Cubic nonlinear wave equation
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Dodson, Benjamin
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.
 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 is really great. But no, and I mean to be honest, it's the weather in a way, you know, it's a hundred 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 an h to the 1 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 a 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 norm that's preserved under that transfer ratio is the h to the 1 half norm of your initial data and h dot minus 1 half norm of your initial velocity and thus it's called a h to the 1 half critical problem. And this is by no means an unimportant thing, as again as if I remind you of what Ron was talking about, he sort of sketched out a way that for example this argument that Christ and Kaliander and Tao used to prove that if you have data that's in fact ill-posed for data less regular than h to the 1 half. So this criticality is really is important. Now of course he mentioned some stuff about how in the negative orders then you sort of a Alice in Wonderland type thing or something, but for as bigger than zero we know that in fact 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 Sog I think, yeah it's by Linblad and Sog who are also from Johns Hopkins and we have that we have local well-posedness for U0 and U1 for U0 in h dot 1 half, oh sorry, so we have local well-posedness for U0 in h dot 1 half and U1 in h dot minus 1 half and just to remind what that is there exists a T of U0 U1 greater than 0 such that a unique solution exists on minus T comma T. U is in L T loke LX, LX just LX and the solution depends continuously on initial data. Okay so this is important because you have this T of U0 U1 because it has to 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. And so then anyway this is the sort of the, I don't want to talk too much about the functional analysis and all that of this but in any event this is sort of the standard definition of local well-posedness. Now there's, the goal that I was hoping to prove and it's not done yet is to prove that there's global well-posedness and scattering for this problem. So global is pretty, hopefully pretty clear from local and then make it all R. But then scattering is that there exists U0 plus or minus in h dot 1 half and U1 in h dot minus 1 half such that U of T minus S of T U0 plus comma U1 plus h dot 1 half cross h dot minus 1 half goes to 0 as T goes to plus infinity and then the same for minus infinity S of T is the solution operator to just the linear solution. So we have our solution, so we have a solution operator, it's basically it starts to look like a linear equation and the nonlinear just perturbation of the linear equation. So this is the goal of the defocusing case. So focusing we know that that doesn't happen, we know that because we saw a talk about that. We saw a talk about some type one blobs solutions and blobs on a pyramid and other things like that. But that was for the focusing problem. For the defocusing there's no such counter example. So this is what I think a lot of people believe the defocusing problem, scottics. And in fact that's equivalent to saying that U is in LTX4, R is to plus 1. So this is equivalent to scattering. Yeah it's a nice exercise to try to show that but it doesn't take very long. I'm not going to do it. So then we have then two types of blow up, two types of ways that 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 that you have the h1 half norm going off to infinity. T plus is the, you know, you've got some maximal interval of existence and T plus is the endpoint of that. And we're saying that the h1 half norm goes to infinity in one of those two time directions. So why is that automatically, automatically I rule that scattering, right? Because you just look at your wave operator and you see that, 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 U1 as well as the U0, 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. But you see that, I mean your wave operator is a unitary operator on subless spaces. So if U of t goes, if this norm goes off to infinity then automatically the game is over, right? You can't scatter. But even still, it might still fail to scatter even if this type one blow up doesn't happen and that's called the type two blow up where you have, it's less than infinity to the soup. But nevertheless you still have the L4 norm equals infinity, yeah. So these are sort of our two types 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 talk about what we have and what we don't have here. So basically we have these, the results for the most part, at least for the defocusing case, ignoring the focusing, there's of course the channels of energy which is I think a little bit different than the, or I should say it's a little bit, it seems to me that it's going to be a new idea. So that's different from how I've done some other ideas. So because it's primarily analyzing linear behavior, right? It's analyzing linear behavior at the exterior. But in general, in any event, it's, so let's say elements in common between the NLS and nonlinear wave equations, let's say. In the elements of proof, 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 norm is h1 cross l2. This is in 40, this is 2D. And then the h dot 1 half critical NLS, which is in 3D. So for these types of problems, the elements of proof have generally centered around conservation laws. And then from that, then you have a, you know, sort of a stress energy tensor along with that. So for example, the energy critical wave equation. So when you solve, 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 combine your conserved quantity in your Morowitz estimate. And then you just use the, for the radial case, and all you would do is just use the radial cell level embedding, right? And then you'd integrate it and you'd get, you know, an L5 bound. And then, but then for the non-radial energy critical wave