you have to show that your energy your h to the one half plus epsilon norm is somehow propagated through and you have to we have to cut off in frequencies and show that and get a Morowitz estimate and show that that energy of your cutoff both one gives you a good Morowitz estimate you know controls the errors pretty well and two also the energy is bounded well how do you do that well that's where the long time Stryker estimates come to save the day and they and and they get you good control on the errors and then you get a nice bootstrap going and you can control you can control this for a Fourier truncated piece and then the long time Stryker estimates just say well then the linear behavior the behavior is basically linear at higher frequencies so you put it together and and you're done so anyway that's probably about it I would I would guess it's probably like at least one over epsilon to a power of some but but it could be even worse yeah yeah I'd have to get back to you on that