equation in the h1 half critical NLS, so these are works by Kennegan Merrill, you have that, you know, you have the profile decomposition, right? So you find a minimal blow up element and then you show, well, this is a minimal blow up element that can't be concentrated because, again, you have your Morowitz estimate, whether that's for the wave equation or the Lin-Strauss Morowitz estimate. So yeah, and then of course then there's the work of the energy critical NLS and mass critical NLS as well. And, you know, a lot of different people have worked on this in the audience and so on and so forth. So I think that the interesting thing to me about these problems is in a way they're very much type 2 blow up results. If you really boil, excuse me, if you really boil down to it, oh, I have to move, okay. This is your cloth, it's straight down against the microphone. It's not feedback. Move it up your collar. No, no, no, no, no, no. Not feedback. Okay, well, maybe we'll get some feedback on that and see if it... It's a wave equation in any case. Yes, that's right. We're studying the wave equation doing an experiment right now. But in any event, this has been very interesting to me because we have these results of, for example, we have these, or let's just say these three results for the energy critical wave equation, energy critical NLS, mass critical NLS. We have a conserved quantity, right? We have a conserved quantity, whether that be the mass in the case of the mass critical NLS, such as u squared, or the energy for the energy critical NLS. And so automatically we know type one blow up doesn't happen, at least in the defocusing case, right? We didn't really do anything. I mean, whoever came up with these conservation laws, of course, did something, but there's no... It doesn't seem to me like to be very... 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... 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 Kennegan-Merrill as well. It just assumes that the H1 half norm was uniformly bounded. But the interesting thing now is that in a way, the only thing that those ever prove is 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 showing no 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 NLS doesn't have a conserved H1 half norm. But when I did the mass critical NLS, I didn't do any work to show that the mass was conserved. I mean, that was easy. I mean, I figured out first day I was doing that problem, right? You just integrate it, right? By the way, so just as a reference to VLOD, we're talking about, in all these cases, we're talking about we can approximate with smooth data, right? So we never have to worry about what VLOD talked about with, do you have a conserved quantity? 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 parts as much as we want. Yeah, but then the other thing that's interesting then about problems where you don't have a conserved quantity is that 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 H1-1-2 critical problem of Kennegan-Merrill, they have a momentum that's conserved. Well, that doesn't control the H1-1-2 norm, but they have conserved momentum which is used for the Lynn-Strauss-Morowitz estimate, right? And that's why I talked about this energy critical radial problem first. That's really easy because in this problem, you have both. In the radial data, it's very clear that you have both. You have conserved H1-1 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 symmetry, you use it, and it's, I mean, this is the proof. This is the whole proof of the scattering for the radial wave equation energy critical. And so then the question is, can we extend these results to other situations where we don't have a conserved quantity anymore? And that's a, to me, that's an interesting question because, and so the type two blob results are in a way removing the harder of the two obstacles, but it's still an obstacle removed because you've removed a way the, you don't have a mechanism by which your conserved quantity is conserved, and thus you have to do some more work, I think, to build up to it. You know, even the mass critical NLS, you have the conserved conservation of the L2 norm, which is extremely useful for the interaction Morowitz estimate, right? It's extremely useful. So, anyway, I hope I haven't said anything controversial, but if not, I will move on. I suppose if I have, I should just move on, too. So from now on, I'm talking about results in the radial case, spot again. In the radial case, and there's a result of myself and Lowry that proved no type two blob, and this is for radial data. So we know that there's, we know there's type one blob for the focusing problem. We don't know about the defocusing, we know about the focusing, we know there's type one blob in the focusing case, but we show that in either case there's no type two blob. And the second result is a, we have the cubic, oh, I was just saying, scatters if, if, if the by estimated value is set, those木rys are half theーン. So, if I can make a brief mention, again, I'm very happy about Caleb going before me because now I can just mention. So, we have data in h to the one-half plus epsilon, but now look, this x to the two-apsilon, 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-apsilon, right? So we're just slightly around the critical regularity, both lower and higher, right? So that's good. I mean, we want to, we don't want to be, because of our scaling, we want to have a norm that somehow interpolates the h to one-half norm or interpolates to the h to one-half norm, but then we also, but if, so we have just an epsilon, right? So if any, if epsilon could go to zero, that'd be great, right? Because that would mean that 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. I guess now it's the hard part. But in any event, you know, so, yeah, so let's see. So let me discuss how these results are proved. So for the first result is, is proved with Lowry, is proved by concentration compactness. So we have a quantity, and this is just a L2 inner product, U sub t equals the energy, right? This is the result of energy. So you've got this, this more, this gives you a nice plus conservation of energy plus additional regularity 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 H1 half, it's an H1. And then from there then you know, okay, that energy is always conserved. And then you've got this nice Morowitz inequality, which says that you can, and then you 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 defogs in case already, you know, then that means U is zero. In the focusing case, then you have a result of Stovall and Kilopin-Vichon that say, well, if its energy is zero, it has to blow up in both time directions. But you've carefully chosen the minimal blow, you've taken the minimal blow up solution that doesn't blow up in both time directions. So as I like to, as an analogy, I like to take an analogy to the Sir Walter Scott who said, oh, 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. And so what it means is that, so basically the idea of these concentration compactness you ought to say, okay, let's assume that it doesn't blow up, that it does blow up. There's a blow up solution. Well, then you've got this, all 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, and 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 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 a good analogy. But this is the standard recipe, right? I mean, but what I want to point out is that for this type of thing, we have to get additional regularity, right, because we don't have any more of its estimate that scales at the h to the 1 half level in the wave equation. So we have to do this work. But then the other proof is a bit of a different argument because this actually uses a lot of the I method quite heavily. But again, so, and along with the hyperbolic coordinates. So this is 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 take our data that we have that's in this norm over here. Well, we know that it, as a radio, so we know that, 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 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, if he draws a hyperbola, it's going to lie in, and Tataru's wasn't a radial least symmetric result. But it's going to lie inside this hyperboloid then, and so if you calculate then the wave equation in hyperbolic coordinates, this is going to get you some nice estimates on the wave equation. But what we, for this result over here, we don't know that it's in, it's compactly supported. So what we do know is we know 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 from reading off of this. We know it's small, small h to the 1 half norm outside that ball. So in particular, out here, the L4 norm 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 Rupang Shen, who realized that on, that in fact, if you were to shift to hyperbolic coordinates, so you have a V of tau, comma, S equals either tau, sinh, S over S, times U of E to the tau, sinh, S, E to the tau, cosh, S. And if you shift to this, and then you want to solve 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 of, and, but then out here, again, are, and I drew the picture wrong because these, these branches should go towards slope one, right? They should have slope one as, as you go along, but I didn't draw it right because I don't draw very well. 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. So then the second thing that, and then there's a, there's a result by Schafflein, then a further result by Rupang Shen, who observed that, in fact, look, look at this. So you have the, 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 U1 of S prime, S prime, D S prime. See plus, and then for, and this, 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, in here, we're going to have more, a little bit more difficulty, but we can handle that. Then, in fact, this wave equation has an energy, E V of tau. So, V to the fourth times S squared D S. So, I might have forgotten an S squared D S somewhere, but that's, that's the polar coordinate integral, right? So, that's just all that, that's, okay, we have this energy, but now if you were to calculate the H1 norm of this, of this energy, right? So you, you calculate the H1 norm of this energy, so you, you would take the derivatives of S times V of S and integrate it from zero to infinity. 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, 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 D R. And then plus integral of R cubed U1 of R squared D R. And, and so, in fact, this is precisely your, if you write it out in, in, in just X's, 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, at least out at the, at the wings is controlled by a norm that scales like H to the one half. Well, 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. 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, 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 the one half. And you have integral of V to the fourth times S squared D S, or S squared times sinh, or Cosh of S over sinh S, D S, 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 the one half norm controlled, H to the one half plus epsilon norm controlled. You want to put, use this Morowitz estimate, which clearly controls the L after you do a, oh, sorry, sorry, S over, after you do a change of variables, this controls the L four 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, well, you 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 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 you're done. So anyway, that's probably about it. Yeah. Thank you. Yes. After you have an idea of the dependence on epsilon about your constant, is it a fairly good idea? I would guess it's probably like at least one over epsilon to a power of epsilon. But it could be even worse. Yeah. Yeah, I'd have to get back to you on that. Yeah. So Hoppe. since you have no other questions, so thank you and... Thank you. Thank you. you
 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 is really great.  But no, and I mean to be honest, it's the weather in a way, you know, it's a hundred  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 an h to the 1 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 a 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 norm that's preserved under that  transfer ratio is the h to the 1 half norm of your initial data and h dot minus 1 half  norm of your initial velocity and thus it's called a h to the 1 half critical problem.  And this is by no means an unimportant thing, as again as if I remind you of what Ron was  talking about, he sort of sketched out a way that for example this argument that Christ  and Kaliander and Tao used to prove that if you have data that's in fact ill-posed for  data less regular than h to the 1 half.  So this criticality is really is important.  Now of course he mentioned some stuff about how in the negative orders then you sort of  a Alice in Wonderland type thing or something, but for as bigger than zero we know that in  fact 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 Sog I think, yeah it's by Linblad and Sog  who are also from Johns Hopkins and we have that we have local well-posedness for U0 and  U1 for U0 in h dot 1 half, oh sorry, so we have local well-posedness for U0 in h dot  1 half and U1 in h dot minus 1 half and just to remind what that is there exists a T of  U0 U1 greater than 0 such that a unique solution exists on minus T comma T. U is in L T loke  LX, LX just LX and the solution depends continuously on initial data.  Okay so this is important because you have this T of U0 U1 because it has to 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.  And so then anyway this is the sort of the, I don't want to talk too much about the functional  analysis and all that of this but in any event this is sort of the standard definition of  local well-posedness.  Now there's, the goal that I was hoping to prove and it's not done yet is to prove that  there's global well-posedness and scattering for this problem.  So global is pretty, hopefully pretty clear from local and then make it all R. But then  scattering is that there exists U0 plus or minus in h dot 1 half and U1 in h dot minus  1 half such that U of T minus S of T U0 plus comma U1 plus h dot 1 half cross h dot minus  1 half goes to 0 as T goes to plus infinity and then the same for minus infinity S of  T is the solution operator to just the linear solution.  So we have our solution, so we have a solution operator, it's basically it starts to look  like a linear equation and the nonlinear just perturbation of the linear equation.  So this is the goal of the defocusing case.  So focusing we know that that doesn't happen, we know that because we saw a talk about that.  We saw a talk about some type one blobs solutions and blobs on a pyramid and other things like  that.  But that was for the focusing problem.  For the defocusing there's no such counter example.  So this is what I think a lot of people believe the defocusing problem, scottics.  And in fact that's equivalent to saying that U is in LTX4, R is to plus 1.  So this is equivalent to scattering.  Yeah it's a nice exercise to try to show that but it doesn't take very long.  I'm not going to do it.  So then we have then two types of blow up, two types of ways that 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 that you have the h1 half norm going off to infinity.  T plus is the, you know, you've got some maximal interval of existence and T plus is the endpoint  of that.  And we're saying that the h1 half norm goes to infinity  in one of those two time directions.  So why is that automatically, automatically I rule that scattering, right?  Because you just look at your wave operator and you see that, 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 U1 as well as the U0, 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.  But you see that, I mean your wave operator is a unitary operator on subless spaces.  So if U of t goes, if this norm goes off to infinity then automatically the game is over,  right?  You can't scatter.  But even still, it might still fail to scatter even if this type one blow up doesn't happen  and that's called the type two blow up where you have, it's less than infinity to the soup.  But nevertheless you still have the L4 norm equals infinity, yeah.  So these are sort of our two types 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 talk about what we have and what we  don't have here.  So basically we have these, the results for the most part, at least for the defocusing  case, ignoring the focusing, there's of course the channels of energy which is I think a  little bit different than the, or I should say it's a little bit, it seems to me that  it's going to be a new idea.  So that's different from how I've done some other ideas.  So because it's primarily analyzing linear behavior, right?  It's analyzing linear behavior at the exterior.  But in general, in any event, it's, so let's say elements in common between the NLS and  nonlinear wave equations, let's say.  In the elements of proof, 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 norm is h1 cross l2.  This is in 40, this is 2D.  And then the h dot 1 half critical NLS, which is in 3D.  So for these types of problems, the elements of proof have generally centered around conservation  laws.  And then from that, then you have a, you know, sort of a stress energy tensor along with that.  So for example, the energy critical wave equation.  So when you solve, 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 combine your conserved quantity in your Morowitz  estimate.  And then you just use the, for the radial case, and all you would do is just use the  radial cell level embedding, right?  And then you'd integrate it and you'd get, you know, an L5 bound.  And then, but then for the non-radial energy critical wave equation in the h1 half critical  NLS, so these are works by Kennegan Merrill, you have that, you know, you have the profile  decomposition, right?  So you find a minimal blow up element and then you show, well, this is a minimal blow  up element that can't be concentrated because, again, you have your Morowitz estimate, whether  that's for the wave equation or the Lin-Strauss Morowitz estimate.  So yeah, and then of course then there's the work of the energy critical NLS and mass critical  NLS as well.  And, you know, a lot of different people have worked on this in the audience and so on and  so forth.  So I think that the interesting thing to me about these problems is in a way they're  very much type 2 blow up results.  If you really boil, excuse me, if you really boil down to it, oh, I have to move, okay.  This is your cloth, it's straight down against the microphone.  It's not feedback.  Move it up your collar.  No, no, no, no, no, no.  Not feedback.  Okay, well, maybe we'll get some feedback on that and see if it...  It's a wave equation in any case.  Yes, that's right.  We're studying the wave equation doing an experiment right now.  But in any event, this has been very interesting to me because we have these results of, for  example, we have these, or let's just say these three results for the energy critical  wave equation, energy critical NLS, mass critical NLS.  We have a conserved quantity, right?  We have a conserved quantity, whether that be the mass in the case of the mass critical  NLS, such as u squared, or the energy for the energy critical NLS.  And so automatically we know type one blow up doesn't happen, at least in the defocusing  case, right?  We didn't really do anything.  I mean, whoever came up with these conservation laws, of course, did something, but there's  no... It doesn't seem to me like to be very... 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... 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 Kennegan-Merrill as well.  It just assumes that the H1 half norm was uniformly bounded.  But the interesting thing now is that in a way, the only thing that those ever prove  is 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 showing no 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 NLS doesn't have a conserved H1 half norm.  But when I did the mass critical NLS, I didn't do any work to show that the mass was conserved.  I mean, that was easy.  I mean, I figured out first day I was doing that problem, right?  You just integrate it, right?  By the way, so just as a reference to VLOD, we're talking about, in all these cases,  we're talking about we can approximate with smooth data, right?  So we never have to worry about what VLOD talked about with, do you have a conserved  quantity?  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 parts as much as we want.  Yeah, but then the other thing that's interesting then about problems where you don't have  a conserved quantity is that 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 H1-1-2 critical problem of Kennegan-Merrill, they have a momentum  that's conserved.  Well, that doesn't control the H1-1-2 norm, but they have conserved momentum which is  used for the Lynn-Strauss-Morowitz estimate, right?  And that's why I talked about this energy critical radial problem first.  That's really easy because in this problem, you have both.  In the radial data, it's very clear that you have both.  You have conserved H1-1 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 symmetry, you use it, and  it's, I mean, this is the proof.  This is the whole proof of the scattering for the radial wave equation energy critical.  And so then the question is, can we extend these results to other situations where we  don't have a conserved quantity anymore?  And that's a, to me, that's an interesting question because, and so the type two blob  results are in a way removing the harder of the two obstacles, but it's still an obstacle  removed because you've removed a way the, you don't have a mechanism by which your conserved  quantity is conserved, and thus you have to do some more work, I think, to build up to  it.  You know, even the mass critical NLS, you have the conserved conservation of the L2 norm,  which is extremely useful for the interaction Morowitz estimate, right?  It's extremely useful.  So, anyway, I hope I haven't said anything controversial, but if not, I will move on.  I suppose if I have, I should just move on, too.  So from now on, I'm talking about results in the radial case, spot again.  In the radial case, and there's a result of myself and Lowry that proved no type two  blob, and this is for radial data.  So we know that there's, we know there's type one blob for the focusing problem.  We don't know about the defocusing, we know about the focusing, we know there's type  one blob in the focusing case, but we show that in either case there's no type two blob.  And the second result is a, we have the cubic, oh, I was just saying, scatters if, if, if  the by estimated value is set, those木rys are half theーン.  So, if I can make a brief mention, again, I'm very happy about Caleb going before me  because now I can just mention.  So, we have data in h to the one-half plus epsilon, but now look, this x to the two-apsilon,  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-apsilon, right?  So we're just slightly around the critical regularity, both lower and higher, right?  So that's good.  I mean, we want to, we don't want to be, because of our scaling, we want to have a norm that  somehow interpolates the h to one-half norm or interpolates to the h to one-half norm,  but then we also, but if, so we have just an epsilon, right?  So if any, if epsilon could go to zero, that'd be great, right?  Because that would mean that 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.  I guess now it's the hard part.  But in any event, you know, so, yeah, so let's see.  So let me discuss how these results are proved.  So for the first result is, is proved with Lowry, is proved by concentration compactness.  So we have a quantity, and this is just a L2 inner product, U sub t equals the energy,  right?  This is the result of energy.  So you've got this, this more, this gives you a nice plus conservation of energy plus additional  regularity 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 H1 half, it's  an H1.  And then from there then you know, okay, that energy is always conserved.  And then you've got this nice Morowitz inequality, which says that you can, and then you 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 defogs in case already, you know, then that means U is zero.  In the focusing case, then you have a result of Stovall and Kilopin-Vichon that say, well,  if its energy is zero, it has to blow up in both time directions.  But you've carefully chosen the minimal blow, you've taken the minimal blow up solution  that doesn't blow up in both time directions.  So as I like to, as an analogy, I like to take an analogy to the Sir Walter Scott who  said, oh, 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.  And so what it means is that, so basically the idea of these concentration compactness  you ought to say, okay, let's assume that it doesn't blow up, that it does blow up.  There's a blow up solution.  Well, then you've got this, all 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, and 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 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 a good analogy.  But this is the standard recipe, right?  I mean, but what I want to point out is that for this type of thing, we have to get additional  regularity, right, because we don't have any more of its estimate that scales at the h  to the 1 half level in the wave equation.  So we have to do this work.  But then the other proof is a bit of a different argument because this actually uses a lot  of the I method quite heavily.  But again, so, and along with the hyperbolic coordinates.  So this is 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 take our data that we have that's in this norm over here.  Well, we know that it, as a radio, so we know that, 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 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, if he draws a hyperbola, it's going to lie in, and Tataru's  wasn't a radial least symmetric result.  But it's going to lie inside this hyperboloid then, and so if you calculate then the wave  equation in hyperbolic coordinates, this is going to get you some nice estimates on the  wave equation.  But what we, for this result over here, we don't know that it's in, it's compactly supported.  So what we do know is we know 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  from reading off of this.  We know it's small, small h to the 1 half norm outside that ball.  So in particular, out here, the L4 norm 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 Rupang Shen,  who realized that on, that in fact, if you were to shift to hyperbolic coordinates, so  you have a V of tau, comma, S equals either tau, sinh, S over S, times U of E to the tau,  sinh, S, E to the tau, cosh, S.  And if you shift to this, and then you want to solve 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 of, and, but then out here, again, are, and  I drew the picture wrong because these, these branches should go towards slope one, right?  They should have slope one as, as you go along, but I didn't draw it right because I don't  draw very well.  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.  So then the second thing that, and then there's a, there's a result by Schafflein, then a  further result by Rupang Shen, who observed that, in fact, look, look at this.  So you have the, 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 U1 of S prime, S prime, D S prime.  See plus, and then for, and this, 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, in here, we're going to have more, a little bit more difficulty, but we can handle  that.  Then, in fact, this wave equation has an energy, E V of tau.  So, V to the fourth times S squared D S. So, I might have forgotten an S squared D S somewhere,  but that's, that's the polar coordinate integral, right?  So, that's just all that, that's, okay, we have this energy, but now if you were to calculate  the H1 norm of this, of this energy, right?  So you, you calculate the H1 norm of this energy, so you, you would take the derivatives  of S times V of S and integrate it from zero to infinity.  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, 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 D R. And then plus integral of R cubed U1 of R squared D R.  And, and so, in fact, this is precisely your, if you write it out in, in, in just X's, 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, at least  out at the, at the wings is controlled by a norm that scales like H to the one half.  Well, 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.  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, 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 the  one half.  And you have integral of V to the fourth times S squared D S, or S squared times sinh, or  Cosh of S over sinh S, D S, 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 the one half norm controlled, H to the one half plus epsilon norm controlled.  You want to put, use this Morowitz estimate, which clearly controls the L after you do  a, oh, sorry, sorry, S over, after you do a change of variables, this controls the L  four 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, well, you 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 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 you're done.  So anyway, that's probably about it.  Yeah.  Thank you.  Yes.  After you have an idea of the dependence on epsilon about your constant, is it a fairly  good idea?  I would guess it's probably like at least one over epsilon to a power of epsilon.  But it could be even worse.  Yeah.  Yeah, I'd have to get back to you on that.  Yeah.  So Hoppe.  since you have no other questions, so thank you and...  Thank you.  Thank you.  you
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